A general SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
Paul M. N. Feehan
We prove an analogue of the Kotschick-Morgan conjecture \cite{KotschickMorgan} in the context of $\SO(3)$ monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the $\SO(3)$-monopole cobordism. Although this article is in a preliminary form, it appears to be of sufficient interest that it is worth now distributing in its present shape.
math0203047.tex
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%  A general SO(3)-monopole cobordism formula relating Donaldson and
%  Seiberg-Witten invariants
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%  Paul M. N. Feehan and Thomas G. Leness
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%  This version:  March 5, 2002
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\begin{document}

\title[A general formula relating Donaldson and Seiberg-Witten invariants]
{A general SO(3)-monopole cobordism formula relating Donaldson and
Seiberg-Witten invariants}
\author[Paul M. N. Feehan]{Paul M. N. Feehan}
\address{Department of Mathematics\\
Rutgers University\\
Piscataway, NJ 08854-8019}
\email{feehan@math.rutgers.edu}
\urladdr{http://www.math.rutgers.edu/$\sim$feehan}
\author[Thomas G. Leness]{Thomas G. Leness}
\address{Department of Mathematics\\
Florida International University\\
Miami, FL 33199}
\email{lenesst@fiu.edu}
\urladdr{http://www.fiu.edu/$\sim$lenesst}
\dedicatory{}
\subjclass{}
\thanks{PMNF was supported in part by NSF grants DMS-9704174 and DMS-0125170.
TGL was supported in part by NSF grant DMS-0103677.}
\date{This version: \today. Preliminary version. math.DG/0203047.}
\keywords{}
\begin{abstract}
We prove an analogue of the Kotschick-Morgan conjecture
\cite{KotschickMorgan} in the context of $\SO(3)$ monopoles, obtaining a
formula relating the Donaldson and Seiberg-Witten invariants of smooth
four-manifolds using the $\SO(3)$-monopole cobordism.
Although this article is in a preliminary form, it appears
to be of sufficient interest that it is worth now
distributing in its present shape.  
\end{abstract}
\maketitle

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\section{Introduction}
\label{sec:intro}
In this article, a sequel to \cite{FL1,FL2a,FL2b,FL3,FL4}, 
we complete the first step in the proof of the Witten and Moore-Witten
formulas relating the Seiberg-Witten and Donaldson invariants \cite{Witten},
\cite{MooreWitten}. In particular, 
we prove that pairings of certain cohomology classes with the links of
lower level Seiberg-Witten moduli subspaces of the Uhlenbeck
compactification of the moduli space of $\SO(3)$ monopoles depends only on
the Seiberg-Witten invariant and the homotopy type of a four-manifold $X$,
through its Euler characteristic $\chi$ and signature $\sigma$ (see Theorem
\ref{thm:LinkPairing}). This leads to the following result.

\begin{thm}
\label{thm:MainThm}
Let $X$ be a closed, connected, oriented smooth four-manifold with
$b_1(X)=0$, odd $b_2^+(X)>1$. Let $\La,w\in H^2(X;\ZZ)$ obey
$w-\La\equiv w_2(X)\pmod 2$. Let $\delta,m$ be non-negative integers for
which $m\leq [\delta/2]$, where $[\,\cdot\,]$ denotes the greatest integer
function, and $\delta\equiv -w^2-\frac{3}{4}(\chi+\sigma)\pmod{4}$, with
$\Lambda$ and $\delta$ obeying $\delta<i(\La)$, where
$i(\La)=\La^2-\frac{1}{4}(\chi+\si)$. Then for any $h\in H_2(X;\RR)$ and
generator $x\in H_0(X;\ZZ)$, we have the following expression for the
Donaldson invariant:
\begin{equation}
\label{eq:MainEquation}
\begin{aligned}
D^w_X(h^{\delta-2m}x^m)
&=
\sum_{\fs\in{\textrm{\em \Spinc}}(X)}
(-1)^{\half(w^2+w\cdot c_1(\fs))}SW_X(\fs)
\\
&\qquad\times\sum_{i=0}^{ \min(\ell,[\delta/2]-m)}
\left(
p_{\delta,\ell,m,i}(c_1(\fs)-\La,\La)Q_X^i
\right)(h),
\end{aligned}
\end{equation}
where $Q_X$ is the intersection form on $H_2(X;\RR)$,
$\ell=\frac{1}{4}(\delta+(c_1(\fs)-\La)^2+\frac{3}{4}(\chi+\si))$
and $p_{\delta,\ell,m,i}(\cdot,\cdot)$ is a degree $\delta-2m-2i$
homogeneous polynomial with coefficients which are universal functions of
$$
\chi,\ \sigma,\ c_1(\fs)^2,\ \La^2,\ c_1(\fs)\cdot\La,\ \delta,\ m, \ \ell.
$$
\end{thm}

Recall that for a `\spinu structure' $\ft$ on $X$ (see \cite{FL2a} or \S
\ref{sec:prelim} here),
the moduli space, $\sM_{\ft}$, of $\SO(3)$ monopoles was a smooth manifold
away from two types of singular subspaces.  The first type of subspace was
identified with the moduli space of anti-self-dual connections
\cite{FL2a}. The second type of subspace, that of reducible $\SO(3)$
monopoles, was identified with the Seiberg-Witten moduli space $M_{\fs}$
where the \spinu structure $\ft$ admits a splitting $\ft=\fs\oplus
\fs\otimes L$ \cite{FL2a}.  The Uhlenbeck compactification 
$\bar\sM_{\ft}/S^1$ of
$\sM_{\ft}/S^1$ provides a cobordism between a link of the moduli space
of anti-self-dual connections and links, $\bar\bL_{\ft,\fs}$, of subspaces
of reducible $\SO(3)$ monopoles.  However, additional reducible $\SO(3)$
monopoles, in the form of subspaces,
\begin{equation}
\label{eq:IntroLowerLevelReducibles}
M_{\fs}\times\Sym^\ell(X),
\end{equation}
appear in the lower levels of the Uhlenbeck compactification.  The
intersection number of certain geometric representatives of cohomology
classes with the link of the anti-self-dual connections yields a multiple
of the Donaldson invariant.  The main technical result of this paper,
Theorem \ref{thm:LinkPairing}, is a formula (which is {\em not\/} completely
explicit) for the intersection number of these geometric representatives
with the link $\bar\bL_{\ft,\fs}$ of the subspace
\eqref{eq:IntroLowerLevelReducibles} of $\bar\sM_{\ft}/S^1$.

We note that a statement similar to Theorem \ref{thm:LinkPairing} can also
be proved by the methods of this article, without the assumptions that
$b_1(X)=0$ or $z=h^{\delta-2m}x^m$, but the resulting expression becomes
considerably more complicated.  While the precise expression for the
intersection number in \eqref{eq:LinkIntersectionFormula} is still unknown,
we also prove the `multiplicity conjecture' in
\cite{FL2b,FKLM, FLLevelOne} (see Theorem \ref{thm:Multiplicity} here),
which holds even without the simplifying assumptions.

\subsection{Outline of the argument}
We have described the program for proving Witten's conjecture relating the
Donaldson and Seiberg-Witten invariants for smooth four-manifolds
extensively elsewhere, \cite{FL1,FL2a,FLGeorgia,FLLevelOne}. The main
technical difficulty with this program has been computing the intersection
pairings \eqref{eq:LinkIntersectionFormula} when the Seiberg-Witten
monopoles lie in a lower level of the Uhlenbeck compactification of the
moduli space of $\SO(3)$ monopoles.  Such a computation was completed in
\cite{FLLevelOne} in the case when the reducibles lie in the first level;
that is, when the Seiberg-Witten stratum has the form $M_{\fs}\times X$.  The
computation behind Theorems \ref{thm:LinkPairing} and
\ref{thm:Multiplicity} is a qualitative understanding
of the intersection pairing for the general case: Seiberg-Witten strata
of the form $M_{\fs}\times\Sym^\ell(X)$.
In this article, we give the topological constructions necessary
to prove Theorem \ref{thm:LinkPairing}; the required analytic results
are established in \cite{FL3, FL4}.

The gluing theorems, \cite{FL3, FL4}, describe neighborhoods
of the strata, $M_{\fs}\times\Si$, of the Seiberg-Witten monopoles
$M_{\fs}\times\Sym^\ell(X)$, where $\Si$ is
a stratum of the symmetric product $\Sym^\ell(X)$. Roughly speaking,
there is a fiber bundle, $M(\Si)\to \Gl(\Si)\to M_{\fs}\times \Si$, with
a well-understood structure
group and fiber $M(\Si)$ given by a product of moduli spaces of
anti-self-dual connections on $S^4$.  There is a section
of a pseudo-vector bundle $\Upsilon(\Si)\to\Gl(\Si)$ and the gluing map
identifies the zero-locus of this section with a neighborhood of
$M_{\fs}\times\Si$ in the moduli space of $\SO(3)$ monopoles.

There are two main sources of the difficulty
in computing the intersection pairings \eqref{eq:LinkIntersectionFormula}
when $\ell>1$.
The first difficulty is the presence of higher charge
moduli spaces of anti-self-dual connections on $S^4$ in the fiber
$M(\Si)$.  The topology of these spaces is in still unknown
although the needed information for the case $\ell=2$ has
been worked out in \cite{LenessWC}.
We do not address this problem here.  Instead, we use a pushforward-pullback
argument (see \S \ref{subsec:Pullbacks}) to isolate the topology of
these fibers into `universal constants' of the type appearing in
\eqref{eq:LinkIntersectionFormula}.
The second difficulty, which we do address here, is the presence of
more than one stratum $\Si\subset\Sym^\ell(X)$, and the resulting need
for more than one gluing map to parameterize a neighborhood
of $M_{\fs}\times\Sym^\ell(X)$.

\subsubsection{The problem of overlaps}
To compute the intersection pairing \eqref{eq:LinkIntersectionFormula} when
$\ell\ge 2$, we must understand the overlap of the images of the gluing
maps for different strata $\Si$ and $\Si'$.  We approach this problem by
first understanding the overlap of the images of the {\em splicing maps\/}
for different strata.  Recall from \cite{FL3} that the gluing map is
defined by a perturbation of a splicing map which is defined, roughly, by
using partitions of unity to piece together pairs on $X$ and on $S^4$.
Essentially, it is easier to compare two different splicing maps because
the splicing maps can be written down explicitly.

Given two strata $\Si$ and $\Si'$ of $\Sym^{\ell}(X)$, with
$\Si\subset \cl\ \Si'$, and splicing maps
$$
\bga'_{\Si'}:\Gl(\Si')\to \bar\sC_{\ft},
\quad
\bga'_{\Si}:\Gl(\Si)\to \bar\sC_{\ft},
$$
we achieve the desired comparison by defining a space of `overlap data',
$\Gl(\Si,\Si')$, and maps
$$
\rho^u_{\Si,\Si'}: \Gl(\Si,\Si')\to \Gl(\Si'),
\quad
\rho^d_{\Si,\Si'}: \Gl(\Si,\Si')\to \Gl(\Si),
$$
such that the diagram
\begin{equation}
\label{eq:IntroOverlapCD}
\begin{CD}
\Gl(\Si,\Si') @> \rho^u_{\Si,\Si'} >> \Gl(\Si')
\\
@V \rho^d_{\Si,\Si'} VV @V \bga'_{\Si'} VV
\\
\Gl(\Si) @> \bga'_{\Si} >> \sC_{\ft}
\end{CD}
\end{equation}
commutes and such that
$$
\Imag(\bga'_{\Si'})\cap\Imag(\bga'_{\Si})
=
\Imag(\bga'_{\Si'}\circ\rho^u_{\Si,\Si'})
=
\Imag(\bga'_{\Si}\circ\rho^d_{\Si,\Si'}).
$$
The diagram \eqref{eq:IntroOverlapCD} is used to define
the space of {\em global splicing data\/} as a pushout of
the spaces of local splicing data $\Gl(\Si)$.
Once this is accomplished, the result of the argument is largely a formal
accounting of choices of cohomology classes with compact support
and a use of the familiar pullback-pushforward technique.

\subsubsection{The commutativity of splicing maps}
Since the majority of the article is taken up with establishing
the diagram \eqref{eq:IntroOverlapCD}, we now describe
in more detail how that is carried out.
The two strata $\Si$ and $\Si'$ are given by
partitions of $\ell$.  If $\Si$ is the lower stratum,
that is, $\Si\subset\cl\ \Si'$, then the partition defining
$\Si'$ is a refinement of the partition $\ell=\ka_1+\dots +\ka_r$ defining
$\Si$.  Write $\ka_i=\ka_{i,1}+\cdots+\ka_{i,r_i}$ for this refinement.
The overlap data space, $\Gl(\Si,\Si')$, is,
approximately, defined by a fiber bundle,
$M(\Si')\to \Gl(\Si,\Si')\to \nu(\Si,\Si')$, where $\nu(\Si,\Si')\to \Si$
can be thought of as the normal bundle of $\Si$ in $\Si'$.
The principal bundle 
underlying $\Gl(\Si,\Si')\to\nu(\Si,\Si')$ is defined by
a pullback of the principal bundle defining $\Gl(\Si)$ by
the projection map $\nu(\Si,\Si')\to\nu(\Si)$.
The `upwards' overlap map, $\rho^u_{\Si,\Si'}$, is then defined by
an isomorphism between the restriction of the bundle
$\Gl(\Si')$ to $\nu(\Si,\Si')$ and $\Gl(\Si,\Si')$.
The `downwards' overlap map, $\rho^d_{\Si,\Si'}$,
is defined by taking the data given by the fiber $M(\Si')$
and the fiber of $\nu(\Si,\Si')\to\Si$ and using it
to splice together connections on $S^4$.
That is, the fiber of  $\nu(\Si,\Si')\to\Si$ can be interpreted
as sets of points in $\RR^4$, while the fiber $M(\Si')$ is defined by
connections $A_{i,j}$ on $S^4$ with instanton numbers $\ka_{i,j}$.
Splicing the connections $A_{i,j}$ to the trivial connection on
$\RR^4$ at the given points in $\RR^4$ yields connections
$A_i$ with instanton number $\ka_i$ and thus a point in the fiber $M(\Si)$.
The downwards overlap map is then defined by applying
the above splicing procedure to the fiber 
of $\Gl(\Si,\Si')\to\nu(\Si,\Si')\to \Si$ and then using the identification
of the principal bundle underlying $\Gl(\Si,\Si')$ with that of $\Gl(\Si)$.


The key technical results in proving the commutativity of diagram
\eqref{eq:IntroOverlapCD} appear in Lemma \ref{lem:CommutingSplicing} and
Proposition \ref{prop:CommutingSplicingSymmQuotient}.  These statements can
be thought of as providing an associativity result for splicing maps on
$S^4$.  In a simple case, they assert that if the connection $A'_1$ is
obtained from splicing the connections $A_{1,1}$ and $A_{1,2}$ to the
trivial connection $\Theta$ at the points $x_1$ and $x_2$ in $\RR^4$,
$$
A'_1=\Theta\#_{x_1}A_{1,1}\#_{x_2} A_{1,2},
$$
then the connection obtained by splicing $A'_1$ and $A_2$ to
the trivial connection at $y_1$ and $y_2$ is equal to the
connection obtained by splicing $A_{1,1}$, $A_{1,2}$, and $A_2$
to the trivial connection at $y_1+x_1$, $y_1+x_2$, and
$y_2$ respectively:
\begin{equation}
\label{eq:R4SplicingEquality}
\begin{aligned}
{}&
\Theta\#_{y_1}\left( \Theta\#_{x_1}A_{1,1}\#_{x_2} A_{1,2}\right)
\#_{y_2} A_2
\\
{}&\quad=
\Theta\#_{y_1+x_1}A_{1,1}\#_{y_1+x_2} A_{1,2} \#_{y_2}A_2.
\end{aligned}
\end{equation}
(Of course, this result only holds for connections which
are suitably concentrated relative to the separation of
the points in $\RR^4$.)  

Proposition \ref{prop:CommutingSplicingSymmQuotient} allows
us to define the `spliced-end moduli space', a deformation
of the moduli space of anti-self-dual connections on $S^4$.
The spliced-end moduli space differs from the actual
moduli space of anti-self-dual connections on $S^4$ only
near the strata where the background connection is trivial.
A neighborhood of these strata in the actual moduli space
is given by the gluing map while a neighborhood of these
strata in the spliced-end moduli space is given by the
splicing maps.  Proposition
\ref{prop:CommutingSplicingSymmQuotient} is used to ensure
that the union of the images of these different splicing
maps is a smoothly stratified space.

Replacing the actual moduli
space with the spliced-ends moduli space allows us
to prove that the diagram \eqref{eq:IntroOverlapCD}
commutes by using 
the results of Proposition \ref{prop:XOverlapControl}, an analogue of
Proposition \ref{prop:CommutingSplicingSymmQuotient}
for splicing onto pairs on $X$.
Because the ends of $M(\Si)$ are replaced by the image
of a splicing map, the commutativity follows
from an expression of the form \eqref{eq:R4SplicingEquality}.
We note that Proposition \ref{prop:CommutingSplicingSymmQuotient}
relies on the triviality of the background connection $\Theta$
and the flatness of the metric on $\RR^4$.
Thus, Proposition \ref{prop:XOverlapControl} only gives
the commutativity of the diagram \eqref{eq:IntroOverlapCD}
for `crude splicing maps' $\bga''_\Si$ which are defined by
flattening the background connection and pair on a larger
subspace than the standard splicing map.  However, once
the space of global splicing data is constructed using the
crude splicing maps, it is a straightforward matter to
construct (see \S \ref{subsec:GlobalSplicingMap}) isotopies
of these crude splicing maps to the standard splicing maps.

\subsection{Outline of the article}
Section \ref{sec:prelim} reviews notation and definitions
from our previous articles in this sequence.
In \S \ref{sec:Diagonals}, we describe a stratification of
$\Sym^\ell(X)$, and define normal bundles
of the strata of $\Sym^\ell(X)$.
In \S \ref{sec:diagTM}, we show that the projection maps
of the normal bundles
introduced in \S \ref{sec:Diagonals} satisfy Thom-Mather type 
equalities on the overlap of their images.
In \S \ref{sec:SplicedEnd}, we define the spliced-ends moduli
space of anti-self-dual connections on $S^4$ and this is used
in \S \ref{sec:GlobalSplicingData} to define the space of
global splicing data.  An analogous construction of an obstruction
bundle over the space of global splicing data
is carried out in \S \ref{sec:SplicedEndsObstruction} and
\S \ref{sec:obstr}.  The cohomological computations are
carried out in
\S \ref{sec:Cohom} where we compute the pullbacks of the relevant cohomology
classes to the space of global splicing data,
in \S \ref{sec:Link} where we define the link $\bar\bL_{\ft,\fs}$,
and in
\S \ref{sec:Comp} where we actually carry out the needed
computations. We discuss the application of the main result of this
article, Equation \eqref{eq:MainEquation}, to the proof of Witten's
conjecture in a separate article \cite{FLWConjecture}.

\subsubsection*{Acknowledgments}
The authors are extremely grateful to Tom Mrowka for his constant
encouragement over the years and in particular to some very fruitful
conversations where he urged us to develop the concept of `spliced-end
moduli spaces' of instantons on $S^4$. He explained this concept to us in
March 1999, and proposed it as a key idea in this part of our work on
Witten's conjecture and the Kotschick-Morgan conjecture, with the latter
following almost immediately from the method developed in the present
article, modulo the requisite gluing theory \cite{FL3, FL4, FLKM1, FLKM2}.


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\section{Preliminaries}
\label{sec:prelim}
We begin in \S \ref{subsec:ModuliSpace} by recalling the definition of the
moduli space of $\SO(3)$ monopoles and its basic properties
\cite{FL1, FeehanGenericMetric}. In \S \ref{subsec:ASDsingularities} we
describe
the stratum of zero-section monopoles, or anti-self-dual connections. In \S
\ref{subsec:Reducibles} we discuss the strata of reducible, or
Seiberg-Witten monopoles, together with their `virtual' neighborhoods and
normal bundles.  In \S
\ref{subsec:Cohomology} we define the cohomology classes which will be
paired with the links of the anti-self-dual and Seiberg-Witten moduli
spaces.  In \S \ref{subsec:Donaldsonseries} we review the definition of the
Donaldson series.  Lastly, in \S \ref{subsec:LinkCobordism} we describe the
basic relation between the pairings with links of the anti-self-dual and
Seiberg-Witten moduli spaces provided by the $\SO(3)$-monopole cobordism.

\subsection{The moduli space of \boldmath{$\SO(3)$} monopoles}
\label{subsec:ModuliSpace}
Throughout this article, $(X,g)$ will denote a closed, connected, oriented,
smooth, Riemannian four-manifold.

\subsubsection{Clifford modules}
\label{subsubsec:SpincuStr}
Let $V$ be a Hermitian vector bundle over $(X,g)$ and let $\rho:T^*X\to
\End_\CC(V)$ be a real-linear map satisfying
\begin{equation}
\label{eq:CliffordMapDefn}
\rho(\alpha)^2 = -g(\alpha,\alpha)\id_{V}
\quad\text{and}\quad
\rho(\alpha)^\dagger = -\rho(\alpha),
\quad \alpha \in C^\8(T^*X).
\end{equation}
The map $\rho$ uniquely extends to a linear isomorphism,
$\rho:\Lambda^{\bullet}(T^*X)\otimes_\RR\CC\to\End_\CC(V)$, and gives $V$
the structure of a Hermitian Clifford module for the complex Clifford
algebra $\CCl(T^*X)$.  There is a splitting $V=V^+\oplus V^-$, where
$V^\pm$ are the $\mp 1$ eigenspaces of $\rho(\vol)$. A unitary connection
$A$ on $V$ is {\em spin \/} if
\begin{equation}
\label{eq:SpinConnection}
[\nabla_A,\rho(\alpha)] =\rho(\nabla\alpha)
\quad\text{on }C^\8(V),
\end{equation}
for any $\alpha\in C^\8(T^*X)$, where $\nabla$ is the Levi-Civita connection.

A Hermitian Clifford module $\fs=(\rho,W)$ is a \spinc structure when $W$
has complex rank four; it defines a class
\begin{equation}
\label{eq:DefineChernClassOfSpinc}
c_1(\fs)=c_1(W^+),
\end{equation}
and every class in $H^2(X;\ZZ)$ lifting the second Stiefel-Whitney class,
$w_2(X)\in H^2(X;\ZZ/2\ZZ)$, arises this way.

We call a Hermitian Clifford module $\ft=(\rho,V)$ a \spinu structure when
$V$ has complex rank eight. Recall that $\fg_{V}\subset\su(V)$ is the
$\SO(3)$ subbundle given by the span of the sections of the bundle $\su(V)$
which commute with the action of $\CCl(T^*X)$ on $V$. We obtain a splitting
\begin{equation}
\label{eq:EndSplitting}
\su(V^+)
\cong
\rho(\Lambda^+)\oplus i\rho(\Lambda^+)\otimes_\RR\fg_{V}
\oplus \fg_{V},
\end{equation}
and similarly for $\su(V^-)$. The fibers $V_x^+$ define complex lines whose
tensor-product square is $\det(V^+_x)$ and thus a complex line bundle over
$X$,
\begin{equation}
\label{eq:CliffordDeterminantBundle}
{\det}^{\frac{1}{2}}(V^+).
\end{equation}
A \spinu structure $\ft$ thus defines classes,
\begin{equation}
\label{eq:SpinUCharacteristics}
c_1(\ft)=\textstyle{\frac{1}{2}} c_1(V^+),
\quad
p_1(\ft) = p_1(\fg_{V}),
\quad
\text{and}\quad
w_2(\ft)=w_2(\fg_{V}).
\end{equation}
Given $W$, one has an isomorphism $V\cong W\otimes_\CC E$ of Hermitian
Clifford modules, where $E$ is a rank-two Hermitian vector bundle
\cite[Lemma 2.3]{FL2a}; then
$$
\fg_V
=
\su(E)
\quad\text{and}\quad
{\det}^{\frac{1}{2}}(V^+)
=
\det(W^+)\otimes_\CC\det(E).
$$

\subsubsection{$\SO(3)$ monopoles}
\label{subsubsec:PU2Monopoles}
We fix a smooth unitary connection $A_\La$ on the line bundle
${\det}^{\frac{1}{2}}(V^+)$, let $k\geq 2$ be an integer, and let $\sA_{\ft}$
be the affine space of $L^2_k$ spin connections on $V$ which induce the
connection $2A_\La$ on $\det(V^+)$. If $A$ is a spin connection on $V$ then
it defines an $\SO(3)$ connection $\hat A$ on the subbundle
$\fg_V\subset\su(V)$
\cite[Lemma 2.5]{FL2a}; conversely, every $\SO(3)$ connection on $\fg_V$
lifts to a unique spin connection on $V$ inducing the connection
$2A_\Lambda$ on $\det(V^+)$ \cite[Lemma 2.11]{FL2a}.

Let $\sG_{\ft}$ denote the group of $L^2_{k+1}$ unitary automorphisms of
$V$ which commute with $\CCl(T^*X)$ and which have Clifford-determinant one
(see \cite[Definition 2.6]{FL2a}). Define
\begin{equation}
\label{eq:SpinUPreConfiguration}
\tsC_\ft(X) = \sA_\ft(X)\times L^2_k(X,V^+)
\quad\text{and}\quad
\sC_\ft = \tsC_\ft/\sG_\ft.
\end{equation}
The action of $\sG_{\ft}$ on $V$ induces an adjoint action on
$\End_\CC(V)$, acting as the identity on
$\rho(\Lambda^\bullet_\CC)\subset\End_\CC(V)$ and inducing an adjoint
action on $\fg_{V}\subset\End_\CC(V)$ (see
\cite[Lemma 2.7]{FL2a}). The space $\tsC_\ft$ and hence $\sC_\ft$ carry
circle actions induced by scalar multiplication on $V$:
\begin{equation}
\label{eq:S1ZAction}
S^1\times V \to V,
\quad (e^{i\theta},\Phi)\mapsto e^{i\theta}\Phi.
\end{equation}
Because this action commutes with that of $\sG_{\ft}$, the
action \eqref{eq:S1ZAction} also defines an action on $\sC_{\ft}$.
Note that $-1\in S^1$ acts trivially on $\sC_{\ft}$.  Let
$\sC^0_\ft\subset\sC_\ft$ be the subspace represented by pairs whose spinor
components are not identically zero, let $\sC^*_\ft\subset\sC_\ft$ be the
subspace represented by pairs where the induced $\SO(3)$ connections on
$\fg_{V}$ are irreducible, and let $\sC^{*,0}_\ft=\sC^*_\ft\cap
\sC^0_\ft$.

We call a pair $(A,\Phi)$ in $\tsC_\ft$ a {\em $\SO(3)$ monopole\/} if
\begin{equation}
\label{eq:UnperturbedMonopoles}
\begin{aligned}
\ad^{-1}(F^+_{\hat A}) - \tau\rho^{-1}(\Phi\otimes\Phi^*)_{00}
&=0,
\\
D_A\Phi + \rho(\vartheta)
&=0.
\end{aligned}
\end{equation}
Here, $D_A=\rho\circ \cov_A:C^\8(X,V^+)\to C^\8(X,V^-)$ is the Dirac
operator; the isomorphism $\ad:\fg_{V}\to \so(\fg_{V})$ identifies the
self-dual curvature $F_{\hat A}^+$, a section of
$\La^+\otimes\so(\fg_{V})$, with $\ad^{-1}(F^+_{\hat A})$, a section of
$\La^+\otimes\fg_{V}$; the section $\tau$ of $\GL(\Lambda^+)$ is a
perturbation close to the identity; the perturbation $\vartheta$ is a
complex one-form close to zero; $\Phi^* \in
\Hom(V^+,\CC)$ is the pointwise Hermitian dual $\langle\cdot,\Phi\rangle$
of $\Phi$; and $(\Phi\otimes\Phi^*)_{00}$ is the component of the section
$\Phi\otimes\Phi^*$ of $i\fu(V^+)$ lying in $\rho(\Lambda^+)\otimes\fg_{V}$
with respect to the splitting $\fu(V^+)=i\underline{\RR}\oplus\su(V^+)$ and
decomposition \eqref{eq:EndSplitting} of $\su(V^+)$.

Equation \eqref{eq:UnperturbedMonopoles} is invariant under the action of
$\sG_\ft$. We let
$\sM_{\ft}\subset \sC_\ft$ be the subspace represented by pairs satisfying
equation
\eqref{eq:UnperturbedMonopoles} and write
\begin{equation}
\label{eq:PU2MonopoleSubspaces}
\sM^*_{\ft} = \sM_{\ft}\cap\sC^*_\ft, \quad
\sM^0_{\ft}= \sM_{\ft}\cap\sC^0_\ft, \quad\text{and}\quad
\sM^{*,0}_\ft = \sM_{\ft}\cap\sC^{*,0}_\ft.
\end{equation}
Since equation \eqref{eq:UnperturbedMonopoles} is invariant under the
circle action induced by scalar multiplication on $V$, the subspaces
\eqref{eq:PU2MonopoleSubspaces} of $\sC_{\ft}$ are also invariant under
this action.

\begin{thm}
\label{thm:Transv}
\cite[Theorem 1.1]{FeehanGenericMetric}, \cite{TelemanGenericMetric}
Let $X$ be a closed, oriented, smooth four-manifold and let $V$ be a
complex rank-eight, Hermitian vector bundle over $X$. Then for parameters
$(\rho,g,\tau,\vartheta)$, which are generic in the sense of
\cite{FeehanGenericMetric}, and $\ft=(\rho,V)$, the space
$\sM^{*,0}_{\ft}$ is a smooth manifold of the expected dimension,
\begin{equation}
\label{eq:Transv}
\begin{aligned}
\dim \sM^{*,0}_{\ft}
= d_a(\ft)+2n_a(\ft),
\quad\text{where }
d_a(\ft)
&=-2p_1(\ft)- \textstyle{\frac{3}{2}}(\chi+\sigma),
\\
n_a(\ft)
&=\textstyle{\frac{1}{4}}(p_1(\ft)+c_1(\ft)^2-\sigma).
\end{aligned}
\end{equation}
\end{thm}

For the remainder of the article, we assume that the perturbation
parameters in \eqref{eq:UnperturbedMonopoles} are chosen as indicated in
Theorem \ref{thm:Transv}. For each non-negative integer $\ell$, let
$\ft_\ell=(\rho,V_{\ell})$, where
$$
c_1(V_\ell)=c_1(V),
\quad
p_1(\fg_{V_\ell})=p_1(\fg_{V})+4\ell,
\quad\text{and}\quad
w_2(\fg_{V_\ell})=w_2(\fg_{V}).
$$
We let $\bar\sM_{\ft}$ denote the closure of $\sM_{\ft}$ in the space of
ideal monopoles,
\begin{equation}
\label{eq:idealmonopoles}
I\sM_{\ft} = \bigsqcup_{\ell=0}^\8(\sM_{\ft_{\ell}}\times\Sym^\ell(X)),
\end{equation}
with respect to an Uhlenbeck topology
\cite[Definition 4.19]{FL1} and call the intersection of $\bar\sM_{\ft}$
with $\sM_{\ft_{\ell}}\times \Sym^\ell(X)$ its {\em $\ell$-th level\/}.

\begin{thm}
\label{thm:Compactness}
\cite[Theorem 1.1]{FL1}
Let $X$ be a Riemannian four-manifold with \spinu structure $\ft$.  Then
there is a positive integer $N$, depending at most on the curvature of the
chosen unitary connection on $\det(V^+)$ together with $p_1(\ft)$, such
that the Uhlenbeck closure $\bar\sM_{\ft}$ of $\sM_{\ft}$ in
$\sqcup_{\ell=0}^N(\sM_{\ft_{\ell}}\times\Sym^\ell(X))$
is a second countable, compact, Hausdorff space. The space $\bar\sM_{\ft}$
carries a continuous circle action,
which restricts to the circle action defined on $\sM_{\ft_\ell}$ on each level.
\end{thm}

\subsection{Stratum of anti-self-dual or zero-section solutions}
\label{subsec:ASDsingularities}
{}From equation \eqref{eq:UnperturbedMonopoles}, we see that the stratum of
$\sM_{\ft}$ represented by pairs with zero spinor is identified with
$$
\{A\in\sA_{\ft}: F_{\hat A}^+ = 0\}/\sG_{\ft} \cong M_\kappa^w(X,g),
$$
the moduli space of $g$-anti-self-dual connections on the $\SO(3)$ bundle
$\fg_{V}$, where $\ka=-\quarter p_1(\ft)$ and $w\equiv w_2(\ft)\pmod 2$.
For a generic Riemannian metric $g$, the space $M_\kappa^w(X,g)$ is a
smooth manifold of the expected dimension $-2p_1(\ft) -
\threehalf(\chi+\sigma)=d_a(\ft)$.

As explained in \cite[\S 3.4.1]{FL2a}, it is desirable to choose
$w\pmod{2}$ so as to exclude points in $\bar\sM_{\ft}$ with associated flat
$\SO(3)$ connections, so we have a {\em disjoint\/} union,
\begin{equation}
\label{eq:StratificationCptPU(2)Space}
\bar\sM_{\ft}
\cong
\bar\sM_{\ft}^{*,0} \sqcup \bar M_\kappa^w \sqcup \bar\sM_{\ft}^{\red},
\end{equation}
where $\bar\sM_{\ft}^*\subset\bar\sM_{\ft}$ is the subspace represented by
triples whose associated $\SO(3)$ connections are irreducible,
$\bar\sM_{\ft}^0\subset\bar\sM_{\ft}$ is the subspace represented by
triples whose spinors are not identically zero,
$\bar\sM_{\ft}^{*,0} = \bar\sM_{\ft}^{*}\cap\bar\sM_{\ft}^{0}$,
while $\bar\sM_{\ft}^{\red}\subset\bar\sM_{\ft}$
is the subspace $\bar\sM_{\ft}-\bar\sM_{\ft}^*$
represented by triples whose associated $\SO(3)$
connections are reducible. We recall the

\begin{defn}
\label{defn:Good}
\cite[Definition 3.20]{FL2b}
A class $v\in H^2(X;\ZZ/2)$ is {\em good\/} if no integral lift of $v$ is
torsion.
\end{defn}

If $w\pmod{2}$ is good, then the union
\eqref{eq:StratificationCptPU(2)Space} is disjoint, as desired. In
practice, rather than constraining $w\pmod{2}$ itself, we replace $X$ by
the blow-up $X\#\overline{\CC\PP}^2$ and $w$ by $w+\PD[e]$ (where $e\in
H_2(X;\ZZ)$ is the exceptional class), noting that $w+\PD[e]\pmod{2}$ is
always good, and define gauge-theoretic invariants of $X$ in terms of
moduli spaces on $X\#\overline{\CC\PP}^2$. When $w\pmod{2}$ is good, we
define \cite[Definition 3.7]{FL2a} the link of $\barM^w_{\ka}$ in
$\bar\sM_{\ft}/S^1$ by
\begin{equation}
\label{eq:DefineASDLink}
\bL^{w}_{\ft,\ka}
=
\{[A,\Phi,\bx]\in\bar\sM_{\ft}/S^1: \|\Phi\|_{L^2}^2=\eps\},
\end{equation}
where $\eps$ is a small positive constant; for generic $\eps$, the link
$\bL^w_{\ft,\ka}$ is a smoothly-stratified, codimension-one subspace of
$\bar\sM_{\ft}/S^1$.

\subsection{Strata of Seiberg-Witten or reducible solutions}
\label{subsec:Reducibles}
We call a pair $(A,\Phi)\in\tsC_\ft$ {\em reducible\/} if the connection
$A$ on $V$ respects a splitting,
\begin{equation}
\label{eq:BasicSplitting}
V = W\oplus W\otimes L = W\otimes (\underline{\CC}\oplus L),
\end{equation}
for some \spinc structure $\fs=(\rho,W)$ and complex line bundle $L$, in
which case $c_1(L)=c_1(\ft)-c_1(\fs)$.
A spin connection $A$ on $V$ is reducible with respect to the splitting
\eqref{eq:BasicSplitting} if and only if $\hat A$ is reducible
with respect to the splitting $\fg_{V}\cong \underline{\RR}\oplus L$,
\cite[Lemma 2.9]{FL2a}. If $A$ is reducible, we can write $A=B\oplus
B\otimes A_L$, where $B$ is a spin connection on $W$ and $A_L$ is a unitary
connection on $L$; then $\hat A= d_{\RR}\oplus A_L$ and $A_L = A_\La\otimes
(B^{\det})^*$, where $B^{\det}$ is the connection on $\det(W^+)$ induced by
$B$ on $W$.


\subsubsection{Seiberg-Witten monopoles}
\label{subsubsec:SWMonopoles}
Given a \spinc structure $\fs=(\rho,W)$ on $X$, let $\sA_{\fs}$ denote the
affine space of $L^2_k$ spin connections on $W$. Let $\sG_{\fs}$
denote the group of $L^2_{k+1}$ unitary automorphisms of $W$, commuting with
$\CCl(T^*X)$, which we identify with $L^2_{k+1}(X,S^1)$. We then define
\begin{equation}
\label{eq:SpincPreConfig}
\tsC_{\fs}= \sA_{\fs}\times L^2_k(W^+)
\quad\text{and}\quad
\sC_{\fs}=\tsC_{\fs}/\sG_{\fs},
\end{equation}
where $\sG_{\fs}$ acts on $\tsC_{\fs}$ by
\begin{equation}
\label{eq:SWGaugeGroupAction}
(s,(B,\Psi))
\mapsto
s(B,\Psi)
=
(B- (s^{-1}ds)\id_W, s\Psi).
\end{equation}
We call a pair $(B,\Psi)\in\tsC_{\fs}$ a Seiberg-Witten monopole if
\begin{equation}
\label{eq:SeibergWitten}
\begin{aligned}
\Tr(F^+_B) - \tau\rho^{-1}(\Psi\otimes\Psi^*)_{0} - F^+(A_{\Lambda})
&=0,
\\
D_B\Psi + \rho(\vartheta)\Psi
&=0,
\end{aligned}
\end{equation}
where $\Tr:\fu(W^+)\to i\RR$ is defined by the trace on $2\times 2$ complex
matrices, $(\Psi\otimes\Psi^*)_0$ is the component of the section
$\Psi\otimes\Psi^*$ of $i\fu(W^+)$ contained in $i\su(W^+)$,
$D_B:C^\8(W^+)\to C^\8(W^-)$ is the Dirac operator, and $A_\Lambda$ is a
unitary connection on a line bundle with first Chern class $\Lambda\in
H^2(X;\ZZ)$.  The perturbations are chosen so that solutions to equation
\eqref{eq:SeibergWitten} are identified with reducible solutions to
\eqref{eq:UnperturbedMonopoles} when $c_1(\ft)=\Lambda$.
Let $\tM_{\fs}\subset\tsC_{\fs}$ be the subspace cut out by equation
\eqref{eq:SeibergWitten} and denote the moduli space of Seiberg-Witten
monopoles by $M_{\fs}=\tM_{\fs}/\sG_{\fs}$.

\subsubsection{Seiberg-Witten invariants}
\label{subsubsec:SWInvariants}
We let $\sC_{\fs}^0\subset\sC_{\fs}$ be the open subspace represented by
pairs whose spinor components which are not identically zero and
define a complex line bundle over $\sC_\fs^0\times X$ by
\begin{equation}
\label{eq:DefineSWUniversal}
\LL_{\fs}= \tsC_\fs^0\times_{\sG_\fs}\underline{\CC},
\end{equation}
where $\underline{\CC}=X\times\CC$ and
$s\in \sG_{\fs}$ acts on $(B,\Psi)\in \tsC_{\fs}$ and $(x,\zeta)\in
\underline{\CC}$ by
\begin{equation}
\label{eq:DefineSWUniversalS1Action}
((B,\Psi),(x,\zeta))
\mapsto
(s(B,\Psi),(x,s(x)^{-1}\zeta)).
\end{equation}
Define
\begin{equation}
\label{eq:DefineAAA2}
\AAA_2(X) = \Sym\left( H_{0}(X;\RR)\right)\otimes\La^\bullet(H_{1}(X;\RR))
\end{equation}
be the graded algebra, with $z=\beta_1\beta_2\cdots\beta_r$ having total
degree $\deg(z) = \sum_p(2-i_p)$, when $\beta_p\in H_{i_p}(X;\RR)$.
Then the map,
$$
\mu_{\fs}:H_\bullet(X;\RR)\to H^{2-\bullet}(\sC^0_{\fs};\RR),
\quad
\beta\mapsto c_1(LL_{\fs})/\beta,
$$
extends in the usual way to
a homomorphism of graded real algebras,
$$
\mu_{\fs}:\AAA_2(X) \to H^{\bullet}(\sC^0_\fs;\RR).
$$
If $x\in H_0(X;\ZZ)$ denotes the positive generator, we set
\begin{equation}
\label{eq:SWClass}
\mu_{\fs} = c_1(\LL_{\fs})/x \in H^2(\sC_\fs^0;\ZZ).
\end{equation}
Equivalently, $\mu_{\fs}$ is the first Chern class of the $S^1$ base-point
fibration over $\sC_\fs^0$. If $b_1(X)=0$, then $c_1(\LL_{\fs})=\mu_{\fs}\times
1$ by \cite[Lemma 2.14]{FL2a}.

For $b_2^+(X)>0$ and generic Riemannian metrics on $X$, the space $M_{\fs}$
contains zero-section pairs if and only if $c_1(\fs)-\Lambda$ is a torsion
class by \cite[Proposition 6.3.1]{MorganSWNotes}. If
$M_{\fs}$ contains no zero-section pairs then, for generic perturbations,
it is a compact, oriented, smooth manifold of dimension
\begin{equation}
\label{eq:DimSW}
d_s(\fs)
=
\dim M_{\fs}
=
\tquarter(c_1(\fs)^2 -2\chi -3\sigma).
\end{equation}
Let $\tilde X=X\#\overline{\CC\PP}^2$ denote the blow-up of $X$ with
exceptional class $e\in H_2(\tilde X;\ZZ)$ and denote its Poincar\'e dual
by $\PD[e]\in H^2(\tilde X;\ZZ)$. Let $\fs^\pm=(\tilde\rho,\tilde W)$
denote the \spinc structure on $\tilde X$ with $c_1(\fs^\pm)=c_1(\fs)\pm
\PD[e]$ obtained by splicing the \spinc structure $\fs=(\rho,W)$ on $X$
with the
\spinc structure on $\overline{\CC\PP}^2$ with first Chern class $\pm \PD[e]$.
(See \cite[\S 4.5]{FL2b} for an explanation of the relation between \spinc
structures on $X$ and $\tilde X$.) Now
$$
c_1(\fs)\pm \PD[e]-\Lambda \in H^2(\tilde X;\ZZ)
$$
is not a torsion class and so --- for $b_2^+(X)>0$, generic Riemannian
metrics on $X$ and related metrics on the connected sum $\tilde X$ --- the
moduli spaces $M_{\fs^\pm}(\tilde X)$ contain no zero-section pairs. Thus,
for our choice of generic perturbations, the moduli spaces
$M_{\fs^\pm}(\tilde X)$ are compact, oriented, smooth manifolds, both of
dimension $\dim M_{\fs}(X)$.

For $b_1(X)=0$ and odd $b_2^+(X)>1$, we define the {\em Seiberg-Witten
invariant\/} by \cite[\S 4.1]{FL2b}
\begin{equation}
\label{eq:DefSW}
  SW_{X}(\fs)
=
\langle\mu_{\fs^+}^{d},[M_{\fs^+}(\tilde X)]\rangle
=
\langle\mu_{\fs^-}^{d},[M_{\fs^-}(\tilde X)]\rangle,
\end{equation}
where $2d=d_s(\fs)=d_s(\fs^\pm)$.
When $b^+_2(X)=1$ the pairing on the right-hand side of definition
\eqref{eq:DefSW} depends on the chamber in the positive cone of
$H^2(\tilde X;\RR)$ determined by the period point of the Riemannian metric
on $\tilde X$. The definition of the Seiberg-Witten invariant for this
case is also given in \cite[\S 4.1]{FL2b}: we assume that the class
$w_2(X)-\Lambda\pmod{2}$ is good to avoid technical difficulties involved
in chamber specification. Since $w\equiv w_2(X)-\Lambda \pmod{2}$, this
coincides with the constraint we use to define the Donaldson invariants in
\S \ref{subsec:Donaldsonseries} when $b_2^+(X)=1$. We refer to \cite[Lemma
4.1 \& Remark 4.2]{FL2b} for a comparison of the chamber structures
required for the definition of Donaldson and Seiberg-Witten invariants when
$b_2^+(X)=1$.

\subsubsection{Reducible $\SO(3)$ monopoles}
\label{subsubsec:RedPU2Monopole}
If $\ft=(\rho,V)$ and $\fs=(\rho,W)$ with $V=W\oplus W\otimes L$,
then there is an embedding
\begin{equation}
\label{eq:DefnOfIota}
\iota: \tsC_{\fs} \embed \tsC_{\ft},
\quad
(B,\Psi)\mapsto (B\oplus B\otimes A_\La\otimes B^{\det,*},\Psi\oplus 0),
\end{equation}
which is gauge-equivariant with respect to the homomorphism
\begin{equation}
\label{eq:GaugeGroupInclusion}
\varrho:\sG_{\fs}\embed \sG_{\ft},
\quad
s\mapsto \id_W\otimes\begin{pmatrix}s & 0 \\ 0 & s^{-1}\end{pmatrix}.
\end{equation}
According to \cite[Lemma 3.13]{FL2a}, the map \eqref{eq:DefnOfIota} defines
a topological embedding $\iota:M_{\fs}^0\embed\sM_{\ft}$, where
$M_{\fs}^0=M_{\fs}\cap\sC_{\fs}^0$ and an embedding of $M_{\fs}$ if
$w_2(\ft)\neq 0$ or $b_1(X)=0$; its image in $\sM_{\ft}$ is represented
by pairs which are reducible with respect to the splitting $V=W\oplus
W\otimes L$. Henceforth, we shall not distinguish between $M_\fs$ and its
image in $\sM_{\ft}$ under this embedding.


\subsubsection{Circle actions}
\label{subsubsec:S1Actions}
When $V=W\oplus W\otimes L$ and $\ft=(\rho,V)$, the space $\tsC_{\ft}$
inherits a circle action defined by
\begin{equation}
\label{eq:DefineS1LAction}
S^1\times V \to V,
\quad (e^{i\theta},\Psi\oplus\Psi')\mapsto \Psi\oplus e^{i\theta}\Psi',
\end{equation}
where $\Psi\in C^\8(W)$ and $\Psi'\in C^\8(W\otimes L)$. With respect to
the splitting $V=W\oplus W\otimes L$, the actions
\eqref{eq:DefineS1LAction} and \eqref{eq:S1ZAction} are related by
\begin{equation}
\label{eq:RelateS1Actions}
\begin{pmatrix}1 & 0 \\ 0 & e^{i2\theta}\end{pmatrix}
=
e^{i\theta}u,
\text{ where }
u = \begin{pmatrix}e^{-i\theta} & 0 \\ 0 & e^{i\theta}\end{pmatrix}
\in\sG_{\ft},
\end{equation}
and so, when we pass to the induced circle actions on the quotient
$\sC_{\ft}=\tsC_{\ft}/\sG_{\ft}$, the actions
\eqref{eq:DefineS1LAction} and \eqref{eq:S1ZAction} on
$\sC_{\ft}$ differ only in their multiplicity. Recall \cite[Lemma
3.11]{FL2a} that the image  in $\tsC_{\ft}$ of the map \eqref{eq:DefnOfIota}
contains all pairs which are fixed by the circle action
\eqref{eq:DefineS1LAction}.


\subsubsection{The virtual normal bundle of the Seiberg-Witten moduli space}
\label{subsubsec:ThickenedNeighborhood}
Suppose $\ft=(\rho,V)$ and $\fs=(\rho,W)$, with $V=W\oplus W\otimes L$, so
we have a topological embedding $M_{\fs}\embed \sM_{\ft}$; we assume
$M_{\fs}$ contains no zero-section monopoles.  Recall from \cite[\S
3.5]{FL2a} that there exist finite-rank, complex vector bundles,
\begin{equation}
\label{eq:VirtualNormalandObstBundles}
\pi_{\Xi}:\Xi_{\ft,\fs}\to M_{\fs}
\quad\text{and}\quad
\pi_{N}:N_{\ft,\fs}\to M_{\fs},
\end{equation}
with $\Xi_{\ft,\fs}\cong M_{\fs}\times\CC^{r_\Xi}$, called the {\em
obstruction bundles\/} and {\em ambient normal bundles\/} of
$M_{\fs}\embed\sM_{\ft}$, respectively.  For a small enough positive radius
$\eps$, there are a topological embedding \cite[Theorem 3.21]{FL2a} of an
open tubular neighborhood,
\begin{equation}
\label{eq:BackgroundConfigEmbedding}
\bga_{\fs}:N_{\ft,\fs}(\eps)
\embed
\sC_{\ft},
\end{equation}
and a smooth section $\bchi_{\fs}$ of the pulled-back
complex vector bundle,
\begin{equation}
\label{eq:PulledBackObstructionBundle}
\pi_{N}^*\Xi_{\ft,\fs} \to N_{\ft,\fs}(\eps),
\end{equation}
such that the tubular map yields a homeomorphism
\begin{equation}
\label{eq:HomeoSWEmbedding}
\bga_{\fs}: \bchi_{\fs}^{-1}(0)\cap N_{\ft,\fs}(\eps)
\cong
\sM_{\ft}\cap \bga_{\fs}(N_{\ft,\fs}(\eps)),
\end{equation}
restricting to a diffeomorphism on the complement of $M_{\fs}$ and
identifying $M_{\fs}$ with its image $\iota(M_{\fs})\subset\sM_{\ft}$.
The image $\bga_{\fs}(N_{\ft,\fs}(\eps))$ is an open subset of an {\em
ambient moduli space\/}, $\sM_{\ft,\fs}^{\stab}\subset \sC_{\ft}$.

Our terminology is loosely motivated by that of \cite{GraberPand} and
\cite{RuanSW}, where the goal (translated to our setting) would be to
construct a {\em virtual fundamental class\/} for $\sM_{\ft}$, given by
the cap product of the fundamental class of an ambient space containing
$\sM_{\ft}$ with the Euler class of a vector bundle over this ambient
space, where $\sM_{\ft}$ is the zero locus of a (possibly
non-transversally vanishing) section. Here, $\sM_{\ft,\fs}^{\stab}$ plays the
role of the ambient space and (the pushforward of) $\Xi_{\ft,\fs}$ the
vector bundle with zero section yielding (an open neighborhood in)
$\sM_{\ft}$.  Then, $N_{\ft,\fs}$ is the normal bundle of $M_{\fs}\embed
\sM_{\ft,\fs}^{\stab}$, while $[N_{\ft,\fs}]-[\Xi_{\ft,\fs}]$ would more
properly be called the `virtual normal bundle' of $M_{\fs}\embed
\sM_{\ft}$, in the language of $K$-theory.

Recall that minus the index of the $\SO(3)$-monopole elliptic deformation
complex \cite[Equations (2.47) \& (3.35)]{FL2a} at a reducible solution can
be written as
\begin{equation}
\label{eq:SWDimRelations}
\dim\sM_{\ft}
=
2n_s(\ft,\fs) + d_{s}(\fs),
\end{equation}
where $d_s(\fs)$ is the expected dimension of the Seiberg-Witten moduli
space $M_{\fs}$ (see equation \eqref{eq:DimSW}), while
$n_s(\ft,\fs)=n_s'(\ft,\fs)+n_s''(\ft,\fs)$ is minus the complex index of
the normal deformation operator \cite[Equations (3.71) \& (3.72)]{FL2a}),
with
\begin{equation}
\label{eq:NormalComponentDims}
\begin{aligned}
n_s'(\ft,\fs)
&= -(c_1(\ft)-c_1(\fs))^2-\textstyle{\frac{1}{2}}(\chi+\sigma),
\\
n_s''(\ft,\fs)
&=
\textstyle{\frac{1}{8}}(c_1(\fs)-2c_1(\ft))^2-\sigma).
\end{aligned}
\end{equation}
If $r_\Xi$ is the complex rank of $\Xi_{\ft,\fs}\to M_{\fs}$, and
$r_N(\ft,\fs)$ is the complex rank of $N_{\ft,\fs}\to M_{\fs}$, then
\begin{equation}
\label{eq:SWBundleRankRelations}
r_N(\ft,\fs)
=
n_s(\ft,\fs) + r_\Xi,
\end{equation}
as we can see from the dimension relation \eqref{eq:SWDimRelations}
and the topological model \eqref{eq:HomeoSWEmbedding}.

The map \eqref{eq:BackgroundConfigEmbedding} is circle equivariant when the
circle acts trivially on $M_{\fs}$, by scalar multiplication on the fibers
of $N_{\ft,\fs}(\eps)$, and by the action
\eqref{eq:DefineS1LAction} on $\sC_{\ft}$. The bundle
\eqref{eq:PulledBackObstructionBundle} and section $\bchi_{\fs}$ are
circle equivariant if the circle acts on $N_{\ft,\fs}$ and the fibers
of $\bga_{\fs}^*\Xi_{\ft,\fs}$ by scalar multiplication.

Let $\tN_{\ft,\fs}\to \tM_{\fs}$ be the pullback of $N_{\ft,\fs}$
by the projection $\tM_{\fs}\to M_{\fs}=\tM_{\fs}/\sG_{\fs}$, so
$\tN_{\ft,\fs}$ is a $\sG_{\fs}$-equivariant bundle,
where $\sG_{\fs}$ acts on the base $\tM_{\fs}$ by the usual gauge group
action \eqref{eq:SWGaugeGroupAction} and the induced action on the total space,
\begin{equation}
\label{eq:VirtualNormalBundleTotalSpace}
\tN_{\ft,\fs}
\subset
\tM_{\fs}\times
L^2_k(\Lambda^1\otimes_\RR L) \oplus L^2(W^+\otimes L)
\subset
\tM_{\fs}\times
L^2_k(\Lambda^1\otimes_\RR \fg_{V}) \oplus L^2(V^+),
\end{equation}
via the embedding \eqref{eq:GaugeGroupInclusion} of $\sG_{\fs}$ into
$\sG_{\ft}$ and the splittings $\fg_{V}\cong\underline{\RR}\oplus L$
\cite[Lemma 3.10]{FL2a} and
$V=W\oplus W\otimes L$.  Thus, $s\in\sG_{\fs}$ acts by scalar multiplication
by $s^{-2}$ on sections of $\Lambda^1\otimes_\RR L$ and by $s^{-1}$ on
sections of $W^+\otimes L$ \cite[\S 3.5.4]{FL2a}.

For a small enough positive $\eps$, there is a smooth embedding \cite[\S
3.5.4]{FL2a} of the open tubular neighborhood $\tN_{\ft,\fs}(\eps)$,
\begin{equation}
\label{eq:BackgroundPreConfigEmbedding}
\tilde\bga_{\fs}:\tN_{\ft,\fs}(\eps)
\to
\tsC_{\ft},
\end{equation}
which is gauge equivariant with respect to the preceding action of
$\sG_{\fs}$, and covers the topological embedding
\eqref{eq:BackgroundConfigEmbedding}.  The map
\eqref{eq:BackgroundPreConfigEmbedding} is circle equivariant, where the
circle acts trivially on $\tM_{\fs}$, by scalar multiplication on the
fibers of $\tN_{\ft,\fs}(\eps)$, and by the action
\eqref{eq:DefineS1LAction} on $\sC_{\ft}$.
We note that the map \eqref{eq:BackgroundPreConfigEmbedding} is also circle
equivariant with respect to the action \eqref{eq:S1ZAction} on $\sC_{\ft}$,
if the circle acts on $\tN_{\ft,\fs}$ by
\begin{equation}
\label{eq:S1ZActionOnN}
(e^{i\theta},(B,\Psi,\beta,\psi))
\mapsto
\varrho(e^{i\theta})( B,\Psi,e^{2i\theta}\beta,e^{2i\theta}\psi)
=
(B,e^{i\theta}\Psi,\beta,e^{i\theta}\psi),
\end{equation}
where $(B,\Psi)\in \tM_{\fs}$,
$(\beta,\psi)\in L^2_k(\Lambda^1\otimes_\RR L) \oplus L^2(W^+\otimes L)$,
so $(B,\Psi,\beta,\psi)\in \tN_{\ft,\fs}$, and
$\varrho:\sG_{\fs}\to\sG_{\ft}$ is the homomorphism
\eqref{eq:GaugeGroupInclusion}. This equivariance follows from
the relation \eqref{eq:RelateS1Actions}
between the actions \eqref{eq:DefineS1LAction}
and \eqref{eq:S1ZAction}.

We note that the Chern character of the bundle $N_{\ft,\fs}$ is
computed in \cite[Theorem 3.29]{FL2a}
while the Segre classes of this bundle are computed, under
some additional assumptions, in \cite[Lemma 4.11]{FL2b}.



\subsection{Cohomology classes on the moduli space of SO(3) monopoles}
\label{subsec:Cohomology}
The identity \eqref{eq:MainEquation} arises as an equality of pairings of
suitable cohomology classes with a link in $\bar\sM_{\ft}^{*,0}/S^1$ of the
anti-self-dual moduli subspace and the links of the Seiberg-Witten moduli
subspaces in $\bar\sM_{\ft}/S^1$. We recall the definitions of these
cohomology classes and their dual geometric representatives given in
\cite[\S 3]{FL2b}.

The first kind of cohomology class is defined on $\sM^*_{\ft}/S^1$, via the
associated $\SO(3)$ bundle,
\begin{equation}
\label{eq:UniversalU2Bundle}
\FF_{\ft}
=
\tsC^{*}_\ft/S^1\times_{\sG_\ft}\fg_{V}
\to
\sC^{*}_\ft/S^1\times X.
\end{equation}
The group $\sG_{\ft}$ acts diagonally in \eqref{eq:UniversalU2Bundle},
with $\sG_{\ft}$ acting on the left on $\fg_{V}$. We define \cite[\S
3.1]{FL2b}
\begin{equation}
\label{eq:DefnMuClasses}
\mu_p: H_\bullet(X;\RR)\to H^{4-\bullet}(\sC^*_\ft/S^1;\RR),
\quad
\beta\mapsto -\tquarter p_1(\FF_{\ft})/\beta.
\end{equation}
On restriction to $M^w_{\ka}\hookrightarrow \sM_{\ft}$, the cohomology classes
$\mu_p(\beta)$ coincide with those used in the definition of Donaldson
invariants \cite[Lemma 3.1]{FL2b}. Define
\begin{equation}
\label{eq:DefineAAA}
\AAA(X) = \Sym\left( H_{\even}(X;\RR)\right)\otimes\La^\bullet(H_{\odd}(X;\RR))
\end{equation}
to be the graded algebra, with $z=\beta_1\beta_2\cdots\beta_r$ having total
degree $\deg(z) = \sum_p(4-i_p)$, when $\beta_p\in H_{i_p}(X;\RR)$.
Then $\mu_p$ extends in the usual way to
a homomorphism of graded real algebras,
$$
\mu_p:\AAA(X) \to H^{\bullet}(\sC^*_\ft/S^1;\RR),
$$
which preserves degrees. Next, we define a complex line bundle over
$\sC^{*,0}_{\ft}/S^1$,
\begin{equation}
\label{eq:DefineDetLineBundle}
\LL_{\ft}= \sC^{*,0}_{\ft}\times_{(S^1,\times -2)}\CC,
\end{equation}
where the circle action is given, for $[A,\Phi]\in\sC^{*,0}_{\ft}$
and $\zeta\in\CC$, by
\begin{equation}
\label{eq:DeterminantS1Action}
([A,\Phi],\zeta) \mapsto ([A,e^{i\theta}\Phi], e^{2i\theta}\zeta).
\end{equation}
Then we define the second kind of cohomology class on $\sM_{\ft}^{*,0}/S^1$ by
\begin{equation}
\label{eq:DefineMuC1}
\mu_c = c_1(\LL_{\ft}) \in H^2(\sC_{\ft}^{*,0}/S^1;\RR).
\end{equation}
For monomials $z\in\AAA(X)$, we constructed
\cite[\S 3.2]{FL2b} geometric representatives $\sV(z)$
dual to $\mu_p(z)$ (following the discussion in \cite{KMStructure})
and $\sW$ dual to $\mu_c$, defined on $\sM_{\ft}^*/S^1$ and
$\sM_{\ft}^{*,0}/S^1$, respectively; their closures in $\bar\sM_{\ft}/S^1$ are
denoted by $\bar\sV(z)$ and $\bar\sW$ \cite[Definition 3.14]{FL2b}. When
\begin{equation}
\label{eq:DimensionConditionPrelim}
\deg(z) +2\eta
=
\dim(\sM^{*,0}_{\ft}/S^1) -1,
\end{equation}
and $\deg(z)\ge \dim M^w_\ka$
it follows that \cite[\S 3.3]{FL2b} the intersection
\begin{equation}
\label{eq:GeomReprIntersection}
\bar\sV(z) \cap \bar\sW^{\eta} \cap \bar\sM^{*,0}_{\ft}/S^1,
\end{equation}
is an oriented one-manifold (not necessarily connected) whose closure in
$\bar\sM_{\ft}/S^1$ can only intersect $(\bar\sM_{\ft}-\sM_{\ft})/S^1$ at
points in $\sM_{\ft}^{\red}\cong\cup(M_{\fs}\times\Sym^\ell(X))$, where the
union is over $\ell\geq 0$ and $\fs\in\Spinc(X)$ \cite[Corollary 3.18]{FL2b}.



\subsection{Donaldson invariants}
\label{subsec:Donaldsonseries}
We first recall the definition \cite[\S 2]{KMStructure} of the Donaldson series
when $b_1(X)=0$ and $b_2^+(X)>1$ is odd, so that, in this case,
$\chi+\sigma\equiv 0\pmod{4}$. See also \S 3.4.2 in \cite{FL2b}, especially
for a definition of the Donaldson invariants when $b_2^+(X)=1$.
For any choice of $w\in H^{2}(X;\ZZ)$, the
Donaldson invariant is a linear function
$$
D^{w}_{X}:\AAA(X) \to \RR.
$$
Let $\tilde X=X\#\overline{\CC\PP}^2$ be the blow-up of $X$ and let $e\in
H_2(\tilde X;\ZZ)$ be the exceptional class, with Poincar\'e dual
$\PD[e]\in H^2(\tilde X;\ZZ)$.  If $z\in\AAA(X)$ is a monomial, we define
$D_X^w(z)=0$ unless
\begin{equation}
    \mathrm{deg}(z)\equiv -2w^{2}-\textstyle{\frac{3}{2}}(\chi+\sigma)\pmod{8}.
    \label{mod8}
\end{equation}
If $\deg(z)$ obeys equation \eqref{mod8}, we let
$\kappa\in\frac{1}{4}\ZZ$ be defined by
$$
\deg(z)=8\ka -\textstyle{\frac{3}{2}}\left(\chi+\si\right).
$$
There exists an $\SO(3)$ bundle over $\tilde X$ with first Pontrjagin
number $-4\ka -1$ and second Stiefel-Whitney class $w+\PD[e]\pmod{2}$.
One then defines the Donaldson invariant on monomials by
\begin{equation}
\label{eq:DefineDonaldson}
D^w_X(z)
=
\#\left( \bar\sV(ze)\cap\barM^{w+\PD[e]}_{\ka+1/4}(\tilde X)\right),
\end{equation}
and extends to a real linear function on $\AAA(X)$. Note that
$w+\PD[e]\pmod{2}$ is good in the sense of Definition
\ref{defn:Good}. If $w'\equiv w \pmod{2}$, then \cite{DonOrient}
\begin{equation}
\label{eq:DonaldsonsSignChange}
D^{w'}_{X}=(-1)^{\frac{1}{4}(w'-w)^{2}}D^{w}_{X}.
\end{equation}
The Donaldson series is a formal power series,
\begin{equation}
\label{eq:DefineDonaldsonSeries}
\bD^{w}_{X}(h) = D^{w}_{X}((1+\textstyle{\frac{1}{2}} x)e^{h}),
\quad h \in H_{2}(X;\RR).
\end{equation}
By equation~\eqref{mod8}, the series $\bD^{w}_{X}$ is even if
$$
-w^{2}-\tthreequarter(\chi+\sigma)\equiv  0 \pmod 2,
$$
and odd otherwise. A four-manifold has KM-simple type if for some $w$ and
all $z\in \AAA(X)$,
$$
D^{w}_{X}(x^{2}z)=4D^{w}_{X}(z).
$$
According to \cite[Theorem 1.7]{KMStructure}, when $X$ has KM-simple type
the series $\bD^{w}_{X}(h)$ is an analytic function of $h$ and there are
finitely many characteristic cohomology classes $K_{1},\ldots,K_{m}$ (the
KM-basic classes) and constants $a_{1},\ldots,a_{m}$ (independent of $w$)
so that
$$
\bD^{w}_X(h)
=
e^{\half h \cdot h}
\sum_{i=1}^{r}(-1)^{\half(w^{2}+K_{i}\cdot w)}a_{i}e^{\langle
K_{i},h\rangle}.
$$
Witten's conjecture, \cite{Witten}, then relates the Donaldson and
Seiberg-Witten series for four-manifolds of simple type.

When $b^+_2(X)=1$ the pairing on the right-hand side of definition
\eqref{eq:DefineDonaldson} depends on the chamber in the positive cone of
$H^2(\tilde X;\RR)$ determined by the period point of the Riemannian metric
on $\tilde X$, just as in the case of Seiberg-Witten invariants described in \S
\ref{subsubsec:SWInvariants}. We refer to \S 3.4.2 in \cite{FL2b} for a
detailed discussion of this case and, as in \cite{FL2b}, we assume that the
class $w\pmod{2}$ is good in order to avoid technical difficulties involved
in chamber specification.

\subsection{Links and the cobordism}
\label{subsec:LinkCobordism}
Since the endpoints of the components of the one-manifold
\eqref{eq:GeomReprIntersection} either lie in $M_\kappa^w$ or
$M_{\fs}\times\Sym^\ell(X)$, for some $\fs$ and $\ell(\ft,\fs)\geq 0$ for
which $\ft_{\ell} = \fs\oplus\fs\otimes L$, we have when $w_2(\ft)$ is good
in the sense of Definition \ref{defn:Good} that
\begin{equation}
\label{eq:RawCobordismSum}
\#\left(\bar\sV(z)\cap \bar\sW^{n_a-1}\cap \bar\bL^w_{\ft,\ka}\right)
=
-\sum_{\fs\in\Spinc(X)}
\# \left( \bar\sV(z)\cap \bar\sW^{n_a-1}\cap \bar\bL_{\ft,\fs}\right),
\end{equation}
where $\bar\bL^w_{\ft,\ka}$ is the link of $\bar M^w_\ka$
in $\bar\sM_{\ft}/S^1$ (see \cite[Definition 3.7]{FL2a})
and
where $\bar\bL_{\ft,\fs}$ is empty if $\ell(\ft,\fs)<0$
and is the boundary of an open neighborhood of the Seiberg-Witten stratum
$M_{\fs}\times\Sym^\ell(X)$ in $\bar\sM_{\ft}/S^1$
if $\ell=\ell(\ft,\fs)\ge 0$.  By construction, the intersection of
$\bar\bL_{\ft,\fs}$ with the top stratum of $\bar\sM_{\ft}/S^1$
is a smooth manifold and the intersection of $\bar\bL_{\ft,\fs}$
with the geometric representatives is in this top stratum.
The precise definition of $\bar\bL_{\ft,\fs}$ is given
in \cite[Definition 3.22]{FL2a} for $\ell=0$ and in Definition
\ref{defn:DefineLink} for $\ell\ge 1$.

When $\deg(z) = \dim M^w_{\ka}$, the intersection of the one-manifold
\eqref{eq:GeomReprIntersection} with the link $\bar\bL^w_{\ft,\ka}$ is given by
\cite[Lemma 3.30]{FL2b}
\begin{equation}
\label{eq:ASDPairing}
2^{1-n_a}\#\left( \bar\sV(z)\cap \bar\sW^{n_a-1}\cap \bar\bL^w_{\ft,\ka}\right)
=
\#\left(\bar\sV(z)\cap\barM^w_{\ka}\right).
\end{equation}
Applying this identity to the blow-up, $X\#\overline{\CC\PP}^2$, when
$n_a(\ft)>0$, we recover the Donaldson invariant $D_X^w(z)$ on the
right-hand side of
\eqref{eq:ASDPairing} via definition \eqref{eq:DefineDonaldson}.

%end of file
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%Article: INTERSECTION
%File: diagonal.tex
%Last update: February 8th, 2002 (Tom)
%Spellcheck:  March 4th, 2002 (Tom)
%Todo:

\section{Diagonals}
\label{sec:Diagonals}
In this section, we describe the stratification of the symmetric
product $\Sym^\ell(X)$ and the normal bundles of the strata.

\subsection{Definitions}
\label{subsec:DiagDefs}
In this section, we introduce some vocabulary for describing the
strata of the symmetric product $\Sym^{\ell}(X)$.

\subsubsection{Subgroups of the symmetric group}
Let $\fS_{\ell}$ denote the symmetric group on $\ell$ elements
which acts on $X^\ell$ by permutation.
We will write partitions of $N_\ell=\{1,\dots,\ell\}$
as $\sP=\{P_1,\dots,P_{r(\sP)}\}$
where $P_i\subset N_\ell$.
Each such partition of $N_\ell$ gives
a partition of $\ell$: $\ell=|P_1|+\dots |P_{r(\sP)}|$ which
we denote by $|\sP|$.
The symmetric group $\fS_{\ell}$ acts on the set of partitions of
$N_\ell$, by:
\begin{equation}
\label{eq:ActionOnPartitions}
\left( \si,\sP\right)
\to \si(\sP):=
\{\si(P_1),\dots,\si(P_{r(\sP)})\}.
\end{equation}
Let $\fS(\sP)<\fS_\ell$ be the stabilizer of a partition $\sP$
with respect to the action \eqref{eq:ActionOnPartitions}.
Note that $\fS(\sP)$ does not preserve the {\em ordering\/} of the sets
$P_1,\dots,P_{r(\sP)}$.
In addition,
let $\Gamma(\sP)$ be the subgroup of $\fS_{\ell}$ defined by the image
of the inclusion:
\begin{equation}
\label{eq:StabilizerInclusion}
\prod_{P\in\sP} \fS_{|P|} \to \fS_{\ell},
\end{equation}
given by $\fS_{|P|}$ acting trivially on the elements of $N_{\ell}$ not
in $P$.

\begin{lem}
\label{lem:CharacterizeGaP}
Let $\sP$ be a partition of $N_\ell$.
Then the subgroup
$\Ga(\sP)$ of $\fS_\ell$ defined by the image of the inclusion
\eqref{eq:StabilizerInclusion}
is the stabilizer of
$\sP$ under the action \eqref{eq:ActionOnPartitions}.
\end{lem}

\begin{proof}
Let $\Ga'(\sP)$ be the stabilizer of $\sP$ under the action
\eqref{eq:ActionOnPartitions}.
There is then an inclusion $\Ga(\sP)\to\Ga'(\sP)$.
This map is surjective because any $\si\in\fS_\ell$
which preserves the subsets $P_1,\dots,P_{r(\sP)}$
is a permutation of each of these subsets and
thus in $\Ga(\sP)$.
\end{proof}

\begin{lem}
\label{lem:RelateSymmetricGroups}
Let $\sP=\{P_1,\dots,P_{r(\sP)}\}$ be a partition of $N_\ell$.
The action \eqref{eq:ActionOnPartitions} defines an action
of $\Ga(\sP)$ on the set $\{P_1,\dots,P_{r(\sP)}\}$.
Let $\fS(|\sP|)$ be the subgroup of $\fS_{r(\sP)}$ which
preserves the partition $|P_1|,\dots,|P_{r(\sP)}|$.
Then $\Ga(\sP)$ is a normal subgroup of
$\fS(\sP)$ and there is an isomorphism:
$$
\fS(|\sP|)\simeq \fS(\sP)/\Ga(\sP).
$$
\end{lem}

\begin{proof}
Let $t: \fS(\sP)\to \fS_{r(\sP)}$ be the homomorphism defined
by the action of $\fS(\sP)$ on the set
$\{P_1,\dots,P_{r(\sP)} \}$.
The kernel of $t$ is the subgroup of elements preserving the sets
$P_1,\dots,P_{r(\sP)}$ which is $\Ga(\sP)$ by Lemma
\ref{lem:CharacterizeGaP}, implying
$\Ga(\sP)$ is a normal subgroup of $\fS(\sP)$.
Because each $\si\in\fS(\sP)$ is a bijection of $N_\ell$,
$t(\si)$ preserves the partition $|\sP|$.  Thus,
$t(\fS(\sP))\subset \fS(|\sP|)$.
To prove surjectivity, given
$\si\in\fS(|\sP|)$, construct $\tilde\si\in\fS(\sP)$ with
$t(\tilde\si)=\si$ as follows.
Define an order relation on $\{P_1,\dots,P_{r(\sP)}\}$
by comparing the smallest elements (with respect to
the natural order on $N_\ell$) of elements of $P_i$ and $P_j$.
If $a\in N_\ell$ is the $k$-th element of $P_i$
and $\si(i)=j$, define $\tilde\si(a)$ to be the
$k$-th element of $P_j$.  One then checks $\tilde\si\in\fS(\sP)$
and $t(\tilde \si)=\si$.
\end{proof}


\subsubsection{Defining the diagonals}
The strata of $\Sym^\ell(X)$ are defined by the quotients
of diagonals in $X^\ell$ by the action of
$\fS_\ell$ on $X^\ell$.  Explicitly, define:
\begin{equation}
\label{eq:DefineDelta}
\begin{aligned}
\Delta^\circ(X^\ell,\sP) & = \{ (x_1,\dots,x_{\ell}) \in X^\ell:
 x_i=x_j \
\text{if and only if there is $P\in\sP$ with $i,j\in P$ } \},
\\
\Delta(X^\ell,\sP) & = \{ (x_1,\dots,x_{\ell}) \in X^\ell:
 x_i=x_j \
\text{if there is $P\in\sP$ with $i,j\in P$ } \}.
\end{aligned}
\end{equation}
The preceding subspaces are related by
\begin{equation}
\label{eq:Closure}
\cl_{X^\ell}\Delta^\circ(X^\ell,\sP)
=
\Delta(X^\ell,\sP).
\end{equation}
We can identify $\Delta^\circ(X^\ell,\sP)$ with the complement of the
big diagonal in $X^{r(\sP)}$ as follows.
Write $\sP$ as an ordered collection $P_1,\dots,P_{r(\sP)}$.
Then define an embedding,
\begin{equation}
\label{eq:DefineDiagonalComplementEmbedding}
\iota_{\sP}:
X^r\less \
\cup_{i<j}\ \{x_i=x_j\}
\to \Delta^\circ(X^\ell,\sP)\subset X^\ell,
\end{equation}
by $\iota(x_1,\dots,x_r)=(y_1,\dots,y_\ell)$ where
$y_k=x_i$ for $k\in P_i$.  The map $\iota_{\sP}$ is
a diffeomorphism.
\begin{lem}
\label{lem:DiagonalStabilizers}
Let $\sP$ be a partition of $N_\ell$.
Then:
\begin{enumerate}
\item
The diagonal $\Delta(X^\ell,\sP)$ is the fixed point set of the group
$\Ga(\sP)$,
\item
If $\bx\in\Delta^\circ(X^\ell,\sP)$, then $\Stab_{\bx}=\Ga(\sP)$,
\item
If $\si\in\fS_\ell$ then $\si(\Delta^\circ(X^\ell,\sP))\subset \Delta^\circ(X^\ell,\sP)$
if and only if $\si\in\fS(\sP)$.
\end{enumerate}
\end{lem}

\begin{proof}
The fixed point set of $\fS_i$ acting on $X^i$ is the small diagonal.
Thus, by the definition of $\Ga(\sP)$,
if $\bx=(x_1,\dots,x_\ell)$ is a fixed point for every $\si\in\Ga(\sP)$,
we have $x_i=x_j$ for all $i,j\in P$ for every $P\in\sP$, implying
the first assertion.

The first assertion implies $\Ga(\sP)\subset\Stab_\bx$
for $\bx\in\Delta^\circ(X^\ell,\sP)$.
If $\si\in\Stab_\bx$ and $\si\notin\Ga(\sP)$, then there is
$P\in\sP$ and $i\in P$ such that $\si(i)\notin P$.
But $\si(\bx)=\bx$ then implies $x_{\si(i)}=x_i$ which
contradicts $\bx\in\Delta^\circ(X^\ell,\sP)$.
This implies the second assertion.

By definition, $\bx\in\Delta^\circ(X^\ell,\sP)$ if and
only if $\si(\bx)\in\Delta^\circ(X,\si(\sP))$.
\end{proof}

\subsubsection{Strata of the symmetric product}
Let $\pi_{\fS}:X^\ell\to\Sym^\ell(X)=X^\ell/\fS_\ell$ be the projection.
Define:
\begin{equation}
\label{eq:DefineStratumOfPartition}
\Si(X^\ell,\sP)= \pi_{\fS}\left(\Delta^\circ(X^\ell,\sP)\right).
\end{equation}
From item (3) of Lemma \ref{lem:DiagonalStabilizers},
we have the relation:
\begin{equation}
\label{eq:StratumQuotientOfDiagonal}
\Si(X^\ell,\sP) = \Delta^\circ(X^\ell,\sP)/\fS(\sP).
\end{equation}
The action of $\fS(\sP)$ on $\Delta^\circ(X^\ell,\sP)$ is not free
as noted in item (2) of Lemma \ref{lem:DiagonalStabilizers}.
From Lemma \ref{lem:RelateSymmetricGroups}, $\fS(|\sP|)$ is
the natural group acting on $\Delta^\circ(X^\ell,\sP)$.  We see
this action as follows.


\begin{lem}
Choose an ordering $P_1,\dots,P_{r(\sP)}$ of the partition
$\sP$ and use the ordering $|P_1|,\dots,|P_{r(\sP)}|$
of $|\sP|$.  Then the map $\iota_{\sP}$
given in \eqref{eq:DefineDiagonalComplementEmbedding}
is equivariant with respect
to the actions of $\fS(|\sP|)$ on $X^r$ and
$\fS(\sP)$ on $\Delta^\circ(X^\ell,\sP)$.
\end{lem}

The proof of the following relations between $\Si(X^\ell,\sP)$ and
$\Delta^\circ(X^\ell,\sP)$ is elementary.


\begin{lem}
\label{lem:PreImageOfSi}
Let $\sP$ be a partition of $N_\ell$ and let
$[\sP]$ denote the orbit of $\sP$ under $\fS_\ell$.
Then,
\begin{enumerate}
\item
If $\sP'=\si(\sP)$, then
$\Ga(\sP)=\si \Ga(\sP')\si^{-1}$,
\item
If $\sP'=\si(\sP)$, then
$\fS(\sP)=\si \fS(\sP')\si^{-1}$,
\item
$\pi_{\fS}^{-1}\left(\Si(X^\ell,\sP)\right)
=
\cup_{\sP'\in[\sP]} \Delta^\circ(X,\sP')$.
\end{enumerate}
\end{lem}




\subsection{Incidence relations}
\label{subsec:IncidenceRelations}
We now describe incidence relations among the diagonals of $X^\ell$
and the resulting relations among the strata
$\Si$ of $\Sym^\ell(X)$.

\begin{defn}
\label{defn:DefineRefinement}
Let $\sP$ and $\sP'$ be partitions of $N_\ell$.
We say $\sP'$ is a {\em refinement\/} of $\sP$, or $\sP<\sP'$,
if for every $P'\in\sP'$ there is $P\in\sP$ with $P'\subseteq P$.
\end{defn}

\begin{lem}
\label{lem:IncidenceRelation}
If $\sP$ and $\sP'$ are partitions of $N_{\ell}$
then $\Delta^\circ(X^\ell,\sP)\subset \Delta(X,\sP')$ if and only
$\sP'$ is a refinement of $\sP$.
\end{lem}

\begin{proof}
If $\sP'$ is a refinement of $\sP$ and $\bx\in\Delta^\circ(X^\ell,\sP)$,
then for every $P'\in\sP$ and $i,j\in P'$, there is $P\in\sP$
with $i,j\in P'\subset P$.  Since $i,j\in P\in\sP$, we have
$x_i=x_j$ so $\bx\in\Delta(X,\sP')$ proving one implication.

Conversely, if $\Delta^\circ(X^\ell,\sP)\subset \Delta(X,\sP')$
and $P'\in\sP'$, then because every $\bx=(x_1,\dots,x_\ell)
\in\Delta^\circ(X^\ell,\sP)$
is also in $\Delta(X,\sP')$ we see that for every $i,j\in P'$,
$x_i=x_j$.  This implies (by the only if in the definition of
$\Delta^\circ(X^\ell,\sP)$ that there must be $P\in\sP$ with $i,j\in P$.
Thus, $P'\subset P$ and $\sP<\sP$.
\end{proof}


Now, we compare the incidence relations with the strata
of $\Sym^\ell(X)$.  Recall that for a partition
$\sP$ of $N_\ell$, $[\sP]$ denoted the orbit of $\sP$
under the action \eqref{eq:ActionOnPartitions} of $\fS_\ell$
on the set of partitions of $N_\ell$.
For $\sP$ and $\sP'$ partitions of $N_\ell$, define
\begin{equation}
\label{eq:ConjugateRefinements}
[\sP<\sP']
=
\{\sP''\in [\sP']: \sP< \sP''\}.
\end{equation}
\begin{rmk}
Observe that $\fS(\sP)$ acts transitively on $[\sP<\sP']$
by the action \eqref{eq:ActionOnPartitions}.
\end{rmk}

The next lemma follows from the definition of
 $\Si(X^\ell,\sP)$ (see \eqref{eq:DefineStratumOfPartition})
as well as
Lemma \ref{lem:IncidenceRelation} and
item (3) in Lemma \ref{lem:PreImageOfSi}.


\begin{lem}
If $\sP$ and $\sP'$ are partitions of $N_{\ell}$
then
$$
\Si(X^\ell,\sP)\subset \cl_{\Sym^\ell(X)} \Si(\sP')
$$
if and only if
there is $\sP''\in [\sP]$ with $\sP'<\sP''$.
\end{lem}




\subsection{Normal bundles}
\label{subsec:DiagonalNormals}
We now introduce the normal bundles of the strata
$\Delta^\circ(X^\ell,\sP)$ of $X^{\ell}$.

\begin{lem}
Let $\sP$ be any partition of
$N_{\ell}$.
The tangent bundle of the submanifold $\Delta^\circ(X^\ell,\sP)$
is given by:
\begin{equation}
\label{eq:TangentToDiagonal}
T\Delta^\circ(X^\ell,\sP)=
\{(v_1,\dots,v_{\ell})\in TX^\ell:
v_i=v_j\ \text{if and only if there is $P\in\sP$ with $i,j\in P$}\}.
\end{equation}
If $g$ is any metric on $X$, then the orthogonal complement of
$T\Delta^\circ(X^\ell,\sP)$ with respect to the metric on $X^\ell$
given by the $\ell$-fold product of $g$
is the following normal bundle of $\Delta^\circ(X^\ell,\sP)$:
\begin{equation}
\label{eq:DiagonalNormal}
\tilde\nu(X^\ell,\sP):=
\{(v_1,\dots,v_{\ell})\in TX^\ell|_{\Delta^\circ(X^\ell,\sP)}:
\
\sum_{i\in P}v_i=0\
\text{for all $P\in\sP$}\}.
\end{equation}
\end{lem}

\begin{proof}
Differentiate any path in $\Delta^\circ(X^\ell,\sP)$ to see the form of
$T\Delta^\circ(X^\ell,\sP)$ appearing in equation \eqref{eq:TangentToDiagonal}.
For $\vec v=(v_1,\dots,v_\ell)\in T\Delta^\circ(X^\ell,\sP)$,
we then write $v_P$ for any $v_i$ with $i\in P$.
Then, if $\vec v=(v_1,\dots,v_\ell)\in T\Delta^\circ(X^\ell,\sP)$
and $\vec w\in \tilde\nu(X^\ell,\sP)$, the equality
$$
\vec v\cdot \vec w=\sum_i v_i\cdot w_i
=
\sum_{P\in\sP} v_P\cdot\left( \sum_{i\in P} w_i\right)
=0,
$$
which holds for any product metric on $X^\ell$, implies
that the bundle in equation \eqref{eq:DiagonalNormal}
is contained in the orthogonal complement of $T\Delta^\circ(X^\ell,\sP)$.
Since the above inclusion must hold for any value of $v_P$,
we see that any element of the orthogonal complement of
$T\Delta^\circ(X^\ell,\sP)$ must satisfy the equations defining
the bundle in \eqref{eq:DiagonalNormal}.
\end{proof}
There is an obvious  $\fS(\sP)$ action on the normal bundle
$\tilde\nu(X^\ell,\sP)$.  We will write $\nu(X^\ell,\sP)$ for
the quotient $\tilde\nu(X^\ell,\sP)/\fS(\sP)$.





\begin{lem}
\label{lem:Normal}
There is an open neighborhood,
$\tilde\sO(X^\ell,\sP)\subset \tilde\nu(X^\ell,\sP)$,
of the zero-section and an $\fS(\sP)$-equivariant
exponential map identifying
$\tilde\sO(X^\ell,\sP)$ with an open neighborhood, $\tilde\sU(X^\ell,\sP)$, of
$\Delta^\circ(X^\ell,\sP)$ in $X^\ell$.
\end{lem}

Because the exponential map is $\fS(\sP)$ equivariant,
equation \eqref{eq:StratumQuotientOfDiagonal} implies:

\begin{lem}
\label{lem:NormalInSymmProd}
Continue the notation of Lemma \ref{lem:Normal}.
If $\sO(X^\ell,\sP)=\tilde\sO(X^\ell,\sP)/\fS(\sP)$,
then a neighborhood,
$$
\sU(X^\ell,\sP)=\tilde\sU(X^\ell,\sP)
$$
of $\Si(X^\ell,\sP)$ in $\Sym^\ell(X)$
is homeomorphic to $\sO(X^\ell,\sP)$.
\end{lem}

We now identify the intersection of $\Delta^\circ(X,\sP')$
with $\tilde\sU(X^\ell,\sP)$.

\begin{lem}
\label{lem:DiagonalInDiagonalNormal}
If $\sP<\sP'$ are partitions of $N_{\ell}$
and
\begin{equation}
\label{eq:DoubleDiagonalNormal}
\begin{aligned}
{}&
\tilde\nu(X^\ell,\sP,\sP')
\\
&\quad
=
\{(v_1,\dots,v_\ell)\in\tilde\nu(X^\ell,\sP):
\text{$v_i=v_j$ if and only if there is $P'\in\sP'$ with $i,j\in P'$}
\}.
\end{aligned}
\end{equation}
then the intersection of $\Delta^\circ(\sP')$ with
$\tilde\sU(X^\ell,\sP)$ is homeomorphic to
$\tilde\nu(X^\ell,\sP,\sP')\cap\tilde\sO(X^\ell,\sP)$.
\end{lem}

\begin{proof}
This follows immediately from the equivariance of the exponential map.
\end{proof}

Thus, a neighborhood of $\Delta^\circ(X^\ell,\sP')$ near the
lower diagonal $\Delta^\circ(X^\ell,\sP)$ can be described
by the bundle $\tilde\nu(X^\ell,\sP,\sP')$.  However,
the end of the stratum $\Si(X^\ell,\sP')$ near the lower
stratum $\Si(X^\ell,\sP)$ can be more complicated than
a quotient of $\tilde\nu(X^\ell,\sP,\sP')$ by a subgroup of $\fS_{\ell}$.
Recall that the pre-image of $\Si(X^\ell,\sP)$ is not just
$\Delta^\circ(X^\ell,\sP)$ but rather all the diagonals given
by a partition in $[\sP]$, the orbit of $\sP$ under the action
\eqref{eq:ActionOnPartitions}.
Thus, while we can cover $\Si(X^\ell,\sP)$ by $\Delta^\circ(X^\ell,\sP)$,
to account for all the ends of $\Delta^\circ(X^\ell,\sP')$ which
get mapped to a neighborhood of the lower stratum, $\Si(X^\ell,\sP)$,
we must consider all diagonals $\Delta^\circ(X^\ell,\sP')$ where
$\sP''\in[\sP']$ and $\sP< \sP'$.  That is, we must consider all
partitions $\sP''\in [\sP<\sP']$, where $[\sP<\sP']$ is
defined in \eqref{eq:ConjugateRefinements}.
The next lemma follows from Lemmas \ref{lem:Normal}
and \ref{lem:DiagonalInDiagonalNormal}.

\begin{lem}
\label{lem:EndOfUpperStratum}
Let $\sP<\sP'$ be partitions of $N_{\ell}$.
Let $[\sP<\sP']$ be the set of partitions defined
in \eqref{eq:ConjugateRefinements}.
Then a neighborhood of $\Si(X^\ell,\sP)$ in $\Si(X^\ell,\sP')$ is
homeomorphic to a neighborhood of the zero-section in
\begin{equation}
\label{eq:EndOfUpperStratum}
\nu(X^\ell,\sP,[\sP'])=
\left(
\sqcup_{\sP''\in [\sP<\sP']}\tilde\nu(X^\ell,\sP,\sP')
\right)
/\fS(\sP).
\end{equation}
\end{lem}




%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Article: INTERSECTION
%File: diagonalTM.tex
%Last update: February 8th, 2002 (Tom)
%Spellcheck:  February 18th, 2002 (Tom)
%Todo:

\section{The Thom-Mather structure of the symmetric product}
\label{sec:diagTM}
In this section, we define a partial Thom-Mather structure on
the symmetric product.
In general, by a partial Thom-Mather structure on a stratified
space $Z$ with strata $\Si_i$, we mean neighborhoods
$U_i$ of $\Si_i$ with {\em tubular neighborhood projections\/} $\pi_i:U_i\to\Si_i$
and {\em tubular distance functions\/} $t_i:U_i\to [0,1)$ satisfying
$t_i^{-1}(0)=\Si_i$ and for $\Si_i\subset\cl_Z(\Si_j)$
the compatability conditions
\begin{equation}
\label{eq:TMProperties}
\begin{aligned}
\pi_i\circ\pi_j&= \pi_i,
\\
t_i\circ \pi_j&=t_i.
\end{aligned}
\end{equation}
If $t$ and $t_r$ are tubular distance functions of a
tubular neighborhood $\pi:U\to \Si$, then we say
$t_r$ is a {\em rescaling\/} of $t$ if there is a non-vanishing
function $f:\Si\to (0,1]$ such that
\begin{equation}
\label{eq:Rescale}
t_r(\cdot)=t(\cdot) f(\pi(\cdot)).
\end{equation}
Note that an actual Thom-Mather structure has a number of additional
requirements on the derivatives of the maps $\pi_i$ and $t_i$.
While it is well-known that
$\Sym^\ell(X)$, being the quotient of the smooth, compact
manifold $X^\ell$ by a finite group $\fS_\ell$, is a
Thom-Mather stratified space, the precise
construction we present here will be useful in following
sections.

We begin by defining subspaces of $\Sym^\ka(\RR^4)$ which
will appear as the fibers of the normal bundles
of \S \ref{subsec:DiagonalNormals}.  We construct partial
Thom-Mather structures on this subspace in \S \ref{subsec:R4Diags}.
To use the normal bundles of \S \ref{subsec:DiagonalNormals} to
construct Thom-Mather structures on $\Sym^\ell(X)$, we first
introduce some deformations of the metric in \S \ref{subsec:FamiliesOfMetric}.
We describe the overlap of the exponential maps from these normal
bundles in \S \ref{subsec:OverlapMaps}.
Finally, we produce the partial Thom-Mather structure on
$\Sym^\ell(X)$ in \S \ref{subsec:TMMapsOnSymX}.

\subsection{Diagonals in $\RR^4$}
\label{subsec:R4Diags}
The fiber of the normal bundles in \S \ref{subsec:DiagonalNormals}
are defined by the subspace of mass-centered points in
$\Sym^\ka(\RR^4)$.
Hence,
for any subset $P$ of $N_\ell$, define
$$
z_P: \oplus_{i\in P}\ \RR^4\to \RR^4,
\quad
z_P((v_i)_{i\in P})=\sum_i v_i.
$$
and
\begin{equation}
\label{eq:DefineZ}
Z_P=z_P^{-1}(0)\subset \oplus_{i\in P}\ \RR^4.
\end{equation}
We will write $z_\ell$ for $z_{N_\ell}$.  We define the
{\em cone point\/}
\begin{equation}
\label{eq:DefineR4DiagonalConePoint}
c_P\in Z_P
\end{equation}
to be the zero vector in $\oplus_{i\in P}\ \RR^4$.
The fiber of $ \tilde\nu(X^\ell,\sP)$ is then given by
(compare \eqref{eq:DiagonalNormal})
\begin{equation}
\label{eq:DiagonalNormalFiber}
\prod_{P\in\sP} Z_P.
\end{equation}
For $\sP<\sP'$ partitions of $N_\ell$, define a partition of each $P\in\sP$ by
\begin{equation}
\label{eq:DefineRefinementPartition}
\sP'_P=\{P'\in\sP': P'\subseteq P\}.
\end{equation}
If $P\in\sP\cap\sP'$, then $\sP'_P=\{P\}$.
If we define
\begin{equation}
\Delta^\circ(Z_P,\sP'_P)
=
\{(v_i)_{i\in P}\in Z_P:
\text{$v_i=v_j$ if
and only if there is $P'\in\sP'_P$ with $i,j\in P'$} \},
\label{eq:DefineCenteredR4Diagonal}
\end{equation}
then the fiber of the bundle $\tilde\nu(X^\ell,\sP,\sP')$
of equation \eqref{eq:DoubleDiagonalNormal} is given by:
\begin{equation}
\label{eq:DoubleDiagNormalFiber}
\prod_{P\in\sP} \Delta^\circ(Z_P,\sP'_P).
\end{equation}
Note that if $\sP'_P$ is given by one set, $P$,
as is the case if $P\in\sP\cap\sP'$, then
$\Delta^\circ(Z_P,\sP'_P)$ is a single point, given by $|P|$ copies
of the zero vector in $\RR^4$.
The following lemma describes the normal bundle
of $\Delta^\circ(Z_P,\sP'_P)\subset z_P^{-1}(0)$.

\begin{lem}
\label{lem:RelatingTrivialStrata}
Let $\sP'_P$ be a partition of $P$.
Then the restriction of the map
\begin{equation}
\label{eq:NormalOfR4Diagonal}
e(Z_P,\sP'_P):
\Delta^\circ(Z_P,\sP'_P) \times
\prod_{P'\in \sP'_P} Z_{P'}
\to
Z_P
\end{equation}
defined by
\begin{equation}
\label{eq:DefineR4DiagonalExponential}
\left(
(v_i)_{i\in P},(z_j)_{j\in P',P'\in\sP_P'}
\right)
\to (v_i+z_i)_{i\in P}
\end{equation}
to a neighborhood $\sO(Z_P,\sP'_P)$ of the subspace
$$
\Delta^\circ(Z_P,\sP'_P) \times \prod_{P'\in \sP'_P}  \{c_{P'}\}
$$
is a homeomorphism onto its image.
\end{lem}

\begin{proof}
We observe the map \eqref{eq:DefineR4DiagonalExponential}
can be inverted by the center of mass map
\begin{equation}
\label{eq:R4CenterOfMassProjection}
\begin{aligned}
(w_i)_{i\in P} \to & \left(
(v_i)_{i\in P},(z_i)_{i\in P',P'\in\sP_P'}
\right)
\\
v_{i}&=\frac{1}{|P'|}\sum_{j\in P'}w_j,
\quad
\text{for $i\in P'$},
\\
z_j&=w_j-v_j.
\end{aligned}
\end{equation}
(Note that the above map is an inverse because $\sum_{j\in P'} z_j=0$
by the definition of $Z_{P'}$.)  However, the map
\eqref{eq:R4CenterOfMassProjection} only takes values
in
$$
\Delta(Z_P,\sP'_P) \times \prod_{P'\in \sP'_P} Z_{P'}
$$
unless we restrict to a
sufficiently small neighborhood $\sO(Z_P,\sP'_P)$.
\end{proof}
Because $Z_P$ is a normal topological space, we can find open neighborhoods
$U(Z_P,\sP'_P)$ of $\Delta^\circ(Z_P,\sP'_P)$
for every partition $\sP'_P$ of $P$ such that
$U(Z_P,\sP'_P)\cap U(Z_P,\sP''_P)$ is empty unless $\sP'_P<\sP''_P$
or $\sP''_P<\sP'_P$.  The normal bundle structure given by the map
\eqref{eq:DefineR4DiagonalExponential} then defines a
tubular neighborhood projection,
\begin{equation}
\label{eq:R4DiagonalProjection}
\pi(Z_P,\sP'_P): U(Z_P,\sP'_P)\to \Delta^\circ(Z_P,\sP'_P),
\end{equation}
given by the $v_i$ component of the map
\eqref{eq:R4CenterOfMassProjection}:
\begin{equation}
\label{eq:DefineR4DiagonalProjection}
\begin{aligned}
{}&\pi(Z_P,\sP'_P)\left((w_i)_{i\in P} \right)
=
(v_i)_{i\in P},\\
{}&\quad
\quad\text{where for $i\in P'\in\sP'_P$, $v_i=\frac{1}{|P'|}\sum_{j\in P'}w_j$}.
\end{aligned}
\end{equation}
The following lemma shows that the maps $\pi(Z_P,\sP'_P)$ satisfy
the first equality of \eqref{eq:TMProperties}.


\begin{lem}
If $\sP'_P<\sP''_P$ are partitions of $P$, then on the intersection
$U(Z_P,\sP'_P)\cap U(Z_P,\sP''_P)$, the equality
\begin{equation}
\label{eq:R4DiagonalTM1}
\pi(Z_P,\sP'_P)\circ \pi(Z_P,\sP''_P) = \pi(Z_P,\sP'_P)
\end{equation}
holds.
\end{lem}

\begin{proof}
The lemma is immediate from the definition
\eqref{eq:DefineR4DiagonalProjection}.
\end{proof}

We use the above set of consistent tubular neighborhoods
to define a scale function which is constant
on the fibers of the maps $\pi(Z_P,\sP'_P)$.

\begin{lem}
\label{lem:R4TMScale}
There is a smooth
function $t(Z_P):Z_P\to [0,\infty)$
and open neighborhoods $U'(Z_P,\sP'_P)\Subset U(Z_p,\sP'_P)$
of the diagonals $\Delta^\circ(Z_P,\sP'_P)$ satisfying
\begin{enumerate}
\item
If $c_P\in Z_P$ is the cone point defined in
\eqref{eq:DefineR4DiagonalConePoint},
$t(Z_P)^{-1}(0)=c_P$,
\item
For every partition $\sP'_P$ of $P$,
$t(Z_P)\circ \pi(Z_P,\sP'_P)=t(Z_P)$,
on $U'(Z_P,\sP'_P)$
\item
The function is invariant with respect to the permutation
action of $\fS_{|P|}$ and the diagonal rotation action of
$\SO(4)$ on $Z_P$.
\end{enumerate}
\end{lem}

\begin{proof}
By induction on $|P|$, we can assume the functions $t(Z_{P'})$
are already defined if $|P'|<|P|$.
(For $|P|=1$, $Z_P=\{c_P\}$ and $t(Z_P)$ must then be the zero function.)
If
$$
\pi_{P'}:\Delta^\circ(Z_P,\sP'_P) \times
\prod_{P'\in \sP'_P} Z_{P'}
\to Z_{P'}
$$
is the projection map, then the map
$t(Z_P,\sP'_P):U(Z_P,\sP'_P)\to [0,\infty)$
defined by
$$
t(Z_P,\sP'_P)=
\left(\sum_{P'\in\sP'_P} |P'|(t(Z_{P'})\circ\pi_{P'})^2\right)
\circ e(Z_P,\sP'_P)^{-1},
$$
where $e(Z_P,\sP'_P)$ is defined in \eqref{eq:NormalOfR4Diagonal},
is a tubular distance function.
By rescaling $t(Z_P,\sP'_P)$ we can obtain a tubular
distance function $t_r(Z_P,\sP'_P):U(Z_P,\sP'_p)\to [0,\infty)$
satisfying
$t_r(Z_P,\sP'_P)^{-1}(0)=\Delta^\circ(Z_P,\sP'_P)$,
and
$t_r(Z_P,\sP'_P)^{-1}([0,\thalf))
=
U'(Z_P,\sP'_P)\Subset U(Z_P,\sP'_P)$.



Enumerate the partitions $\sP'_P$ of $P$ as $\sP_1,\dots,\sP_r$
such that if $\sP_i<\sP_j$, then $i<j$.
Let $\beta:[0,1]\to[0,1]$ be a smooth function with
$\beta([0,\tquarter])=1$ and $\beta([\thalf,1])=0$.
If we define $\beta_i=\beta\circ t_r(Z_P,\sP'_i)$,
then
$U'(Z_P,\sP_i) \subset \beta_i^{-1}(1)$,
and $\beta_i$ is compactly
supported on $\sU(Z_P,\sP_i)$.

We construct
functions $t_i:Z_P\to [0,\infty)$ using upwards induction on $i$.
Begin with the standard scale, $t_0((v_i)_{i\in P})=\sum_i |v_i|^2$.
Assume $t_{i-1}$ satisfies
\begin{equation}
\label{eq:InductiveAssumption1}
t_{i-1}\circ \pi(Z_P,\sP_j)=t_{i-1}\quad\text{for all $j<i$}
\end{equation}
and the invariance conditions in the conclusion of the lemma.
We construct $t_i$ by
\begin{equation}
\label{eq:InterpolateTubular}
t_i(\cdot)= \beta_i(\cdot)t_{i-1}\circ \pi(Z_P,\sP_i)(\cdot)
+
\left( 1-\beta_i(\cdot)\right)t_{i-1}(\cdot).
\end{equation}
Observe that by the inclusion $U'(Z_P,\sP_i) \subset \beta_i^{-1}(1)$,
$t_i=t_{i-1}\circ \pi(Z_P,\sP_i)$ on $U'(Z,\sP_i)$
and thus, because
$\pi(Z_P,\sP_i)\circ\pi(Z_P,\sP_i)=\pi(Z_P,\sP_i)$,
$t_i\circ\pi(Z_P,\sP_i)=t_i$ on $U'(Z,\sP_i)$ on $U'(Z_P,\sP_i)$.
By construction, $t_i$ is $\SO(4)$-invariant.
We make $t_i$ $\fS(\sP)$-invariant by defining all $t_{i'}$
for $\sP_{i'}$ conjugate to $\sP_i$ by the obvious conjugation.
Finally, we observe that for $j<i$, the equality
$$
\pi(Z_P,\sP_j)\circ\pi(Z_P,\sP_i)=\pi(Z_P,\sP_j)
$$
(see \eqref{eq:R4DiagonalTM1}) and the inductive assumption
\eqref{eq:InductiveAssumption1} imply that
on $U'(Z_P,\sP_j)$,
$$
t_{i-1}\circ\pi(Z_P,\sP_i)
=
t_{i-1}\circ \pi(Z_P,\sP_j)\circ\pi(Z_P,\sP_i)
=
t_{i-1}\circ\pi(Z_P,\sP_j)
=
t_{i-1}.
$$
The interpolation between $t_{i-1}$ and
$t_{i-1}\circ \pi(Z_P,\sP_i)$ in \eqref{eq:InterpolateTubular}
thus
leaves $t_i$ equal to $t_{i-1}$ on $U'(Z_P,\sP'_j)$.
Hence, for $j<i$,
$$
t_i\circ \pi(Z_P,\sP_j)|_{U'(Z_P,\sP_j)}
=
t_{i-1} \pi(Z_P,\sP_j)|_{U'(Z_P,\sP_j)}
=
t_{i-1}|_{U'(Z_P,\sP_j)},
$$
completing the induction.
\end{proof}


\subsection{Families of metrics}
\label{subsec:FamiliesOfMetric}
A metric $g$ on $X$ determines a product metric on $X^\ell$
and thus exponential maps of the normal bundles of the diagonal.
We will use a non-standard diffeomorphism of
the normal bundle $\tilde\nu(X^\ell,\sP)$ with a neighborhood
of $\Delta^\circ(X^\ell,\sP)$ in $X^\ell$.  This diffeomorphism
will be defined
by varying the metric on $X$ and thus the resulting product metric on $X^\ell$.
We begin by noting that it is possible to define an exponential
map with a varying metric.



\begin{lem}
If $g_{\sP}$ is a smooth family of metrics on $X$ parameterized
by $\bx\in\Delta^\circ(X^\ell,\sP)$, then there is an open
neighborhood $\tilde\sO(X^\ell,\sP,g_{\sP})\subset\tilde\nu(X^\ell,\sP)$ of the zero-section
and a smooth embedding,
\begin{equation}
\label{eq:VaryingMetricExponentialMap}
e(X^\ell,g_{\sP}):\tilde\sO(X^\ell,\sP,g_{\sP})\subset\tilde\nu(X^\ell,\sP)\to X^\ell,
\end{equation}
parameterizing an open neighborhood,
$\tilde\sU(X^\ell,\sP,g_{\sP})$, of $\Delta^\circ(X^\ell,\sP)$
in $X^\ell$ such that the restriction of the map
$e(X^\ell,g_{\sP})$ to a fiber $\tilde\sO(X^\ell,\sP,g_{\sP})|_\bx$ is
given by the exponential map defined by the $\ell$-fold product
of the metric $g_{\sP,\bx}$.
\end{lem}

\begin{proof}
For any metric $g'$ on $X$, the derivative of the exponential
map at a point in the zero-section is the identity.
Thus, the derivative of this exponential map with varying product metric
is also the identity and this map defines an embedding of a neighborhood
of the zero-section.
\end{proof}



The image of the exponential
map in \eqref{eq:VaryingMetricExponentialMap}
is a tubular neighborhood.
Such a tubular neighborhood, $\tilde\sU(X^\ell,\sP,g_{\sP})\subset X^\ell$, defines a
projection $\pi(X^\ell,g_{\sP}):\tilde\sU(X^\ell,\sP,g_{\sP})\to \Delta^\circ(X^\ell,\sP)$ by
the composition of  $e(X^\ell,g_{\sP})^{-1}$ and the projection
$\tilde\nu(X^\ell,\sP)\to\Delta^0(X,\sP)$.

For any smooth n-manifold $M$, let $\Fr_{\GL}(TM)$ denote the
$\GL(n)$-bundle of frames of tangent vectors.
Define $\Fr_{\Gl}(TX^\ell,\sP)$ as:
$$
\Fr_{\GL}(TX^\ell,\sP)=
\{(F_1,\dots,F_\ell)\in \Fr(TX^\ell)|_{\Delta^\circ(X^\ell,\sP)}:
F_i=F_j\ \text{for all $i,j\in P\in\sP$}\}.
$$
If $g_{\sP}$ is a family of metrics parameterized by
$\Delta^\circ(X^\ell,\sP)$, then we define
$\Fr(TX^\ell,\sP,g_{\sP})$ to be the subbundle of $\Fr_{\GL}(TX^\ell,\sP)$
of frames such that if
$(F_1,\dots,F_\ell)\in \Fr(TX^\ell,\sP,g_{\sP})|_{\bx}$ then
$F_i$ is orthonormal with respect to $g_{\sP,\bx}$.
The structure group of $\Fr(TX^\ell,\sP,g_{\sP})$ is
$$
G(T,\sP)=\SO(4)^{r(\sP)},
\quad\text{where $\sP=\{P_1,\dots,P_{r(\sP)}\}$}.
$$
Then we have the identity:
\begin{equation}
\label{eq:DiagonalNormalAssocBundle}
\tilde\nu(X^\ell,\sP)
=
\Fr(TX^\ell,\sP,g_{\sP})\times_{G(T,\sP)}
\prod_{P\in\sP}Z_P.
\end{equation}
Then Lemma \ref{lem:DiagonalInDiagonalNormal}
implies the following.

\begin{lem}
\label{lem:RestricedExponentialMapDomain}
Let $\sP<\sP'$ be partitions of $N_\ell$.
Then,
the restriction of the exponential map $e(X^\ell,g_{\sP})$ to
\begin{equation}
\label{eq:RestrictedExponentialMapDomain}
\tilde\sO(X^\ell,\sP,g_{\sP})
\cap
\left(
\Fr(TX^\ell,\sP,g_{\sP})\times_{G(T,\sP)}
\prod_{P\in\sP}
\Delta^\circ(Z_P,\sP'_P)
\right),
\end{equation}
takes values in $\Delta(X,\sP')$.
\end{lem}
Define:
$$
G(T,\sP'_P)=\prod_{P'\in\sP'_P}\SO(4).
$$
Then, we have:

\begin{lem}
\label{lem:RelateTXFiberBundles}
Let $\sP<\sP'$ be partitions of $N_\ell$.
Let $\tilde\sU(X,\sP,\sP')\subset\Delta^\circ(X^\ell,\sP')$ be the image
of the restriction of the exponential map in
\eqref{eq:RestrictedExponentialMapDomain}.
Let $g_{\sP}$ and $g_{\sP'}$ be smooth families of metrics parameterized
by $\Delta^\circ(X^\ell,\sP)$ and $\Delta^\circ(X,\sP')$ respectively.
Assume that $g_{\sP',\bx'}=g_{\sP,\bx}$ if $\pi(X^\ell,g_{\sP})(\bx')=\bx$.
Then,
\begin{equation}
\label{eq:TangentBundleOnEnd}
\Fr(TX^\ell,\sP',g_{\sP'})|_{\tilde\sU(X,\sP,\sP')}
\simeq
\Fr(TX^\ell,\sP,g_{\sP})
\times_{G(T,\sP)}
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_P,\sP'_P)
\times
G(T,\sP'_P)
\right).
\end{equation}
\end{lem}

\begin{proof}
By Lemma \ref{lem:RestricedExponentialMapDomain},
a neighborhood of $\Delta^\circ(X^\ell,\sP)$ in
$\Delta^\circ(X^\ell,\sP')$ is diffeomorphic to
a neighborhood of the zero-section in
\begin{equation}
\label{eq:NormalData}
\tilde\nu(X^\ell,\sP,\sP')
\cong
\Fr(TX^\ell,\sP,g_{\sP})
\times_{G(T,\sP)}
\prod_{P\in\sP}
\Delta^\circ(Z_P,\sP'_P)
\end{equation}
by the exponential map $e(X^\ell,g_{\sP})$.
The isomorphism \eqref{eq:TangentBundleOnEnd} is
then defined by the obvious parallel translation
of the frames
in $\Fr(TX^\ell,\sP,g_{\sP})$ to the point
in $\tilde\sU(X^\ell,\sP,\sP')$ given by the
data in \eqref{eq:NormalData}.
\end{proof}

\subsection{Overlap maps}
\label{subsec:OverlapMaps}
Let $\sP<\sP'$ be partitions of $N_\ell$.
To define tubular neighborhoods and projection maps
which satisfy the conditions \eqref{eq:TMProperties},
we define
a space to control the overlap of the tubular neighborhoods
$\sU(X,\sP,g_{\sP})$ and $\sU(X,\sP',g_{\sP'})$ as follows.
\begin{equation}
\label{eq:OverlapNormalBundle}
\tilde\nu(X^\ell,\sP,\sP',g_{\sP})=
\Fr(TX^\ell,\sP,g_{\sP})
\times_{G(T,\sP)}
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_P,\sP'_P)
\times
\prod_{P'\in\sP'_P} Z_{P'}
\right).
\end{equation}
We define an open subspace $\tilde\sO(X^\ell,\sP,\sP')$
of $\tilde\nu(X^\ell,\sP,\sP')$ and maps
$\rho^{X,u}_{\sP,\sP'}$ and $\rho^{X,d}_{\sP,\sP'}$
so that the diagram
\begin{equation}
\label{eq:OverlapEquation}
\begin{CD}
\tilde\sO(X^\ell,\sP,\sP',g_{\sP})
@> \rho^{X,u}_{\sP,\sP'} >>
\tilde\sO(X^\ell,\sP',g_{\sP'})
\\
@V \rho^{X,d}_{\sP,\sP'} VV
@V e(X^\ell,g_{\sP'}) VV
\\
\tilde\sO(X^\ell,\sP,g_{\sP})
@> e(X^\ell,g_{\sP'}) >>
X^\ell
\end{CD}
\end{equation}
commutes and controls the overlaps of the images of the exponential maps
$e(X^\ell,g_{\sP})$ and $e(X^\ell,g_{\sP'})$.

\subsubsection{The downwards overlap map}
The downwards overlap map,
\begin{equation}
\label{eq:DefineXlDownwardsOverlap}
\rho^{X,d}_{\sP,\sP'}:\tilde\sO(X^\ell,\sP,\sP',g_{\sP})
\to
\tilde\sO(X^\ell,\sP,g_{\sP})
\end{equation}
is given by the fiberwise inclusions of the map of
equation \eqref{eq:NormalOfR4Diagonal},
$$
\Delta^\circ(Z_P,\sP'_P)
\times
\prod_{P'\in\sP'_P} Z_{P'}
\to
Z_P.
$$
Because this map is $\SO(4)$ equivariant, it extends
to the domain given in \eqref{eq:DefineXlDownwardsOverlap}.
The open subspace $\tilde\sO(X^\ell,\sP,\sP',g_{\sP})$
must satisfy
\begin{equation}
\label{eq:XlOverlapCondition1}
\tilde\sO(X^\ell,\sP,\sP',g_{\sP})
\subseteq
(\rho^{X,d}_{\sP,\sP'})^{-1}
\left(
\tilde\sO(X^\ell,\sP,g_{\sP})
\right)
\end{equation}
for the composition in the diagram \eqref{eq:OverlapEquation}
to be defined.

\subsubsection{The upwards overlap map}
\label{subsubsec:XUpwardsOverlap}
We define the upwards overlap map,
\begin{equation}
\label{eq:DefineXlUpwardsOverlap}
\rho^{X,u}_{\sP,\sP'}:\tilde\nu(X^\ell,\sP,\sP',g_{\sP})
\to
\tilde\sO(X^\ell,\sP,g_{\sP})
\end{equation}
as follows.
Let
\begin{equation}
\label{eq:ProjectionToOverlapNormal}
\pi_1: \tilde\nu(X^\ell,\sP,\sP')
\to
\tilde\nu(X^\ell,\sP,\sP')
\end{equation}
be the projection implicit in
\eqref{eq:NormalData} and
\eqref{eq:OverlapNormalBundle}.
We assume the open set $\tilde\sO(X^\ell,\sP,\sP',g_{\sP})$
satisfies
\begin{equation}
\label{eq:XlOverlapCondition2}
\tilde\sO(X^\ell,\sP,\sP',g_{\sP})
\subseteq
\pi_1^{-1}\left(
\tilde\sO(X^\ell,\sP\to\sP')
\right).
\end{equation}
Then, the composition $e(X^\ell,g_{\sP})\circ \pi_1$
is defined on $\tilde\sO(X^\ell,\sP,\sP',g_{\sP})$
and takes values in $\Delta^\circ(X^\ell,\sP')$.

We now define the map $\rho^{X,u}_{\sP,\sP'}$.
We make the assumption on the families of metrics
$g_{\sP}$ and $g_{\sP'}$ appearing in
Lemma \ref{lem:RelateTXFiberBundles}.
Given a point in $\tilde\nu(X^\ell,\sP,\sP',g_{\sP})$,
lying over $\by=(y_P)\in \Delta^\circ(X^\ell,\sP)$,
define a point $\by'=(y_{P'})_{P'\in\sP'}\in\Delta^\circ(X^\ell,\sP')$
by the composition
$$
e(X^\ell,g_{\sP})\circ \pi_1:
\tilde\nu(X^\ell,\sP,\sP',g_{\sP})
\to
\Delta^\circ(X^\ell,\sP').
$$
Then, for $P'\in\sP'_P$ (where $P\in\sP$), define
$F_{P'}$ to be the
parallel translate of the frame $F_P\in \Fr(TX)|_{y_P}$
from $y_P$ to $y'_{P'}$ using the
Levi-Civita connection defined by the metric $g_{\sP,\by}$.
Leave the data in $Z_{P'}$ alone.  The result is an element of
$$
\Fr(TX^\ell,sP',g_{\sP'})|_{\by'}\times_{G(T,\sP')} \prod_{P'\in\sP'} Z_{P'}
=
\tilde\nu(X^\ell,\sP',g_{\sP'})|_{\by'},
$$
thus defining the map $\rho^{X,u}_{\sP,\sP'}$.

\subsubsection{Commuting overlaps maps}
We now define conditions on the metrics $g_{\sP}$
and $g_{\sP'}$ which ensure that the diagram
\eqref{eq:OverlapEquation} commutes.The diagram will only commute if we chose the
families of metrics $g_{\sP}$ and $g_{\sP'}$
in such a way as to eliminate holonomy.
To do this, we introduce the notion of a locally
flattened metric.


\begin{defn}
If $A\subset X^\ell$, then the {\em support\/} of $A$ in $X$ is
the union of the images of $A$ under the projection maps
$X^\ell\to X$.

Let $g_{\sP}$ be a smooth family of metrics on $X$ parameterized
by $\Delta^\circ(X^\ell,\sP)$.
Let $\tilde\sU'\subseteq\tilde\sU(X^\ell,\sP,g_{\sP})$ be a neighborhood
of $\Delta^\circ(X^\ell,\sP)$.
The family of metrics $g_{\sP}$ is {\em locally flat with respect to $\tilde\sU'$
at  $\bx\in\Delta^\circ(X^\ell,\sP)$\/}
if the metric $g_{\sP,\bx}$ is flat on
the support of $\tilde\sU'\cap\pi_{\sP}^{-1}(\bx)$ in $X$.
The family of metrics is
{\em locally flat with respect to $\tilde\sU'$\/} if this holds for
all $\bx\in \Delta^\circ(X^\ell,\sP)$.
\end{defn}

The following lemma shows that the diagram
\eqref{eq:OverlapEquation} commutes when the metrics are flat
in the sense of the preceding definition.

\begin{lem}
\label{lem:CommutingExponentialMaps}
Let $\sP<\sP'$ be partitions of $N_\ell$.
Assume that the smooth families of
metrics $g_{\sP}$ and $g_{\sP'}$
satisfy
\begin{enumerate}
\item
The metrics $g_{\sP}$ are flat with respect to a
neighborhood $\tilde\sU'\subseteq\tilde\sU(X^\ell,\sP,g_{\sP})$,
\item
For $\by'\in \tilde\sU(X^\ell,\sP,g_{\sP'})\cap\Delta^\circ(X^\ell,\sP')$
with $\pi(X^\ell,g_{\sP})(\by')=\by$, the metrics satisfy
$g_{\sP,\by}=g_{\sP',\by'}$.
\end{enumerate}
Then there is a neighborhood
$\tilde\sO'(X,\sP,\sP',g_{\sP})$ of
the zero-section of the bundle
\eqref{eq:OverlapNormalBundle} such that the diagram
\eqref{eq:OverlapEquation} commutes when
restricted to $\tilde\sO'(X,\sP,\sP',g_{\sP})$.
\end{lem}

\begin{proof}
We observe that for the compositions,
$$
e(X^\ell,g_{\sP})\circ\rho^{X,d}_{\sP,\sP'}
\quad\text{and}\quad
e(X^\ell,g_{\sP'})\circ\rho^{X,u}_{\sP,\sP'}
$$
to be defined, the neighborhood $\tilde\sO'(X,\sP,\sP',g_{\sP})$
must satisfy the inclusion relations
\eqref{eq:XlOverlapCondition1} and
\eqref{eq:XlOverlapCondition2}.
However, we replace the condition \eqref{eq:XlOverlapCondition1}
with the stronger requirement that
\begin{equation}
\label{eq:XOverlapAssumption}
\tilde\sO'(X,\sP,\sP',g_{\sP})
\subseteq
(e(X^\ell,g_{\sP})\circ\rho^{X,d}_{\sP,\sP'})^{-1}( \tilde\sU').
\end{equation}
On any open subset satisfying the above condition,
the composition
$$
e(X^\ell,g_{\sP})\circ\rho^{X,d}_{\sP,\sP'}
$$
is defined by the exponential map of the metric $g_{\sP,\by}$
of the vectors obtained by the appropriate adding of the
vectors in the fiber of $\tilde\nu(X^\ell,\sP,\sP',g_{\sP'})$.
The composition
$$
e(X^\ell,g_{\sP'})\circ\rho^{X,u}_{\sP,\sP'}
$$
is defined by first parallel translating the vectors
defined by the elements of $Z_{P'}$ from $\by$
to the nearby points in the support of $\by'$,
and then exponentiating.  Because, by \eqref{eq:XOverlapAssumption}
and the first assumption on the metrics $g_{\sP}$ and $g_{\sP'}$,
the $g_{\sP,\by}$ is flat where this parallel translation takes
place, the two compositions are equal.
\end{proof}



\subsection{Constructing the Thom-Mather maps}
\label{subsec:TMMapsOnSymX}
We now show how to construct the families of metrics
which satisfy the conditions of Lemma
\ref{lem:CommutingExponentialMaps} and thus
yield projection maps $\pi(X^\ell,g_{\sP})$ which satisfy
the first condition of \eqref{eq:TMProperties}.
The construction of tubular distance functions is
then quite straightforward.


\subsubsection{The projection maps}
The maps $\pi(X^\ell,g_{\sP})$ were defined in the previous section.
For $\sP<\sP'$ partitions of $N_\ell$, the first Thom-Mather
property
\begin{equation}
\label{eq:DiagonalsTM1}
\pi(X^\ell,g_{\sP})\circ\pi(X^\ell,g_{\sP'})=\pi(X^\ell,g_{\sP})
\end{equation}
will follow from the commutativity of the diagram
\eqref{eq:OverlapEquation}.  Explicitly:

\begin{lem}
\label{lem:DiagonalsTM1}
Continue the hypotheses and notation of Lemma \ref{lem:CommutingExponentialMaps}.
Then, restricted to $\tilde\sU'\cap\Delta^\circ(X^\ell,\sP')$,
the equality \eqref{eq:DiagonalsTM1} holds.
\end{lem}

\begin{proof}
The subspace $\tilde\sU'\cap\Delta^\circ(X^\ell,\sP')$
is in the image of the composition
$e(X^\ell,g_{\sP})\circ\rho^{X,d}_{\sP,\sP'}$.
The equality follows from the observation that the pullback of
$\pi(X^\ell,g_{\sP'})$ by $\rho^{X,d}_{\sP,\sP'}$ is given by
the projection $\pi_1$ of \eqref{eq:ProjectionToOverlapNormal}.
\end{proof}




Thus,
we must go about constructing families of metrics
$g_{\sP}$ for each partition of $N_\ell$ which
satisfy the conditions of Lemma \ref{lem:CommutingExponentialMaps}.
The first step is showing how
to flatten a family of metrics.

\begin{lem}
\label{lem:FlatteningMetrics}
If $g_{\sP}$ is a family of metrics smoothly parameterized by
$\Delta^\circ(X^\ell,\sP)$, then there is a family of smooth
metrics $F(g_{\sP})$ smoothly parameterized by $\Delta^\circ(X^\ell,\sP)$
such that
\begin{enumerate}
\item
$F(g_{\sP})$ is locally flat with respect to a tubular neighborhood
$\tilde\sU'\subseteq\tilde\sU(X^\ell,\sP,g_{\sP})$,
\item
For every $\bx\in \Delta^\circ(X^\ell,\sP)$, $F(g_{\sP})_{\bx}$ is equal to
$g_{\sP,\bx}$ on the
complement of the support of
$\tilde\sU(X^\ell,\sP,g_{\sP})\cap\pi(X^\ell,g_{\sP})^{-1}(\bx)$,
\item
$g_{\sP}$ and $F(g_{\sP})$ are $C^1$ close within support of
$\tilde\sU'$,
\item
The exponential maps $e(X^\ell,g_{\sP})$ and $e(X^\ell,F(g_{\sP}))$
are equal.
\item
If $g_{\sP}$ is already flat with respect to $\tilde\sU(X^\ell,\sP,g_{\sP})$
at $\bx\in \Delta^\circ(X^\ell,\sP)$, 
then $F(g_{\sP,\bx})=g_{\sP,\bx}$.
\end{enumerate}
\end{lem}

\begin{proof}
Let $\tilde\sU(X^\ell,\sP,g_{\sP})$ be the tubular neighborhood
of $\Delta^\circ(X^\ell,\sP)$
defined by the family of metrics $g_{\sP}$, so
$e(X^\ell,g_{\sP}):\tilde\sO(X^\ell,\sP,g_{\sP})\subset\tilde\nu(X^\ell,\sP)\simeq\tilde\sU(X^\ell,\sP,g_{\sP})$.
For $P\in\sP$, let $s_P:\Delta^\circ(X^\ell,\sP)\to[0,\infty)$ be
a smooth function such that
\begin{equation}
\label{eq:DefineSepFunctions}
\{(v_1,\dots,v_\ell)\in\tilde\nu(X^\ell,\sP)|_{\bx}: |v_i|\le 4s_P(\bx)
\quad\text{for all $i\in P$ and for all $P\in\sP$}\}
\subset
\sO(\sP)
\end{equation}
For $\bx\in\Delta^\circ(\sP)$, let $(F_P)_{P\in\sP}$ be frames in
$\Fr(TX^\ell,\sP,g_{\sP})$ lying over $\bx$.
Use the frame $F_P$ and
the exponential map of the metric $g_{\sP,\bx}$ to identify
$B(0,4 s_P(\bx))\subset \RR^4$ with
$B_{g_{\sP,\bx}}(x_P,4s_P(\bx))$. Let $\delta_{x_P}$ denote the
flat metric on $B_{g_{\sP,\bx}}(x_P,4s_P(\bx))$ given by the
pushforward of the Euclidean metric by the exponential map.  Let
$\chi:\RR\to \RR$ satisfy: $\chi(x)=0$ for $x\le \half$ and
$\chi(x)=1$ for $x\ge 1$. Then, we define $F(g)_{\sP,\bx}$ by:
\begin{equation}
\label{eq:FlatFamily1} F(g)_{\sP,\bx}=
  \begin{cases}
    g_{\sP,\bx} & \text{on $X\less \cup_i B_{g_{\sP,\bx}}(x_P,4s_P(\bx))$},\\
    \chi_{4s_P(\bx)}(|x|)g_{\sP,\bx} + (1-\chi_{4s_P(\bx)}(|x|))\delta_{x_P}
      &\text{on $\Omega_{g_{\sP,\bx}}(x_P,2s_P(\bx),4s_P(\bx)))$},\\
    \delta_{x_P} & \text{on $B_{g_{\sP,\bx}}(x_P,2s_P(\bx))$}
  \end{cases}
\end{equation}
where $x$ refers to the coordinates given by the $g_{\sP,\bx}$
exponential map.

The resulting family $F(g)_{\sP}$ is locally flat with respect
to the tubular neighborhood $\tilde\sU'$ defined by replacing
$4s_P(\bx)$ with $2s_P(\bx)$ in \eqref{eq:DefineSepFunctions}.

Because the metric $g_{\sP,\bx}$ has not been changed at $x_P$,
the frame bundles do not change.  If the metric $g_{\sP,\bx}$ is
flat on $B(x_P,4s_P(\bx))$, then it is given by the Euclidean
metric in exponential coordinates, by \cite[Vol. II, Ch. 7, Thm.
11, p. 306]{Spivak}.  Thus, $\delta_{x_P}=g_{\sP,\bx}$ and the
interpolation in \eqref{eq:FlatFamily1} does not change the
metric.
\end{proof}
We now construct the desired families of metrics.

\begin{lem}
\label{lem:FlatteningMetrics2}
For each partition $\sP$ of $N_\ell$,
there is a smooth family of metrics on $X$, $g_{\sP}$,
defining tubular neighborhoods $\tilde\sU(X^\ell,\sP,g_{\sP})$ by their
exponential maps, such that
\begin{enumerate}
\item
The family $g_{\sP}$ is locally flat with respect
to a tubular neighborhood
$\tilde\sU'(X^\ell,\sP,g_{\sP})\subset \tilde\sU(X^\ell,\sP,g_{\sP})$,
\item
If $\sP<\sP'$, then for all
$\bx'\in\Delta^\circ(X^\ell,\sP')\cap\tilde\sU'(X^\ell,\sP,g_{\sP})$
$g_{\sP',\bx'}=g_{\sP,\bx}$ where $\bx=\pi(X^\ell,g_{\sP})(\bx')$.
\end{enumerate}
\end{lem}

\begin{proof}
The proof is by induction on the partitions $\sP$.

Fix a metric $g$ on $X$.  To every partition $\sP$ of
$N_\ell$, define an initial family of metrics $g_{\sP}$ by
$g_{\sP,\bx}=g$.

Let $\sP_0$ be the crudest partition.
Redefine $g_{\sP_0}$ by applying
the flattening procedure of Lemma \ref{lem:FlatteningMetrics}.
Then the redefined family of metrics $g_{\sP_0}$ is locally
flat with respect to the tubular neighborhood $\tilde\sU'(X^\ell,\sP_0,g_{\sP_0})$
given in Lemma \ref{lem:FlatteningMetrics}.

Assume families of metrics $g_{\sP}$ and tubular
neighborhoods $\tilde\sU'(X^\ell,\sP,g_{\sP})$ satisfying the
conclusions of the Lemma have been defined
for all $\sP<\sP'$.  There are two families of metrics
on $X$ parameterized by the intersection
\begin{equation}
\Delta^\circ(X^\ell,\sP')\cap \sU(X^\ell,\sP,g_{\sP}).
\end{equation}
There is the initial family $g_{\sP'}$ and the family
$\pi(X^\ell,g_{\sP})^*g_{\sP}$ which is defined by
$$
\pi(X^\ell,g_{\sP})^*g_{\sP,\bx'}=g_{\sP,\bx}
\quad\text{where}\quad
\pi(X^\ell,g_{\sP})(\bx')=\bx.
$$
Observe that, by shrinking the tubular neighborhoods
if necessary, for $\sP_1<\sP'$ and $\sP_2<\sP'$ if
the intersection
\begin{equation}
\label{eq:TripleOverlap}
\Delta^\circ(X^\ell,\sP')\cap \tilde\sU(X^\ell,\sP_1,g_{\sP_1})
\cap \tilde\sU(X^\ell,\sP_2,g_{\sP_2})
\end{equation}
is non-empty, then there is a relation $\sP_1<\sP_2<\sP'$.
Then, the relations
$$
\pi(X^\ell,\sP_1)\circ \pi(X^\ell,\sP_2)
=
\pi(X^\ell,\sP_1)
$$
(which follows by induction and Lemma \ref{lem:DiagonalsTM1})
and
$$
g_{\sP_2}=\pi(X^\ell,\sP_1)^*g_{\sP_1}
$$
imply that the families of metrics
\begin{equation}
\label{eq:PullbackMetrics}
\pi(X^\ell,g_{\sP_1})^*g_{\sP_1}
\quad
\text{and}
\quad
\pi(X^\ell,g_{\sP_1})^*g_{\sP_1}
\end{equation}
are equal on \eqref{eq:TripleOverlap}.  Thus we can form
a new family of metrics by interpolating between $g_{\sP'}$ on
$$
\Delta^\circ(X,\sP') - \cup_{\sP<\sP'}\tilde\sU'(X^\ell,\sP,g_{\sP}),
$$
and the pullback metrics \eqref{eq:PullbackMetrics} .
By further shrinking the tubular neighborhoods
$\tilde\sU'(X^\ell,\sP,g_{\sP})$ for
$\sP<\sP'$ and taking a sufficiently small neighborhood
$\tilde\sU'(X^\ell,\sP',g_{\sP'})$
we can assume this interpolated family is locally flat with respect
to $\tilde\sU'(X^\ell,\sP',g_{\sP'})$ on
\begin{equation}
\label{eq:NghOfLowerDiagonals}
\Delta^\circ(X^\ell,\sP')\cap\left(\cup_{\sP<\sP'}\tilde\sU(X^\ell,\sP,g_{\sP})\right).
\end{equation}
Then, apply the flattening procedure of Lemma \ref{lem:FlatteningMetrics}
to this interpolated family.  Because this family is already flat with
respect to $\tilde\sU'(X^\ell,\sP',g_{\sP'})$ for $\bx$ in
the open space \eqref{eq:NghOfLowerDiagonals}, the flattening procedure
does not change the interpolated family there.  Thus, the new, flattened
family satisfies both properties of the conclusion of the lemma,
completing the induction.
\end{proof}

\subsubsection{The tubular distance function}
Given the family of metrics constructed in the previous section,
the tubular distance functions $t(\sP)$ are defined by the composition
of a bundle norm on $\tilde\nu(X^\ell,\sP)$ with the inverse of the exponential map
$e(X^\ell,g_{\sP})^{-1}$.  These do not satisfy the
second Thom-Mather identity, \eqref{eq:TMProperties}, immediately.


We define a new tubular distance function $t(X^\ell,\sP)$
and a rescaling of it $t_r(X^\ell,\sP)$ on a
possibly smaller tubular neighborhoods.  These new functions will
satisfy \eqref{eq:TMProperties}.


\begin{lem}
\label{lem:XTMTubularDistance}
For $\sP$ a partition of $N_\ell$, let
$g_{\sP}$ be the smooth family of metrics parameterized
by $\Delta^\circ(X^\ell,\sP)$
and let $\tilde\sU'(X^\ell,\sP,g_{\sP})$ be the tubular
neighborhood of $\Delta^\circ(X^\ell,\sP)$
constructed in
Lemma \ref{lem:FlatteningMetrics2}.
Then there are tubular neighborhoods
$\tilde\sU''(X^\ell,\sP,g_{\sP})$ and
$\fS(\sP)$-invariant
smooth functions
\begin{equation}
\label{eq:XTubularDistFunc}
t(X^\ell,\sP):\tilde\sU''(X^\ell,\sP,g_{\sP})\to [0,1],
\quad\text{and}\quad
t_r(X^\ell,\sP):\tilde\sU''(X^\ell,\sP,g_{\sP})\to [0,1],
\end{equation}
such that
\begin{enumerate}
\item
$t_r(X^\ell,\sP)$ is a rescaling of $t(X^\ell,\sP)$,
\item
$t(X^\ell,\sP)^{-1}(0)=t_r(X^\ell,\sP)^{-1}(0)=\Delta^\circ(X^\ell,\sP)$,
\item
$t_r(X^\ell,\sP)^{-1}([0,\thalf)) \Subset\tilde\sU''(X^\ell,\sP,g_{\sP})$
\item
For all partitions $\sP<\sP'$, the equalities
\begin{equation}
\label{eq:XTMTubularDistance}
\begin{aligned}
t(X^\ell,\sP)\circ \pi(X^\ell,\sP',g_{\sP'})&=t(X^\ell,\sP),
\\
t_r(X^\ell,\sP)\circ \pi(X^\ell,\sP',g_{\sP'})&=t_r(X^\ell,\sP),
\end{aligned}
\end{equation}
hold on $\sU''(X^\ell,g_{\sP})\cap \sU''(X^\ell,\sP',g_{\sP'})$.
\end{enumerate}
\end{lem}

\begin{proof}
Because the functions $t(Z_P):Z_P\to [0,\infty)$
defined in Lemma \ref{lem:R4TMScale} are invariant
under the $\SO(4)$ action on $Z_P$, the function
$\sum_{P\in\sP} |P|t(Z_P)^2$ is defined on
$$
\Fr(TX^\ell,\sP,g_{\sP}\times_{G(T,\sP)}\prod_{P\in\sP}Z_P,
$$
and hence
\begin{equation}
\label{eq:DefineXTubDistance1}
t(X^\ell,\sP)=\left( \sum_{P\in\sP} |P|t(Z_P)^2\right)
\circ e(X^\ell,g_{\sP})^{-1}
\end{equation}
is defined on $\tilde\sU'(X^\ell,\sP,g_{\sP})$ and satisfies
item (2) of the lemma.
By definition, $t(X^\ell,\sP)$ is invariant under the action of
$\fS(\sP)$.
By the construction of the tubular
neighborhoods $\tilde\sU'(X^\ell,\sP,g_{\sP})$ in
Lemma \ref{lem:FlatteningMetrics2},  for $\sP<\sP'$ the
intersection
$\tilde\sU'(X^\ell,\sP,g_{\sP})\cap \tilde\sU'(X^\ell,\sP',g_{\sP'})$
is contained in the image of the neighborhood
$\tilde\sO(X^\ell,\sP,\sP',g_{\sP})$ under the maps in the
diagram \eqref{eq:OverlapEquation}.
Under the identification of
$\tilde\sU'(X^\ell,\sP,g_{\sP})\cap \tilde\sU'(X^\ell,\sP',g_{\sP'})$
with a subspace of
$$
\Fr(TX^\ell,\sP,g_{\sP})
\times_{G(T,\sP)}
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_P,\sP'_P)
\times
\prod_{P'\in\sP'_P} Z_{P'}
\right)
$$
(see \eqref{eq:OverlapNormalBundle}), the projection map
$\pi(X^\ell,\sP',g_{\sP'})$ is identified with
the projection
$$
\begin{CD}
\Fr(TX^\ell,\sP,g_{\sP})
\times_{G(T,\sP)}
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_P,\sP'_P)
\times
\prod_{P'\in\sP'_P} Z_{P'}
\right)
\\
@VVV
\\
\Fr(TX^\ell,\sP,g_{\sP})
\times_{G(T,\sP)}
\prod_{P\in\sP}
\Delta^\circ(Z_P,\sP'_P)
\end{CD}
$$
By item (2) in Lemma \ref{lem:R4TMScale},
by the definition of the map
$\rho^{X,d}_{\sP,\sP'}$, and by the definition
of $t(X^\ell,\sP)$ in \eqref{eq:DefineXTubDistance1}
as a sum of the functions $t(Z_P)$,
the function $t(X^\ell,\sP)$  satisfies
\eqref{eq:XTMTubularDistance}.

We define $t_r(X^\ell,\sP)$ to be a rescaling of
$t(X^\ell,\sP)$.  The function $f:\Delta^\circ(X^\ell,\sP)$
used to rescale $t(X^\ell,\sP)$ is chosen to ensure
item (3) holds.  Then, the equality
$$
t_r(X^\ell,\sP)(\cdot)
=
t(X^\ell,\sP)(\cdot) f(\pi(X^\ell,\sP,g_{\sP})(\cdot)),
$$
and the equality
$$
f\circ \pi(X^\ell,\sP,g_{\sP})
=
f\circ \pi(X^\ell,\sP,g_{\sP})\circ\pi(X^\ell,\sP',g_{\sP'})
$$
(which follows from \eqref{eq:DiagonalsTM1}),
and the fact that \eqref{eq:XTMTubularDistance} holds
for $t(X^\ell,\sP)$
ensure
that $t_r(X^\ell,\sP)$ also satisfies
\eqref{eq:XTMTubularDistance}.
\end{proof}


\begin{rmk}
All the constructions of this section can be made $\fS_\ell$
equivariant and thus descend to the symmetric product
$\Sym^\ell(X)$.
\end{rmk}

%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Article: INTERSECTION
%File: splicedends.tex
%Last update: February 13th, 2002 (Tom)
%Spellcheck:  February 18th, 2002 (Tom)
%Todo:


\section{The instanton moduli space with spliced ends}
\label{sec:SplicedEnd}
The first step in constructing the space of global splicing data
is to define the moduli space with spliced ends.
This moduli space will be identical
to $\barM^{s,\natural}_{\ka}(\delta)$, the Uhlenbeck compactification of
the mass-centered, charge $\ka$, framed instantons on $\RR^4$
with scale less than $\delta$,
except on an Uhlenbeck neighborhood of the punctured, centered symmetric product,
\begin{equation}
\label{eq:PuncturedTrivialStrata}
[\Theta]\times \left(\Sym^{\ka,\natural}_\delta(\RR^4)\less c_{\ka}\right)
\end{equation}
where $\Theta$ is the trivial connection,
$\Sym^{\ka,\natural}_\delta(\RR^4)$ is the subspace of
mass-centered points in
the $\ka$-th symmetric product of $\RR^4$ with a
scale constraint (see \eqref{eq:SingularPointScale}),
and $c_\ka\in \Sym^{\ka}(\RR^4)$ is the point totally
concentrated at the origin.

A neighborhood of the strata \eqref{eq:PuncturedTrivialStrata} in
$\barM^{s,\natural}_\ka(\delta)$ is parameterized by the union of the images
of the gluing maps $\bga_{\Theta,\Si}$, as $\Si$ varies over the strata
in \eqref{eq:PuncturedTrivialStrata}.  Roughly speaking,
if the stratum $\Si$ of $\Sym^\ka(\RR^4)$
is given by the partition $\ka=\ka_1+\dots +\ka_r$,
and $\sP$ is a partition of $N_\ka$ corresponding to $\Si$
then the gluing map is an smoothly stratified embedding
$$
\bga_{\Theta,\sP}: \Si \times \prod_{i=1}^r \bar M^{s,\natural}_{\ka_i}(\delta_i)
\to \barM^{s,\natural}_{\ka}(\delta).
$$
The scale parameters on the moduli spaces $\bar M^{s,\natural}_{\ka_i}(\delta)$
make the image of $\bga_{\Theta,\Si}$ into a kind of tubular neighborhood
of $\{[\Theta]\}\times\Si$ in $\barM^{s,\natural}_\ka(\delta)$.
The moduli space with spliced ends will be defined, roughly
speaking, by replacing the
image of $\bga_{\Theta,\Si}$ with the image of the splicing map
$\bga'_{\Theta,\Si}$ on the interior of this tubular neighborhood.
We summarize the properties of the spliced-ends moduli space in
the following theorem.


\begin{thm}
\label{thm:ExistenceOfSplicedEndsModuli}
Let $\bar\sB^s_\ka(\delta)$ be the Uhlenbeck extension of the
framed quotient space defined in
\eqref{eq:ExtendedFramedQuotientSpace}.
There exists a
smoothly-stratified moduli space with spliced ends,
$\barM^{s,\natural}_{spl,\ka}(\delta)\subset\bar\sB^s_\ka(2\delta)$,
closed under the
$\SO(3)\times\SO(4)$ action, satisfying the following properties
\begin{enumerate}
\item
There is a smoothly-stratified, $\SO(3)\times\SO(4)$-equivariant,
homeomorphism,
$\barM^{s,\natural}_{spl,\ka}(\delta)\cong \barM^{s,\natural}_{\ka}(\delta)$,
which is the identity on
the trivial strata,
$$
[\Theta]\times \Sym^{\ka,\natural}_\delta(\RR^4),
$$
where $\Sym^{\ka,\natural}_\delta(\RR^4)\subset\Sym^\ka(\RR^4)$ is
defined in \eqref{eq:SingularPointScale}.
\item
There is an Uhlenbeck neighborhood $W_{\ka}$ of the punctured
centered symmetric product, \eqref{eq:PuncturedTrivialStrata},
in $\bar{\sB}^s_{\ka}(2\delta)$ such that
$$
\barM^{s,\natural}_{spl,\ka}(\delta)\less W_{\ka}
 =
\barM^{s,\natural}_{\ka}(\delta) \less W_{\ka}.
$$
\item
To each stratum $\{[\Theta]\}\times\Si$ in the punctured,
centered symmetric product, \eqref{eq:PuncturedTrivialStrata}, there is
an Uhlenbeck neighborhood $W(\Si)\subset\bar{\sB}^s_{\ka}(\delta)$ of the
trivial stratum
\begin{equation}
\label{eq:TrivialStratumP}
[\Theta]\times \Si,
\end{equation}
and an open neighborhood $\sO^{\asd}_1(\Theta,\sP,\delta)\subset
\nu(\Theta,\sP,\delta)$
of the trivial stratum \eqref{eq:DefineASDSplicingDomain}
in the domain of the splicing map $\bga'_{\Theta,\sP}$
defined in \eqref{eq:R4TrivialSplicing},
such that
\begin{equation}
\label{eq:EqualsSplicingImage}
\barM^{s,\natural}_{spl,\ka}(\delta)\cap
W(\Si) = \bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right).
\end{equation}
\item
For all $[A,\bx]\in \barM^{s,\natural}_{spl,\ka}$,
$\|F^+_{A}\|_{L^{\sharp,2}}\le M$ where $M$ is the constant appearing
in \cite[Proposition 7.6]{FLKM1}.
\end{enumerate}
\end{thm}

We construct $\barM^{s,\natural}_{spl,\ka}(\delta)$ using
induction on $\ka$.  We first define the
{\em spliced end\/} of the moduli space.  The spliced end
will be an Uhlenbeck neighborhood of the punctured, centered symmetric product in
$\barM^{s,\natural}_{spl,\ka}(\delta)$.
It is defined as the union of the images of splicing maps
$\bga'_{\Theta,\Si}$ where $\Si$ is
a stratum in $\Sym^{\ka,\natural}_\delta(\RR^4)-\{c_{\ka}\}$
and where the domain of the splicing map is not defined by
the lower charge moduli spaces $\barM^{s,\natural}_{\ka_i}(\delta_i)$
but rather by the lower charge {\em spliced ends moduli spaces\/}
$\barM^{s,\natural}_{\ka_i}(\delta_i)(\delta_i)$.
A technical result on the equality of two splicing maps,
Proposition \ref{prop:CommutingSplicingSymmQuotient}
controls the overlaps of the images of the different splicing maps,
proving that this union is a smoothly stratified space.

In \S \ref{subsec:R4Diagonals} we introduce some
vocabulary to describe the trivial strata.
In \S \ref{subsec:SplicingMapsR4}, we define the relevant
splicing maps.
In \S \ref{subsec:OverlapSpliceMapsR4}, we prove the
technical result Proposition \ref{prop:CommutingSplicingSymmQuotient}
controlling the overlap of two splicing maps.
Then in \S \ref{subsec:SplicedEnd}, we finish the first stage of
the proof of Theorem \ref{thm:ExistenceOfSplicedEndsModuli}
by constructing the spliced end in
Proposition \ref{prop:ExistenceOfSplicedEnd}.

The second stage of the proof of Theorem
\ref{thm:ExistenceOfSplicedEndsModuli} appears in
\S \ref{subsec:Collar}.  There, we show how to isotope
the complement of a neighborhood of the punctured, centered symmetric
product in the spliced end to the actual moduli space
$\barM^{s,\natural}_{\ka}(\delta)$.
This isotopy is defined by the composition of the isotopy defining the gluing
map and the isotopy defined by the centering map.  That such an isotopy exists
follows immediately from the estimates on
$\|F^+_{A}\|_{L^{\sharp,2}}$ for $A$ in the spliced end; most of the
work in \S \ref{subsec:Collar} lies in constructing the parameter
on the spliced end for the isotopy.  The existence of this parameter
is established in \S \ref{subsec:TMOfSplicedEnd} by constructing a partial
Thom-Mather structure on the spliced end.



\subsection{The trivial strata}
\label{subsec:R4Diagonals}
We begin with some vocabulary,
mostly adapted from \S \ref{sec:Diagonals} to
describe the trivial strata \eqref{eq:PuncturedTrivialStrata}.

The symmetric product, $\Sym^{\ka,\natural}(\RR^4)$ is the quotient
of the zero locus of the center of mass map $z_\ka:
\oplus_{i\in N_{\ka}}\RR^4\to\RR^4$
(defined following \eqref{eq:DefineZ}) by the symmetric group:
$$
\Sym^{\ka,\natural}(\RR^4)=z_\ka^{-1}(0)/\fS_\ka.
$$
We shall abbreviate $z_{\ka}^{-1}(0)$ by $Z_\ka$.
We define a scale for elements of $\Sym^\ka(\RR^4)$ by
\begin{equation}
\label{eq:SingularPointScale}
\la([v_1,\dots,v_\ka])^2= \sum_i |v_i|^2,
\end{equation}
Then, we define $Z_\ka(\delta)=z_{\ka}^{-1}(0)\cap \la^{-1}([0,\delta))$
and
\begin{equation}
\Sym^{\ka,\natural}_\delta(\RR^4)
=
Z_\ka(\delta)/\fS_\ka.
\end{equation}
We describe the strata of $\Sym^{\ka,\natural}_\delta(\RR^4)$
by the quotients of the diagonals of $Z_\ka(\delta)$.
For any partition $\sP$ of $N_\ka$,
define
\begin{equation}
\begin{aligned}
\Delta^\circ(Z_\ka(\delta),\sP)&=
\Delta^\circ(Z_\ka,\sP)\cap Z_{\ka}(\delta),
\\
\Si(Z_\ka(\delta),\sP)&=\Delta^\circ(Z_\ka(\delta),\sP)/\fS(\sP)
\subset\Sym^{\ka,\natural}_\delta(\RR^4)
\end{aligned}
\end{equation}
where $\Delta^\circ(Z_\ka,\sP)$ is defined in
\eqref{eq:DefineCenteredR4Diagonal}.
Let $\Si(Z_\ka(\delta),\sP)$ be the image of $\Delta^\circ(Z_\ka(\delta),\sP)$
under the projection $Z_{\ka}(\delta)\to\Sym^{\ka,\natural}_\delta(\RR^4)$.
We then have:

\begin{lem}
\label{lem:DiagonalsInZ}
If $\sP$ and $\sP'$ are partitions of $N_{\ka}$ and
$\pi_{\ka}: Z_{\ka}\subset
\oplus_{i\in N_{\ka}}\RR^4\to\Sym^{\ka,\natural}(\RR^4)$ is
the projection, then:
\begin{enumerate}
\item
$\Si(Z_\ka(\delta),\sP) = \Delta^\circ(Z_\ka(\delta),\sP)/\fS(\sP)$,
\item
$\pi_{\ka}^{-1}\left( \Si(Z_\ka(\delta),\sP)\right)
=
\sqcup_{\sP'\in[\sP]} \Delta^\circ(Z_\ka(\delta),\sP')$.
\item
$\Si(Z_\ka(\delta),\sP) \subset \cl\left( \Si(Z_{\ka}(\delta),\sP')\right)$
if and only if there is $\sP''\in [\sP]$ such that $\sP''<\sP$.
\end{enumerate}
\end{lem}

If $(v_1,\dots,v_{\ka})\in
\Delta^\circ(Z_{\ka},\sP)$ and $P\in\sP$, then $v_i=v_j$ for $i,j\in
P$.  Thus, we will write $v_P:=v_i$ for any $i\in P$.  With this
notation, we will write $(y_1,\dots,y_\ka)=(y_P)_{P\in\sP}$ for an
element of $\Delta^\circ(Z_{\ka},\sP)$.


\subsubsection{Tubular neighborhoods}
\label{subsubsec:TubNgh}


A tubular neighborhood of a diagonal in
$Z_\ka(\delta)$ can be obtained using
the exponential map with respect to the product of the
flat metrics on $\oplus_{i\in N_{\ka}}\RR^4$:

\begin{lem}
\label{lem:R4DiagonalsNormal}
The normal bundle of the diagonal $\Delta^\circ(Z_\ka(\delta),\sP)$
in $Z_{\ka}$ is
\begin{equation}
\label{eq:R4DiagonalNormal}
\begin{aligned}
\nu(Z_{\ka}(\delta),\sP)
&=
\{((x_1,\dots,x_\ka),
(v_1,\dots,v_{\ka}))\in
\Delta^\circ(Z_\ka(\delta),\sP)\times \oplus_{i\in N_{\ka}}\RR^4:
\\
&\qquad\text{ for all $P\in\sP$ \ $\sum_{i\in P} v_i=0$}\}.
\end{aligned}
\end{equation}
There is an open neighborhood, $\tilde\sO(Z_{\ka}(\delta),\sP)$,
of the zero section in $\nu(Z_{\ka}(\delta),\sP)$
such that the restriction of
the exponential map $e(Z_{\ka},\sP)$ defined by
\begin{equation}
\label{eq:R4Exponential}
e(Z_{\ka},\sP)\left(
(x_1,\dots,x_\ka),(v_1,\dots,v_{\ka})
\right)
=
( x_1+v_1,\dots,x_\ka+v_\ka)
\end{equation}
to $\tilde\sO(Z_{\ka}(\delta),\sP)$ is injective and $\fS(\sP)$ equivariant.
\end{lem}

Let $\tilde\sU(Z_{\ka}(\delta),\sP)$ be the image of
$\tilde\sO(Z_{\ka}(\delta),\sP)$ under the exponential map
\eqref{eq:R4Exponential}.
By the $\fS(\sP)$ equivariance of the exponential map
$e(Z_{\ka}(\delta),\sP)$, there is a homeomorphism,
\begin{equation}
\label{eq:R4DiagonalTubNghSymmQuotient}
\sO(Z_{\ka}(\delta),\sP)\cong \sU(Z_{\ka}(\delta),\sP)
\end{equation}
where $\sO(Z_{\ka}(\delta),\sP)=\tilde\sO(Z_{\ka}(\delta),\sP)/\fS(\sP)$
and $\sU(Z_{\ka}(\delta),\sP)=\tilde\sU(Z_{\ka}(\delta),\sP)/\fS(\sP)$.



To describe the pre-image of $\Delta^\circ(Z_\ka(\delta),\sP')$
under $e(Z_{\ka},\sP)$ where $\sP<\sP'$, we introduce the
following notation.  First, for $P\subseteq N_\ka$
let $Z_P\subset \oplus_{i\in P}\RR^4$ be the mass-centered
subspace defined in \eqref{eq:DefineZ}.


\begin{lem}
\label{lem:R4DiagonalExpPreImage}
Let $\sP<\sP'$ be partitions of $N_\ka$.
For $P\in\sP$, let $\sP'_P$ be the partition of $P$
defined in \eqref{eq:DefineRefinementPartition}.
Then there is an inclusion
\begin{equation}
\label{eq:DefineR4NormalPP'}
\nu(Z_{\ka}(\delta),\sP,\sP')
=
\Delta^\circ(Z_\ka(\delta),\sP)
\times
\prod_{P\in\sP} \Delta^\circ(Z_P,\sP'_P)
\to
\nu(Z_{\ka}(\delta),\sP)
\end{equation}
such that if we define
\begin{equation}
\label{eq:DefineTubNghR4Normal}
\tilde\sO(Z_{\ka}(\delta),\sP,\sP')
=
\tilde\sO(Z_{\ka}(\delta)(\delta),\sP)
\cap
\nu(Z_{\ka}(\delta),\sP,\sP'),
\end{equation}
then $\tilde\sO(Z_{\ka}(\delta),\sP,\sP')=
e(Z_{\ka},\sP)^{-1}(\Delta^\circ(Z_\ka(\delta),\sP'))$.
\end{lem}

\begin{proof}
We observe that
$$
\Delta^\circ(Z_P,\sP'_P)
=\{(v_i)_{i\in P}\in Z_P: v_i=v_j\quad
\text{if and only if there is $P'\in\sP'_P$ with $i,j\in P'$}\}.
$$
Because $\sP$ is a partition of $N_{\ka}$, there
is an isomorphism
$$
\oplus_{P\in\sP}\oplus_{i\in P}\RR^4\simeq
\oplus_{i\in N_{\ka}}\RR^4,
$$
which induces an inclusion
\begin{equation}
\label{eq:NormalBundleInclusion}
\prod_{P\in\sP}\Delta^\circ(Z_P,\sP'_P)
\to Z_{\ka}
\quad
((v_i)_{i\in P,P\in\sP}) \mapsto (v_i)_{i\in N_{\ka}}.
\end{equation}
By the $\fS(\sP)$ equivariance of the exponential map
$e(Z_{\ka},\sP)$, an element of the pre-image of
$\Delta^\circ(Z_\ka(\delta),\sP')$ under $e(Z_{\ka},\sP)$
can be written as
$$
((x_1,\dots,x_{\ka}),(v_1,\dots,v_{\ka}))\in
\Delta^\circ(Z_\ka,\sP)\times
\oplus_{i\in N_{\ka}}\RR^4
$$
where $(v_1,\dots,v_{\ka})\in \oplus_{i\in N_{\ka}}\RR^4$
satisfies
\begin{enumerate}
\item
$\sum_i v_i=0$ for all $P\in\sP$,
\item
For all $i,j$, $v_i=v_j$ if and only if there is
$P'\in\sP'$ with $i,j\in P'$.
\end{enumerate}
Then one observes that the elements
$(v_1,\dots,v_{\ka})\in \oplus_{i\in N_{\ka}}\RR^4$
satisfying these two conditions are precisely the
image of the inclusion \eqref{eq:NormalBundleInclusion}.
\end{proof}

By the second and third items of Lemma \ref{lem:DiagonalsInZ},
to describe the end of  $\Si(Z_{\ka}(\delta),\sP')$ near
$\Si(Z_\ka(\delta),\sP)$, we must describe not only the
end of $\Delta^\circ(Z_\ka(\delta),\sP')$ near
$\Delta^\circ(Z_\ka(\delta),\sP)$ as is done in Lemma
\ref{lem:R4DiagonalExpPreImage}, but also the ends of
$\Delta^\circ(Z_\ka(\delta),\sP'')$ near
$\Delta^\circ(Z_\ka(\delta),\sP)$ for all $\sP''\in [\sP']$
with $\sP<\sP''$.

Thus, define
\begin{equation}
\label{eq:R4DiagonalOverlap}
\nu(Z_{\ka}(\delta),[\sP<\sP'])
=
\Delta^\circ(Z_\ka(\delta),\sP)
\times
\coprod_{\sP''\in [\sP<\sP']}
\left(
\prod_{P\in\sP}
\Delta^\circ(Z_P,\sP''_P)
\right).
\end{equation}
There is an action of $\fS(\sP)$ on $\nu(Z_{\ka},[\sP<\sP'])$
by the standard action on $\Delta^\circ(Z_\ka,\sP)$
and the action of $\fS(\sP)$ on $[\sP<\sP']$.
Although $\Delta^\circ(Z_\ka(\delta),\sP)$
is not a subset of $\nu(Z_{\ka}(\delta),[\sP<\sP'])$,
we use the phrase {\em a neighborhood of $\Delta^\circ(Z_\ka(\delta),\sP)$
in $\nu(Z_{\ka}(\delta),[\sP<\sP'])$\/} to refer to the obvious parallel to
subspaces of $\nu(Z_{\ka}(\delta),\sP)$.

\begin{lem}
\label{lem:SymmProdNeigh}
Let $\sP$ and $\sP'$ be partitions of $N_{\ka}$ with
$\sP<\sP'$.  Then a neighborhood of $\Si(Z_\ka(\delta),\sP)$ in
$\Si(Z_{\ka}(\delta),\sP')$ is homeomorphic to a neighborhood
of the zero section in
$$
\nu(Z_{\ka}(\delta),[\sP<\sP'])
/\fS(\sP).
$$
Denote this neighborhood by $\sO(Z_\ka(\delta),[\sP<\sP'])$.
\end{lem}

\begin{proof}
The bundle $\nu(Z_{\ka}(\delta),[\sP<\sP'])$ is the union
of the bundles $\nu(Z_{\ka}(\delta),\sP,\sP'')$ defined
in \eqref{eq:DefineR4NormalPP'} as $\sP''$ varies in
$[\sP<\sP']$ along the subspace $\Delta^\circ(Z_\ka(\delta),\sP)$
of each of these bundles.
Thus a neighborhood of
$\Delta^\circ(Z_\ka(\delta),\sP)$ in
\begin{equation}
\label{eq:UnionOfDiags}
\cup_{\sP''\in [\sP<\sP']} \Delta^\circ(Z_\ka(\delta),\sP'')
\end{equation}
is homeomorphic to a neighborhood of the zero-section in
$\nu(Z_{\ka}(\delta),[\sP<\sP'])$.  By the characterization of
the pre-image of $\Si(Z_\ka(\delta),\sP)$ under the projection
$Z_{\ka}(\delta)\to\Sym^{\natural,\ka}_\delta(\RR^4)$ in Item (2) of Lemma
\ref{lem:DiagonalsInZ}, a neighborhood of
$\Si^\natural(Z_\ka(\delta),\sP)$ in $\Si^\natural(Z_\ka(\delta),\sP')$
is homeomorphic to the quotient of \eqref{eq:UnionOfDiags}
by $\fS(\sP)$, proving the lemma.
\end{proof}


\subsection{Splicing to the trivial on $\RR^4$}
\label{subsec:SplicingMapsR4}
In this section, we define the
basic operation of splicing as used in \cite{FL3}.  The main
result is in Lemma \ref{lem:CommutingSplicing} where we show that
the composition of two splicing maps is equal to a single
splicing map under suitably small open sets.

Let $\sB_\ka$ denote the quotient space of $\SU(2)$ connections
on a bundle $E_\ka\to S^4$ with $c_2(E_\ka)=\ka$.
Let
$$
\bar\sB_{|P|}
=
\cup_{i=0}^p \ \left( \sB_{|P|-i}\times \Sym^i(\RR^4)\right),
$$
denote the space of ideal connections on $S^4$, given the topology
induced by Uhlenbeck convergence.  We write elements of this
space as $[A,\bx]$ where $[A]\in \sB_{|P|-\ell}$ and
$\bx\in \Sym^\ell(\RR^4)$.
We define the {\em center of mass\/} $z([A,\bx])$,
for $\bx=(x_1,\dots,x_\ell)$ by:
\begin{equation}
\label{eq:CenterOfMass}
z\left( [A,\bx]\right)
= \frac{1}{8\pi^2|P|} \int_{\RR^4} y |F_A|^2 d^4y
+ \sum_{i=1}^\ell x_i
\end{equation}
and the {\em scale\/}, $\la([A,\bx])$ by
\begin{equation}
\label{eq:ConnectionScale}
\la\left( [A,\bx]\right)^2
=
\frac{1}{8\pi^2} \int_{\RR^4} | y -z([A,F^s])|^2|F_A|^2 d^4y
+
\sum_{i=1}^\ell |x_i|^2.
\end{equation}
Define
$$
\bar\sB_{|P|}(\eps)= \la^{-1}([0,\eps]).
$$
If $[A,\bx]\in \bar\sB_{|P|}(\eps)$ and $\eps<<1$, then the support
of $\bx$ is disjoint from the south pole.  Thus, we can extend the
framed quotient space $\sB^s_{|P|}(\eps)\to \sB_{|P|}(\eps)$ over the quotient
space of ideal connections:
\begin{equation}
\label{eq:ExtendedFramedQuotientSpace}
\bar\sB^s_{|P|}(\eps)
=
\la^{-1}([0,\eps])
\cap
\left(
\sqcup_{i=0}^p \ \left( \sB^s_{|P|-i}\times \Sym^i(\RR^4)\right)\right).
\end{equation}
We now define the operation of splicing connections to
the trivial connection.
Let $\sP$ be a partition of $N_{\ka}$. We define the splicing map
on an open subspace of
$$
\Delta^\circ(Z_\ka(\delta),\sP) \times_{\fS(\sP)}
\prod_{P\in\sP}\bar{\sB}^{s,\natural}_{|P|}(\delta_P),
$$
where the constants $\delta_P$ are invariant under the action of
$\fS(\sP)$ and
where $\fS(\sP)$ acts on the generalized connections by permuting
them, sending and element of $\bar{\sB}^{s,\natural}_{|P|}(\delta_P)$
to an element of $\bar{\sB}^{s,\natural}_{|\si(P)|}(\delta_{|\si(P)|})$.
Note that elements of $\Delta^\circ(Z_\ka(\delta),\sP)$
can be written as $(y_P)_{P\in\sP}$, where
$y_P\in\RR^4$ while elements of
$\prod_{P\in\sP}\bar{\sB}^{s,\natural}_{|P|}(\delta_P)$ can be written
as $([A_P,F^s_P,\bx_P])_{P\in\sP}$, where
$[A_P,F^s_P,\bx_P]\in \bar{\sB}^{s,\natural}_{|P|}(\delta_P)$.
The open subspace on which the splicing map will be defined is
\begin{align}
\notag \tilde\sO(\Theta,\sP,\delta) &:=
\{ \left( y_P,[A_P,F^s_P,\bx_P]\right)_{P\in\sP}\in
\Delta^\circ(Z_\ka(\delta),\sP) \times \prod_{P\in\sP}\bar{\sB}^s_{|P|}(\delta_P) :
\\
\label{eq:DefineSplicingDataR4} & \quad \text{for all $P\neq P'$,
$8\sqrt{\la([A_P,\bx_P])}+8\sqrt{\la([A_{P'},\bx_{P'})} < \dist(x_P,x_{P'})$} \}.
\end{align}
Then,  we define
\begin{equation}
\label{eq:R4TrivialSplicing}
\bga'_{\Theta,\sP}:\sO(\RR^4,\sP)
=\tilde\sO(\RR^4,\sP)/\fS(\sP)\to \bar{\sB}^s_{\ka}(2\delta),
\end{equation}
as follows.

For $P\in\sP$, let $E_P\to S^4$ be the bundle supporting the connection
$A_P$.
The framed connection $[A_P,F^s_P]$ defines a section of
$\Fr(E_P)$, $\phi(A_P,F^s_P)$, over the complement of the north pole
by parallel translation along great circles from the south pole. Let
$\Theta(A_P,F^s_P)$ be the flat connection defined by this
section.  Note that if $A_P$ is flat on the domain of
$\phi(A_P,F^s_P)$, then $\Theta(A_P,F^s_P)=A_P$.

If $\Theta$ denotes the trivial connection on $\RR^4\times
\SO(3)$, then the flat connection $\Theta(A_P,F^s_P)$ is
canonically identified with the restriction of $\Theta$ to the
domain of the $\phi(A_P,F^s_P)$ as follows. The section
$\phi(A_P,F^s_P)$ gives a trivialization of the bundle $E$ and
thus identifies it with the restriction of the trivial bundle
$\RR^4\times\SO(3)$ to the domain of the section. Under this
identification the connection $\Theta$ is identified with
$\Theta(A_P,F^s_P)$.

We also note that given any two connections, $A_1$ and $A_2$, a
convex linear combination of them, $tA_1 + (1-t)A_2$ also defines
a connection.  For example, if $A_1$ is a connection which is
trivial in a particular trivialization and $A_2=A_1+a_2$, we
would write the connection one form for $tA_1 + (1-t)A_2$ in this
trivialization as $(1-t)a_2$.

Let $c_{y,\la}:\RR^4\to \RR^4$ be defined by
$c_{\la,y}(z)=(z-y)/\la$.  Thus, $c_{y,\la}$ maps the ball
$B(y,\la)$ to $B(0,1)$.

Let $\chi:\RR\to [0,2)$ be a smooth function satisfying
$\chi(x)=0$ for $x\le \half$ and $\chi(x)=1$ for $x\ge 1$. Define
$\chi_{y,\la}:\RR^4\to [0,2)$ by
$\chi_{y,\la}(x)=\chi(|x-y|/\la)$.

Then, for
$\by=(y_P)_{P\in\sP}\in \Delta^\circ(Z_\ka(\delta),\sP)$,
$\la_P=\la([A_P,F^s_P,\bx_P])$, we define:
\begin{equation}
\label{eq:SplicedGenConn}
\bga'_{\Theta,\sP}
\left(
\bx,([A_P,F^s_P,\bx_P])_{P\in\sP}
\right)
=
[A',F^s_P,\bx']
\end{equation}
where the framed connection $[A',F^s_P]$ is defined by
\begin{align}
\label{eq:SplicedConnection}
A'
&=
\begin{cases}
\Theta & \text{on $\RR^4\less \cup_{P\in\sP}B(y_P,4\sqrt{\la_P})$}
 \\
(1-\chi_{y_P,4\sqrt{\la_P}})c_{y_P,1}^*A_P +
\chi_{y_P,4\sqrt{\la_P}}c_{y_P,1}^*\Theta(A_P,F^s_P) &\text{on
$\Omega(y_P,2\sqrt{\la_P},4\sqrt{\la_P}))$}
\\
c_{y_P,1}^*A_P & \text{on $B(y_P,2\sqrt{\la_P})$}.
\end{cases}
\end{align}
and the frame in $[A',F^s_P]$ is given by the
canonical frame for the trivial connection $\Theta$.

The point  $\bx'$ in the expression
\eqref{eq:SplicedGenConn} is defined by translating
the singular support (and multiplicity) of the point
$\bx_P$ to $c_{y_P}(\bx_P)$.

Note that we have not computed the behavior of the function
$\la\circ\bga'_{\Theta,\sP}$ precisely so we cannot assert
that the image of $\bga'_{\Theta,\sP}$ is contained in
$\sB^s_{\ka}(2\delta)$.  However, that containment will follow
from the continuity of $\la$ and of $\bga'_{\Theta,\sP}$
with respect to Uhlenbeck limits if we shrink the domain
$\sO(\Theta,\sP,\delta)$ by requiring the scales $\la_P$ to
be sufficiently small.


We note the following result on the behavior of this
splicing map at the cone point of $\bar\sB^s_{|P|}(\delta_P)$.

\begin{lem}
\label{lem:UhlebeckLimitCont}
For any partition $\sP$ of $N_{\ka}$,
the map $\bga'_{\Theta,\sP}$ is continuous with
respect to Uhlenbeck limits.
If $c_{|P|}\in\Sym^{|P|,\natural}_\delta(\RR^4)$
denotes the cone point, then for any
$(y_P)_{P\in\sP}\in\Delta^\circ(Z_\ka(\delta),\sP)$,
\begin{equation}
\label{eq:SplicingConePoint}
\bga'_{\Theta,\sP}
\left(
(y_P)_{P\in\sP},([\Theta,c_{|P|})_{P\in\sP})
\right)
=
[\Theta,(y_p)_{P\in\sP}]
\end{equation}
\end{lem}


It is important that our constructions in \S
\ref{subsec:SplicedEnd} be equivariant with respect to the
$\SO(3)\times\SO(4)$ actions on the space of connections so that
those constructions can be used in the space of gluing data which
is defined with this action. Thus, we prove the following result
on the equivariance of the splicing map.

\begin{lem}
\label{lem:GroupEquiv}
Let $\sP$ be a partition of $N_{\ka}$.
Let
$\SO(3)\times\SO(4)$ act on
$$
\sO(\Theta,\sP,\delta) \subset \Delta^\circ(Z_\ka,\sP)
\times_{\fS(\sP)}\prod_{P\in\sP}\bar{\sB}^s_{|P|}(\delta_P),
$$
by, for $A\in\SO(3)$ and $R\in \SO(4)$,
\begin{equation}
\label{eq:SO3SO4Actions} \left( (x_P)_{P\in\sP},
[A_P,F^s_P]_{P\in\sP},A,R \right) \to \left( (Rx_P)_{P\in\sP},
[(R^{-1})^*A_P,RF^s_PA^{-1}]_{P\in\sP} \right),
\end{equation}
where $R\in\SO(4)$ acts on the frame by a lift to the principal
bundle. Then, the splicing map
$$
\bga_{\Theta,\sP}:\sO(\Theta,\sP,\delta)\to \bar{\sB}^s_{\ka}(2\delta),
$$
is $\SO(3)\times\SO(4)$ equivariant.
\end{lem}

\begin{proof}
The equivariance of the $\SO(4)$ action follows from noting that
the inclusion
$$
\coprod_{P\in\sP} B(0,4\sqrt{\la_P})\to \RR^4,
$$
given by identifying $B(0,4\sqrt{\la_P})$ with
$B(y_P,4\sqrt{\la_P})$ is $\SO(4)$ equivariant if $R\in\SO(4)$
acts on $(x_P)_{P\in\sP}$ by
$$
(x_P)_{P\in\sP}\to (R x_P)_{P\in\sP}.
$$
(This follows because the action of $\SO(4)$ on a connection is
given by taking any bundle map lift of the action on $\RR^4$.)
The equivariance of the $\SO(3)$ action follows from observing
that splicing in $A_P$ by identifying the section
$\phi(A_P,F^s_P)$ with the identity section of
$\RR^4\times\SO(3)$ gives the same connection as splicing in
$A_P$ by identifying the section $\phi(A_P,F^s_PA^{-1})$ with the
section $x\to (x,A^{-1})$ of $\RR^4\times\SO(3)$.
\end{proof}

We will also require the following observation about centering
maps.

\begin{lem}
\label{lem:ComposeCenterings}
Let $c_{y,\la}:\RR^4\to \RR^4$ be defined by
$c_{\la,y}(z)=(z-y)/\la$.
For any $\la_1,\la_2>0$ and
$x_1,x_2\in\RR^4$, we have:
\begin{align*}
c_{x_2,\la_2}\circ c_{x_1,\la_1} &= c_{x_1+\la_1x_2,\la_1\la_2},
\\
c_{x_2,\la_2}^*\chi_{x_1,\la_1} &= \chi_{x_2+\la_2x_1,\la_1\la_2}.
\end{align*}
\end{lem}

\begin{proof}
Elementary algebra.
\end{proof}

If one splices in a mass-centered connection of charge $\ka$ (to
the trivial background) and scale $\la$ at a point $y\in\RR^4$,
it is not immediately clear what the scale of the resulting connection will
be. However, the limit of the scale of the spliced up connection
(as $\la$ goes to zero)  will be $\sqrt{\ka |y|^2}$. The following
exploitation of that fact will be used in Lemma
\ref{lem:CommutingSplicing} to prove that, on a suitable open set
of splicing data,  when composing two splicing maps, the
cutting-off operation of the second splicing map does not
interact with that of the first splicing map.



\begin{lem}
\label{lem:ScaleLimit}
Define $T(\Theta,\sP,\delta)\subset
\sO(\Theta,\sP,\delta)$ by
\begin{equation}
\label{eq:DefineTrivialStrataInR4GluingSpace}
T(\Theta,\sP,\delta)=
\{(x_P)_{P\in\sP},[\Theta,c_{|P|}]_{P\in\sP})\in \sO(\Theta,\sP,\delta)\}
\end{equation}
Then,
\begin{equation}
\label{eq:SplicingOnTrivialStratum}
\bga'_{\Theta,\sP}(T(\Theta,\sP,\delta))
=
\{[\Theta]|\times \Si(Z_\ka(\delta),\sP)
\end{equation}
and there is an open neighborhood, $D(\Theta,\sP,\delta)$ of
$T(\Theta,\sP,\delta)$ in $\sO(\Theta,\sP,\delta)$ such that for every
$$
\bA:= \left( x_P,[A_P,F^s_P,\bx_P] \right)_{P\in\sP} \in
D(\Theta,\sP,\delta),
$$
and for every $P\in\sP$, if
$\la_P=\la([A_P,F^s_P,\bx_P])$ and $\la=\la(\bga'_{\Theta,\sP}(\bA))$,
then
\begin{equation}
\label{eq:BallContainment}
B(x_P,4\la_P^{1/2})\Subset B(0,2\la^{1/2})
\end{equation}
and
\begin{equation}
\label{eq:BallContainment2}
B(x_P,\tquarter\la_P^{1/3})\Subset B(0,\teighth\la^{1/2}).
\end{equation}
\end{lem}


\begin{proof}
The equality \eqref{eq:SplicingOnTrivialStratum} follows immediately
from Lemma \ref{lem:UhlebeckLimitCont}.


The functions $\la_P$ vanish on $T(\Theta,\sP,\delta)$
while $\la\circ\bga'_{\Theta,\sP}$ is equal to the square root of
$\sum_{P\in\sP}|P||x_P|^2$, and thus non-zero.  The
continuity of the functions involved then implies the
existence of the desired open subspace.
\end{proof}



\subsection{Composing splicing maps}
\label{subsec:OverlapSpliceMapsR4}
To construct the moduli space with spliced ends we need to show
the images of the splicing maps $\bga'_{\Theta,\sP}$
form a smoothly stratified
space. Let $\sP,\sP'$ be partitions of $N_{\ka}$ with $\sP<\sP'$.
For each $P\in\sP$, let $\sP'_P$ be the partition of $P$
defined in \eqref{eq:DefineRefinementPartition}. We will need to
understand the overlaps of the images of the maps $\bga'_{\Theta,\sP}$ and
$\bga'_{\Theta,\sP'}$. To that end,  we define a space
$\tilde\sO(\Theta,\sP,\sP',\delta)$ with maps to $\tilde\sO(\Theta,\sP,\delta)$
and $\tilde\sO(\Theta,\sP',\delta)$ so that the following diagram commutes:
\begin{equation}
\label{eq:CommutingSplice0}
\begin{CD}
\tilde\sO(\Theta,\sP,\sP',\delta)
@>\rho^{\Theta,d}_{\sP,\sP'} >>
\tilde\sO(\Theta,\sP,\delta)
\\
@V \rho^{\Theta,u}_{\sP,\sP'}   VV @V \bga'_{\Theta,\sP} VV
\\
\tilde\sO(\Theta,\sP',\delta)
@> \bga'_{\Theta,\sP'} >>
\bar{\sB}^s_{\ka}(2\delta)
\end{CD}
\end{equation}
We note that the $d$ and $u$ appearing in the maps
$\rho^{\Theta,d}_{\sP,\sP'}$ and $\rho^{\Theta,d}_{\sP,\sP'}$
are supposed to imply ``down'' and ``up'' respectively as
$\sP$ and $\sP'$ correspond to the lower and upper strata.

\subsubsection{Defining the overlap data}
We now define the objects appearing in the diagram
\eqref{eq:CommutingSplice0}.
The open set
$\tilde\sO(\Theta,\sP,\sP',\delta)$ will be defined to be a subspace of
\begin{equation}
\label{eq:R4OverlapSpace}
\nu(\Theta,\sP,\sP',\delta)=
\Delta^\circ(Z_\ka(\delta),\sP) \times
\prod_{P\in\sP}
\left(
    \Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)
    \times
    \prod_{P'\in\sP_P'} \bar{\sB}^s_{|P'|}(\delta_{P'})
\right),
\end{equation}
where the constants $\delta_P$ and $\delta_{P'}$ will not need to be
defined explicitly.
We introduce the following  projections from $\nu(\Theta,\sP,\sP',\delta)$:
\begin{equation}
\label{eq:DefineProjections}
\begin{aligned}
 \pi_{\sP,\bx} &: \nu(\Theta,\sP,\sP',\delta) \to
 \Delta^\circ(Z_\ka(\delta),\sP),
\\
\pi_{\sP'_P,\bx} &: \nu(\Theta,\sP,\sP',\delta) \to
\Delta^{\natural}(Z_{|P|}(\delta_P),\sP'_P),
\\
\pi_{\bx} &: \nu(\Theta,\sP,\sP',\delta) \to
\Delta^\circ(Z_\ka(\delta),\sP)\times
    \prod_{P\in\sP}\Delta^{\natural}(Z_{|P|}(\delta_P),\sP'_P),
\\
\pi_{P} &: \nu(\Theta,\sP,\sP',\delta) \to
\Delta^{\natural}(Z_{|P|}(\delta_P),\sP'_P)\times \prod_{P'\in\sP'_P}\bar{\sB}^s_{|P'|},
\\
\pi_{P'}&: \nu(\Theta,\sP,\sP',\delta) \to \bar{\sB}^s_{|P'|}(\delta_P).
\end{aligned}
\end{equation}
Use the inclusion given in \eqref{eq:DefineR4NormalPP'},
$$
\nu(Z_\ka(\delta),\sP,\sP')
=\Delta^\circ(Z_\ka(\delta),\sP)
\times
\prod_{P\in\sP} \Delta^\circ(Z_P(\delta_P),\sP'_P)
\to
\nu(Z_{\ka}(\delta),\sP),
$$
to consider $\nu(Z_\ka(\delta),\sP,\sP')$ as a subspace of
$\nu(Z_\ka(\delta),\sP)$.  We may then apply the exponential map
$e(Z_\ka,\sP)$ of \eqref{eq:R4Exponential}
to elements of $\nu(Z_\ka,\sP,\sP')$.
We desire to define the map $\rho^{\Theta,u}_{\sP,\sP'}$
appearing in the diagram
\eqref{eq:CommutingSplice0} by
\begin{equation}
\label{eq:R4TrivialSplicingUpwardsOverlapMap}
\rho^{\Theta,u}_{\sP,\sP'}= (e(Z_\ka,\sP)\circ\pi_{\bx}) \times
\prod_{P'\in\sP'} \pi_{P'}.
\end{equation}
That is, the map $\rho^{\Theta,u}_{\sP,\sP'}$ leaves the connection
data in $\nu(\Theta,\sP,\sP',\delta)$
alone and maps the points in $\nu(Z_{\ka}(\delta),\sP,\sP')$ to
the diagonal $\Delta^\circ(Z_\ka(\delta),\sP')$.


For the composition $e(Z_\ka,\sP)\circ\pi_{\bx}$
in \eqref{eq:R4TrivialSplicingUpwardsOverlapMap}
to be defined, we must restrict the domain of
$\rho^{\Theta,u}_{\sP,\sP'}$ to an open subspace
$\tilde\sO(\Theta,\sP,\sP',\delta)$ of $\nu(\Theta,\sP,\sP',\delta)$ satisfying
\begin{equation}
\label{eq:OverlapCondition1}
\tilde\sO(\Theta,\sP,\sP',\delta) \subset
\pi_{\bx}^{-1}\left( \sO(Z_{\ka}(\delta),\sP,\sP') \right),
\end{equation}
where $\sO(Z_{\ka}(\delta),\sP,\sP')$ is defined in \eqref{eq:DefineTubNghR4Normal}.

To define the composition
$\bga'_{\Theta,\sP'}\circ\rho^{\Theta,u}_{\sP,\sP'}$
appearing in the diagram \eqref{eq:CommutingSplice0},
the open subspace $\tilde\sO(\Theta,\sP,\sP',\delta)$ must satisfy
\begin{equation}
\label{eq:OverlapCondition2}
\tilde\sO(\Theta,\sP,\sP',\delta) \subset
(\rho^{\Theta,u}_{\sP,\sP'})^{-1}\left( \tilde\sO(\Theta,\sP',\delta)\right).
\end{equation}

Before defining $\rho^{\Theta,d}_{\sP,\sP'}$, we note that
if $P\in\sP\cap\sP'$ then the partition $\sP'_P$ is just the set $P$
and $\Delta^\natural(Z_{|P|},\sP'_P)$ is
a single point (the zero vector).
In this case, we define the splicing map
$$
\bga'_{\Theta,\sP'_P}:
\Delta^\natural(Z_{|P|}(\delta),\sP'_P)\times \bar\sB^s_{|P|}(\delta_P)
\to \bar\sB^s_{|P|}(2\delta),
$$
to be the projection onto
$\bar\sB^s_{|P|}(\delta_P)\subset\bar\sB^s_{|P|}(2\delta)$.
Then,
the map $\rho^{\Theta,d}_{\sP,\sP'}$ of \eqref{eq:CommutingSplice0} will
be defined by:
\begin{equation}
\label{eq:DefineUpwardTransition}
\rho^{\Theta,d}_{\sP,\sP'}:=
\pi_{\sP,\bx}\times
\left(\prod_{P\in\sP}\bga'_{\Theta,\sP'_P}\circ\pi_{P}\right)
\end{equation}
That is, the map $\rho^{\Theta,d}_{\sP,\sP'}$ leaves the point
in $\Delta^\circ(Z_\ka,\sP)$ alone and for each $P\in\sP$
applies the splicing map $\bga'_{\Theta,\sP'_P}$ to the space
$$
\Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)\times
\prod_{P'\in\sP'_P} \bar\sB^s_{|P'|}(\delta_{P'}).
$$
For this map to be defined on $\tilde\sO(\Theta,\sP,\sP',\delta)$ we need
the image of $\pi_P$ to be contained in $\sO(\Theta,\sP'_P,\delta_P)$, the
domain of the splicing map $\bga'_{\Theta,\sP'_P}$.
However, we will make a stronger requirement (to be used in Lemma
\ref{lem:CommutingSplicing}):
\begin{equation}
\label{eq:OverlapCondition3}
\tilde\sO(\Theta,\sP,\sP',\delta) \subset
\pi_P^{-1}\left( D(\Theta,\sP'_P,\delta_P)\right) \quad \text{for all
$P\in\sP\less\sP'$},
\end{equation}
where $D(\Theta,\sP'_P,\delta_P)$ is the open subspace of
$\tilde\sO(\Theta,\sP'_P,\delta_P)$ defined in Lemma \ref{lem:ScaleLimit}.  Since
$D(\Theta,\sP'_P,\delta_P)\subset \sO(\Theta,\sP'_P,\delta_P)$, the condition
\eqref{eq:OverlapCondition3} implies $\rho^{\Theta,d}_{\sP,\sP'}$ is
defined on $\tilde\sO(\Theta,\sP,\sP',\delta)$.

Finally, for the composition $\bga'_{\Theta,\sP}\circ \rho^{\Theta,d}_{\sP,\sP'}$
to be well defined, the open subspace
$\tilde\sO(\Theta,\sP'_P,\delta_P)$ must satisfy
\begin{equation}
\label{eq:OverlapCondition4}
\tilde\sO(\Theta,\sP,\sP',\delta) \subset
(\rho^{\Theta,d}_{\sP,\sP'})^{-1} \left( \tilde\sO(\Theta,\sP,\delta) \right).
\end{equation}


We now record a result showing that the image of the restriction
of $\bga'_{\Theta,\sP'}\circ\rho^{\Theta,u}_{\sP,\sP'}$ to
an appropriate open subspace
$\tilde\sO(\Theta,\sP,\sP',\delta)$ of the overlap
data $\nu(\Theta,\sP,\sP',\delta)$ contains
\begin{equation}
\label{eq:TrivialStrataInImageOfDiagonalOverlapMap}
\{[\Theta,\bx]: \bx\in e(Z_\ka,\sP)(\sO(Z_\ka(\delta),\sP,\sP'))\}.
\end{equation}
Thus, the image of
$\bga'_{\Theta,\sP'}\circ\rho^{\Theta,u}_{\sP,\sP'}$
will contain all of
the trivial stratum
$\{[\Theta]\}\times\Si(Z_\ka(\delta),\sP')$ in the image
of $\bga'_{\Theta,\sP}$.

\begin{lem}
\label{lem:OverlappingTrivial}
Let $\sP,\sP'$ be partitions of
$N_{\ka}$ satisfying $\sP<\sP'$. Let $T(\Theta,\sP,\sP',\delta)$ be the
subspace of the space $\nu(\Theta,\sP,\sP',\delta)$ given by
\begin{equation}
\label{eq:DefineTrivialOverlap}
T(\Theta,\sP,\sP',\delta):=
\pi_{\bx}^{-1}\left(\sO(Z_{\ka},\sP,\sP')\right) \cap \left(
\cap_{P'\in\sP'} \pi_{P'}^{-1}([\Theta,c_{|P'|}])\right).
\end{equation}
Then, there is an open subspace, $\tilde\sO(\Theta,\sP,\sP',\delta)$, of
$\nu(\Theta,\sP,\sP',\delta)$ satisfying the conditions
\eqref{eq:OverlapCondition1},\eqref{eq:OverlapCondition2},
\eqref{eq:OverlapCondition3}, and \eqref{eq:OverlapCondition4}
and also satisfying
\begin{enumerate}
\item
$T(\Theta,\sP,\sP',\delta)\subset\tilde\sO(\Theta,\sP,\sP',\delta)$,
\item
$\bga'_{\Theta,\sP'}\circ\rho^{\Theta,u}_{\sP,\sP'} \left( T(\Theta,\sP,\sP',\delta)
\right) =\{[\Theta,\bx]: \bx\in
e(Z_\ka,\sP)\left(\sO(Z_{\ka}(\delta),\sP,\sP')\right)\}$,
\item
$(\bga'_{\Theta,\sP})^{-1} \left([\Theta]\times
e(Z_\ka,\sP)(\sO(Z_{\ka}(\delta),\sP,\sP'))\right) =
\rho^{\Theta,d}_{\sP,\sP'}\left( T(\Theta,\sP,\sP',\delta)\right)$.
\end{enumerate}
\end{lem}

\begin{proof}
First,
\begin{equation}
\label{eq:piXofT}
\pi_{\bx}(T(\Theta,\sP,\sP',\delta)) = \sO(Z_\ka(\delta),\sP,\sP'),
\end{equation}
by definition so there is an open neighborhood of
$T(\Theta,\sP,\sP',\delta)$ in $\nu(\Theta,\sP,\sP',\delta)$ satisfying
\eqref{eq:OverlapCondition1}.  Second, for every $P'\in\sP'$,
$$
\pi_{P'}(T(\Theta,\sP,\sP',\delta))=[\Theta,c_{|P'|}]
$$
so
$\pi_P(T(\Theta,\sP,\sP',\delta))\subset T(\Theta,\sP'_P,\delta_P)
\subset D(\Theta,\sP,\delta)$ for
all $P\in\sP$ and
there is an open neighborhood of
$T(\Theta,\sP,\sP',\delta)$ in $\nu(\RR^4,\sP,\sP')$ satisfying
\eqref{eq:OverlapCondition3}.

The second item in the conclusions of the lemma
follows from the definitions of the spaces and maps
and the value of the splicing map given
in \eqref{eq:SplicingConePoint}.
The second item  implies that
$\rho^{\Theta,u}_{\sP,\sP'}(T(\Theta,\sP,\sP',\delta)) \subset
\tilde\sO(\Theta,\sP',\delta)$, so there is an open neighborhood of
$T(\Theta,\sP,\sP',\delta)$ in $\nu(\Theta,\sP,\sP',\delta)$ satisfying
\eqref{eq:OverlapCondition2}.

Next,  equation \eqref{eq:SplicingConePoint} implies that
$$
\rho^{\Theta,d}_{\sP,\sP'}|_{T(\Theta,\sP,\sP',\delta)}=\pi_{\bx}|_{T(\Theta,\sP,\sP',\delta)}.
$$
This equality and \eqref{eq:piXofT} imply the third item of
the conclusions of the lemma.  This third item implies
that $\rho^{\Theta,d}_{\sP,\sP'}(T(\Theta,\sP,\sP',\delta))$ is contained in
the domain of $\bga'_{\Theta,\sP'}$,
$\sO(\Theta,\sP,\delta)$, so there is an open neighborhood of
$T(\Theta,\sP,\sP',\delta)$ in $\nu(\Theta,\sP,\sP',\delta)$ satisfying
\eqref{eq:OverlapCondition4}.

By taking the intersection of these open neighborhoods of
$T(\Theta,\sP,\sP',\delta)$ in $\nu(\Theta,\sP,\sP',\delta)$, we obtained
the desired open neighborhood $\tilde\sO(\Theta,\sP,\sP',\delta)$.
\end{proof}

\subsubsection{Equality of splicing maps}
We now are in a position to prove that the diagram
\eqref{eq:CommutingSplice0} commutes. The key point is the
restriction \eqref{eq:OverlapCondition3} on the domain
$\tilde\sO(\Theta,\sP,\sP',\delta)$.  This ensures that in the iterated
splicing construction defining the composition
$\bga'_{\Theta,\sP}\circ \rho^{\Theta,d}_{\sP,\sP'}$ (first the splicings
$\bga'_{\Theta,\sP'_P}$ defining the map $\rho^{\Theta,d}_{\sP,\sP'}$ are
performed, then the splicing $\bga'_{\Theta,\sP}$ is performed),
the cut-off function of the second splicing has no effect. This
follows from Lemma \ref{lem:ScaleLimit} which ensures that the
connections given by $\rho^{\Theta,d}_{\sP,\sP'}$ are {\em already\/}
trivial on the annuli on which the splicing map
$\bga'_{\Theta,\sP}$ interpolates between them and the trivial
connection.  Thus, this iterated splicing procedure is equivalent
to a single splicing, that given by the composition
$\bga'_{\Theta,\sP'}\circ \rho^{\Theta,u}_{\sP,\sP'}$.




\begin{lem}
\label{lem:CommutingSplicing}
Let $\sP$ and $\sP'$ be partitions
of $N_{\ka}$ with $\sP<\sP'$. Let $\tilde\sO(\Theta,\sP,\sP',\delta)$ be the
open set defined in Lemma \ref{lem:OverlappingTrivial}
of the space $\nu(\Theta,\sP,\sP',\delta)$
defined in \eqref{eq:R4OverlapSpace}. Then, the following diagram
commutes:
\begin{equation}
\label{eq:CommutingSplicing}
\begin{CD}
\tilde\sO(\Theta,\sP,\sP',\delta)
@> \rho^{\Theta,u}_{\sP,\sP'} >>
\tilde\sO(\Theta,\sP',\delta)
\\
@V \rho^{\Theta,d}_{\sP,\sP'} VV @V \bga'_{\Theta,\sP'} VV
\\
\tilde\sO(\Theta,\sP,\delta)
@> \bga'_{\Theta,\sP} >>
\bar{\sB}^s_{\ka}(2\delta)
\end{CD}
\end{equation}
\end{lem}

\begin{proof}
To simplify the notation, we write $Q$ for
an element of the partition $\sP'$.
Let $\bA\in \tilde\sO(\Theta,\sP,\sP',\delta)$ be given by:
$$
\bA= \left( (y_P)_{P\in\sP},
((x_Q)_{P'\in\sP'_P},[A_Q,F^s_Q]_{Q\in\sP'_P})\right),
$$
where  $(y_P)_{P\in\sP}\in \Delta^\circ(Z_\ka,\sP)$ and
$((x_Q)_{Q\in\sP'_P},[A_Q,F^s_Q]_{Q\in\sP'_P})\in
D(\Theta,\sP'_P,\delta_P)$.
(For simplicity of exposition, we assume
$[A_Q,F^s_Q]\in\bar\sB^s_{|Q|}(\delta_Q)$ has no ideal points;
the proof is no more difficult without that assumption, but
the notation becomes more opaque.)

The image of $\bA$ under $\rho^{\Theta,u}_{\sP,\sP'}$ is given by the same
connections, $[A_Q,F^s_Q]$ as given in $\bA$, but by the point
$(x'_Q)_{Q\in\sP'}\in \Delta^\circ(Z_\ka(\delta),\sP')$, where
\begin{equation}
\label{eq:DefinePsi}
x'_Q=y_P+x_Q \quad \text{if $Q\in\sP'_P$.}
\end{equation}
(Note that
$x'_Q=y_{P}$ if $Q\in\sP\cap\sP'$ as $x_Q$ is then the zero vector.)
If we set $\la_Q=\la([A_Q])$, then
\begin{align}
\label{eq:SplicedConn0} &
\bga'_{\Theta,\sP'}\circ\rho^{\Theta,u}_{\sP,\sP'}(\bA)
\\
\notag &=
\begin{cases}
\Theta & \text{on $\RR^4 \less \cup_{Q\in\sP'_P,P\in\sP}
B(x_Q',4\sqrt{\la_Q})$},
\\
(1-\chi)c_{x'_Q,4\sqrt{\la_Q}}^*A_Q +
\chi_{x'_Q,4\sqrt{\la_Q}}c_{x'_Q}^*\Theta(A_Q,f_{E_Q}) & \text{on
$\Omega(x'_Q,2\sqrt{\la_Q},4\sqrt{\la_Q})$},
\\
c_{x'_Q,1}^*A_Q & \text{on $B(x'_Q,2\sqrt{\la_Q})$}.
\end{cases}
\end{align}
We now compare \eqref{eq:SplicedConn0} with
$\bga'_{\Theta,\sP}\circ\rho^{\Theta,d}_{\sP,\sP'}(\bA)$.

If, for $P\in\sP\less\sP'$, we define
$$
[A_P,F^s_P] = \bga'_{\Theta,\sP'_P}(\pi_P(\bA)) =
\bga'_{\Theta,\sP'_P} \left(
(x_Q)_{Q\in\sP'_P},[A_Q,f_{E_Q}]_{Q\in\sP'_P} \right),
$$
and $[A_P,F^s_P]=[A_Q,F^s_Q]$ for $P=Q\in\sP\cap\sP'$, then
$$
\rho^{\Theta,d}_{\sP,\sP'}(\bA)
=
\left(
(y_P)_{P\in\sP},([A_P,F^s_P])_{P\in\sP}
\right)
\in \sO(\Theta,\sP,\delta).
$$
The composition
$\bga'_{\Theta,\sP}\circ\rho^{\Theta,d}_{\sP,\sP'}$ is then given by:
\begin{align}
\label{eq:SplicedConn1} & \bga'_{\Theta,\sP}\circ\rho^{\Theta,d}_{\sP,\sP'}
\\
\notag &=
\begin{cases}
\Theta & \text{on $\RR^4\less \cup_{P\in\sP} B(y_P,4\sqrt
{\la_P})$},
\\
(1-\chi_{y_P,4\sqrt{\la_P}})c_{y_P,1}^*A_P +
\chi_{y_P,4\sqrt{\la_P}}c_{y_P,1}^*\Theta(A_P,f_{E_P}) & \text{on
$\Omega(y_P,2\sqrt{\la_P},4\sqrt{\la_P})$},
\\
c_{y_P,1}^*A_{P} & \text{on $B(y_P,2\sqrt{\la_P})$},
\end{cases}
\end{align}
where $\la_P=\la([A_P])$. On the balls $B(x_P,4\sqrt{\la_P})$ for
$P\in\sP\cap\sP'$, and on $\RR^4\less
\cup_{P\in\sP}B(x_P,4\sqrt{\la_P})$ the connections
\eqref{eq:SplicedConn0} and \eqref{eq:SplicedConn1} are
identical.  We thus focus our attention on the balls
$B(x_P,4\sqrt{\la_P})$ for $P\in\sP\less \sP'$.  For such $P$,
\begin{align}
\label{eq:SplicedConn2} A_P&=\bga'_{\Theta,\sP'_P}(\pi_P(\bA))
\\
\notag &=
\begin{cases}
\Theta & \text{on $\RR^4\less \cup_{Q\in\sP'_P}
B(x_Q,4\sqrt{\la_Q})$},
\\
(1-\chi_{x_Q,\sqrt{4\la_Q}})c_{x_Q,1}^*A_Q +
\chi_{x_Q,\sqrt{4\la_Q}}c_{x_Q,1}^*\Theta(A_Q,f_{E_Q}) & \text{on
$\Omega(x_Q,2\sqrt{\la_Q},4\sqrt{\la_Q})$},
\\
c_{x_Q,1}^*A_Q & \text{on $B(z_Q,2\sqrt{\la_Q})$}.
\end{cases}
\end{align}
Because the points $(x_Q)_{Q\in\sP'_P}\in
\Delta^{\natural}(\RR^4,\sP'_P)$ and the connections
$[A_Q,f_{E_Q}]\in\bar{\sB}^s_{|Q|}$ form a point in
$D(\Theta,\sP'_P,\delta_P)$ by the construction of $\tilde\sO(\Theta,\sP,\sP',\delta)$,
Lemma \ref{lem:ScaleLimit} and equation \eqref{eq:BallContainment}
imply that for all $Q\in\sP'_P$,
$$
B(x_Q,4\sqrt{\la_Q}) \subset B(0,2\sqrt{\la_P}).
$$
Thus, the construction in \eqref{eq:SplicedConn2} implies $A_P$
is already equal to $\Theta$ on
$$
\RR^4\less B(0,2\sqrt{\la_P}),
$$
so $c_{y_P,1}^*A_P$ is equal to $\Theta$ on
$$
\RR^4\less B(y_P,2\sqrt{\la_P}).
$$
Thus, the convex combination in the second line of
\eqref{eq:SplicedConn1} is equal to $\Theta$. We examine the
final line of \eqref{eq:SplicedConn1} by applying Lemma
\ref{lem:ComposeCenterings}:
\begin{align}
\label{eq:SplicedConn3} &c_{y_P,1}^*A_P
\\
\notag &=
\begin{cases}
\Theta & \text{on $\RR^4 \less \cup_{Q\in\sP'_P}
B(y_P+x_Q,4\sqrt{\la_Q})$},
\\
(1-\chi)c_{y_P+x_Q,4\sqrt{\la_Q}}^*A_Q +
\chi_{y_P+x_Q,4\sqrt{\la_Q}}c_{y_P+x_Q}^*\Theta(A_Q,f_{E_Q}) &
\text{on $ \Omega(y_P+x_Q,2\sqrt{\la_Q},4\sqrt{\la_Q})$},
\\
c_{y_P+x_Q,1}^*A_Q & \text{on $B(y_P+x_Q,2\sqrt{\la_Q})$}.
\end{cases}
\end{align}
Thus, we see that $\bga'_{\Theta,\sP}\circ\rho^{\Theta,d}_{\sP,\sP'}(\bA)$
is given by:
\begin{equation}
\label{eq:SplicedConn4}
\begin{cases}
\Theta & \text{on $\RR^4 \less \cup_{Q\in\sP'_P,P\in\sP}
B(y_P+x_Q,4\sqrt{\la_Q})$},
\\
(1-\chi)c_{y_P+x_Q,4\sqrt{\la_Q}}^*A_Q +
\chi_{y_P+x_Q,4\sqrt{\la_Q}}c_{y_P+x_Q}^*\Theta(A_Q,f_{E_Q}) &
\text{on $\Omega(y_P+x_Q,2\sqrt{\la_Q},4\sqrt{\la_Q})$},
\\
c_{y_P+x_Q,1}^*A_Q & \text{on $B(y_P+x_Q,2\sqrt{\la_Q})$}.
\end{cases}
\end{equation}
Then comparing \eqref{eq:SplicedConn4} with
\eqref{eq:SplicedConn0} and noting the definition of $x_Q$ in
\eqref{eq:DefinePsi} completes the proof.
\end{proof}

\subsubsection{Symmetric group quotients}
Finally, we make some observations on the action of the symmetric
group necessary to define
a quotient of the diagram \eqref{eq:CommutingSplice0} by the
symmetric group.
Recall from Lemma \ref{lem:SymmProdNeigh}
that for $\sP<\sP'$, to describe a neighborhood of
$\Si(Z_\ka(\delta),\sP)$ in $\Si(Z_\ka,\sP)$,
we must describe all the diagonals
$\Delta^{\natural}(Z_\ka(\delta),\sP'')$ where $\sP''\in [\sP<\sP']$.
We define
\begin{equation}
\label{eq:SplicingOverlapSpaceR4SymmQuotient}
\nu(\Theta,\sP,[\sP'],\delta)
=\sqcup_{\sP''\in [\sP<\sP']}
\nu(\Theta,\sP,\sP'',\delta)
\end{equation}
The group $\fS(\sP)$ acts on $\nu(\Theta,\sP,[\sP'],\delta)$ by
the obvious permutation.
Note that if $\si\in\fS(\sP)$ and
$P'\in\sP'$, then it is possible that $\si(P')\in\sP''\in [\sP<\sP']$
with $\sP''\neq \sP'$, thus explaining the need for the additional
partitions $\sP''\in [\sP<\sP']$.
Let $\sO(\Theta,\sP,[\sP'],\delta)\subset\nu(\Theta,\sP,[\sP'],\delta)$
be the analogue of the open subspace $\sO(\Theta,\sP,\sP',\delta)$
defined in Lemma \ref{lem:OverlappingTrivial}.
We define (analogously to the definition of
$T(\Theta,\sP,\sP',\delta)$ in \eqref{eq:DefineTrivialOverlap})
\begin{equation}
\label{eq:TrivialStrataOverlapSymmQuotient}
\begin{aligned}
T(\Theta,\sP,[\sP'],\delta)
=&
\{
\left(
    (y_P)_{P\in\sP},
    (x_Q)_{Q\in\sP''},
    ([\Theta,c_{|Q|}])_{Q\in\sP''}
\right) \in \nu(\Theta,\sP,[\sP'],\delta):
\\
&\quad \left(
    (y_P)_{P\in\sP},
    (x_Q)_{Q\in\sP''}
\right)\in \sO(Z_\ka(\delta),[\sP<\sP'])
\},
\end{aligned}
\end{equation}
where $\sO(Z_\ka(\delta),[\sP<\sP'])$ is defined in
Lemma \ref{lem:SymmProdNeigh}.




The maps $\rho^{\Theta,u}_{\sP,\sP''}$ for $\sP''\in [\sP<\sP']$
define a map
\begin{equation}
\label{eq:SymmetricProductNormalBundlesOfHigherStratum}
\rho^{\Theta,u}_{\sP,[\sP']}:
\sO(\Theta,\sP,[\sP'],\delta)
\to
\nu(\Theta,[\sP<\sP'],\delta)
=
\left(\coprod_{\sP''\in [\sP<\sP']} \nu(\Theta,\sP'',\delta)\right)/\fS(\sP).
\end{equation}
Similarly, the coproduct of the maps $\rho^{\Theta,d}_{\sP,\sP''}$
is invariant under the action of $\fS(\sP)$ and thus
defines a map
\begin{equation}
\rho^{\Theta,d}_{\sP,[\sP']}:
\sO(\Theta,\sP,[\sP'],\delta)
\to
\sO(\Theta,\sP,\delta).
\end{equation}
The coproduct of the splicing maps $\bga'_{\Theta,\sP''}$, over
$\sP''\in [\sP<\sP']$, is invariant under the action of $\fS(\sP)$
and thus defines a map
\begin{equation}
\label{eq:SplicingOverSeveralPartitions}
\bga'_{\Theta,\sP,[\sP']} =
\left(\coprod_{\sP''\in [\sP<\sP']}\bga'_{\Theta,\sP''}\right)/\fS(\sP)
:
\left(\coprod_{\sP''\in [\sP<\sP']}\sO(\RR^4,\sP'',\delta)\right)/\fS(\sP)
\to
\bar{\sB}^s_{\ka}(2\delta),
\end{equation}
which has the same image as $\bga'_{\Theta,\sP'}$.
Lemma \ref{lem:CommutingSplicing} then implies the following.

\begin{prop}
\label{prop:CommutingSplicingSymmQuotient}
Let $\sP$ and $\sP'$ be partitions
of $N_{\ka}$ with $\sP<\sP'$.
Let $\sO(\Theta,\sP,\sP',\delta)$ be the open subspace defined
following\eqref{eq:SplicingOverlapSpaceR4SymmQuotient}.
Then, the following diagram
commutes:
\begin{equation}
\label{eq:CommutingSplicingQuotient}
\begin{CD}
\sO(\Theta,\sP,[\sP'],\delta) @> \rho^{\Theta,u}_{\sP,[\sP']} >>
\left(\coprod_{\sP''\in [\sP<\sP']}\sO(\RR^4,\sP'')\right)/\fS(\sP)
\\
@V \rho^{\Theta,d}_{\sP,[\sP']} VV @V \bga'_{\Theta,\sP,[\sP']} VV
\\
\sO(\Theta,\sP,\delta) @> \bga'_{\Theta,\sP} >> \bar{\sB}^s_{\ka}(2\delta)
\end{CD}
\end{equation}
where $\bga'_{\Theta,[\sP<\sP']}$ is defined in
\eqref{eq:SplicingOverSeveralPartitions}.
\end{prop}

\subsection{The spliced end}
\label{subsec:SplicedEnd}
We now construct the spliced end, given by the union of the
images of the splicing maps $\bga'_{\Theta,\sP}$ restricted
to appropriate subspaces of $\sO(\Theta,\sP,\delta)$.

This construction is inductive.  We therefore assume that
the spliced ends moduli space $\barM^{s,\natural}_{spl,|P|}(\delta)$
with the properties stated in Theorem \ref{thm:ExistenceOfSplicedEndsModuli}
has already been constructed for $|P|<\ka$.  Note that because
we only need to describe Uhlenbeck neighborhoods of the {\em punctured\/}
trivial strata, we only need to use splicing maps $\bga'_{\Theta,\sP}$
where $|\sP|>1$.  For a partition $\sP$ of $N_\ka$ with
$|\sP|>1$, $|P|<\ka$ for all $P\in\sP$.  Therefore, the
inductive hypothesis suffices for this construction.

The splicing maps will be restricted to the following spaces
(where the precise choice of the constants $\delta_P>0$ will not be relevant
to the rest of the discussion)
\begin{equation}
\label{eq:DefineASDSplicingDomain}
\begin{aligned}
\tilde\sO^{\asd}(\Theta,\sP,\delta)
&=
\tilde\sO(\Theta,\sP,\delta)\cap \left(
    \Delta^\circ(Z_\ka(2\delta),\sP)
    \times
    \prod_{P\in\sP} M^{s,\natural}_{spl,|P|}(\delta_P)
\right),
\\
\sO^{\asd}(\Theta,\sP,\delta)&=\tilde\sO^{\asd}(\Theta,\sP,\delta)/\fS(\sP).
\end{aligned}
\end{equation}
Observe that $T(\Theta,\sP,\delta)\subset \tilde\sO^{\asd}(\Theta,\sP,\delta)$ and
the dimension of $\sO^{\asd}(\Theta,\sP,\delta)$ is equal to that of
$\barM^{s,\natural}_{\ka}(\delta)$.


The spliced end is constructed by the following proposition.

\begin{prop}
\label{prop:ExistenceOfSplicedEnd}
Assume that the spliced end moduli space $\barM^{s,\natural}_{spl,\ka'}(\delta)$
with the properties enumerated in Theorem
\ref{thm:ExistenceOfSplicedEndsModuli} has already been constructed.
Then for every partition $\sP$ of
there is an $\SO(3)\times\SO(4)$-invariant,
open neighborhood $\sO^{\asd}_1(\Theta,\sP,\delta)$ of
$T(\Theta,\sP,\delta)$ in $\sO^{\asd}(\Theta,\sP,\delta)$ such that
the space,
\begin{equation}
\label{eq:DefineSplicedEnd}
W_\ka = \cup_{\sP} \ \bga'_{\Theta,\sP}\left(\sO^{\asd}_1(\Theta,\sP,\delta)\right),
\end{equation}
is a smoothly-stratified subspace of $\bar\sB^s_{\ka}(\delta)$
with every point $[A,F^s,\bx]\in W_\ka$ satisfying the estimate
$\|F^+_{A}\|_{L^{\sharp,2}}\le M$.
\end{prop}

We prove Proposition \ref{prop:ExistenceOfSplicedEnd} by using
Proposition \ref{prop:CommutingSplicingSymmQuotient} to
control the overlaps of the images of the splicing maps.
The transition maps $\rho^{\Theta,u}_{\sP,[\sP']}$ and
$\rho^{\Theta,d}_{\sP,[\sP']}$ will be restricted to the
following subspace of $\sO(\Theta,\sP,[\sP'],\delta)$
\begin{equation}
\label{eq:SplicedModuliOverlapR4}
\begin{aligned}
{}&\tilde\sO^{\asd} (\Theta,\sP,[\sP'],\delta)
\\
&\quad=
\tilde\sO(\Theta,\sP,[\sP'],\delta)\\
 &\quad \cap \left(
    \Delta^\circ(Z_\ka(\delta),\sP)
    \times
    \coprod_{\sP''\in [\sP<\sP']}
    \prod_{P\in\sP}
            \left(
            \Delta^\circ(Z_P(\delta_P),\sP_P'')
            \times
            \prod_{Q\in\sP_P''} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
            \right)
\right),
\\
{}&\sO^{\asd}(\Theta,\sP,[\sP'],\delta)=\tilde\sO^{\asd} (\Theta,\sP,[\sP'],\delta)/\fS(\sP).
\end{aligned}
\end{equation}
(Compare the definition
\eqref{eq:SplicingOverlapSpaceR4SymmQuotient}).
Observe that $T(\Theta,\sP,[\sP'],\delta)\subset
\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$
where $T(\Theta,\sP,[\sP'],\delta)$ is defined in
\eqref{eq:TrivialStrataOverlapSymmQuotient} because
$[\Theta,c_{|Q|}]\in \barM^{s,\natural}_{spl,|Q|}(\delta_Q)$ for
all $|Q|<\ka$.

We must first verify that 
the transition maps $\rho^{\Theta,u}_{\sP,[\sP']}$ and
$\rho^{\Theta,d}_{\sP,[\sP']}$ map
$\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$ to
$\sO^{\asd}(\Theta,\sP',\delta)$ and $\sO^{\asd}(\Theta,\sP,\delta)$ respectively.


\begin{lem}
\label{lem:SplicedEndsEmbeddingR4d}
Assume $\barM^{s,\natural}_{spl,\ka'}(\delta)$
has been constructed satisfying the properties enumerated in
Theorem \ref{thm:ExistenceOfSplicedEndsModuli} for all
$\ka'<\ka$.  Let $T(\Theta,\sP,[\sP'],\delta)$ be the subspace of
$\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$ defined in
\eqref{eq:TrivialStrataOverlapSymmQuotient}.
Then there is an open neighborhood
$\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$
of $T(\Theta,\sP,[\sP'],\delta)$ in $\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$
such that  the restriction of $\rho^{\Theta,d}_{\sP,[\sP']}$
to $\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$ is an open embedding
of $\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$ into
$\sO^{\asd}(\Theta,\sP,\delta)$.
%
%Then, for every open neighborhood $\sO^{\asd}_1(\Theta,\sP,\delta)$ of
%$T(\Theta,\sP,\delta)$ in $\sO^{\asd}(\Theta,\sP,\delta)$, there exists an open
%neighborhood, $\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$, of
%$$
%T(\Theta,\sP,[\sP'],\delta) \cap (\rho^{\Theta,d}_{\sP,[\sP']})^{-1}\left(
%\sO^{\asd}_1(\Theta,\sP,\delta) \right)
%$$
%in $\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$ such that
%the restriction of $\rho^{\Theta,d}_{\sP,[\sP']}$
%to $\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$ defines an open
%embedding of $\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$ into
%$\sO^{\asd}(\Theta,\sP,\delta)$
%and
%$$
% \rho^{\Theta,d}_{\sP,[\sP']} \left(\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta) \right)
%$$
%is an open neighborhood of
%\begin{equation}
%\label{eq:OverlapTrivialImage} \sO^{\asd}_1(\Theta,\sP,\delta) \cap
%\rho^{\Theta,d}_{\sP,[\sP']} \left( T(\Theta,\sP,[\sP'],\delta) \right)
%\end{equation}
%in $\sO^{\asd}_1(\Theta,\sP,\delta)$.
\end{lem}

\begin{proof}
Induction and the property \eqref{eq:EqualsSplicingImage} in
Theorem \ref{thm:ExistenceOfSplicedEndsModuli} imply
that for each $P\in\sP\less\sP'$, there is an open
neighborhood $\sO^{\asd}_1(\Theta,\sP'_P,\delta_P)$ of
$T(\Theta,\sP'_P,\delta)$ in
$\sO^{\asd}(\Theta,\sP'_P,\delta)$ such that
$$
\bga'_{\Theta,\sP'_P}\left( \sO^{\asd}_1(\Theta,\sP'_P,\delta_P)\right) \subset
\barM^{s,\natural}_{spl,|P|}(\delta_P).
$$
Thus, if in the definition of $\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$,
\eqref{eq:SplicedModuliOverlapR4},
we replace the factor
$$
\Delta^\circ(Z_P(\delta_P),\sP_P'')
            \times
            \prod_{Q\in\sP_P''} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
$$
with the open subspace
$$
\sO^{\asd}_1(\Theta,\sP'_P,\delta_P)
 \subset
\Delta^\circ(Z_P(\delta_P),\sP_P'')
            \times
            \prod_{Q\in\sP_P''} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
$$
the resulting open subspace $\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$
of $\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$ will satisfy
\begin{equation}
\label{eq:RestrictedUpwardsTransitionInclusion}
\rho^{\Theta,d}_{\sP,[\sP']}
\left(\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)\right)
\subset
\sO^{\asd}(\Theta,\sP,\delta).
\end{equation}
(Note that we are not concerned with the subsets $P\in\sP\cap\sP'$
because $\rho^{\Theta,d}_{\sP,\sP'}$ is the identity on these factors
and thus they do not affect the inclusion
\eqref{eq:RestrictedUpwardsTransitionInclusion}.)

That the restriction of $\rho^{\Theta,d}_{\sP,[\sP']}$
to $\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$ is
an open embedding into $\sO^{\asd}(\Theta,\sP,\delta)$
follows from the assumption that
$\bga'_{\Theta,\sP'_P}$ gives an open embedding of
$\sO^{\asd}_1(\Theta,\sP'_P,\delta_P)$ into
$\barM^{s,\natural}_{spl,|P|}(\delta_P)$.
\end{proof}


We now prove an analogy of Lemma \ref{lem:SplicedEndsEmbeddingR4d}
for the map $\rho^{\Theta,u}_{\sP,[\sP']}$.
First, we must define the appropriate range of $\rho^{\Theta,u}_{\sP,[\sP']}$,
to take into account the appearance of the conjugate partitions
$\sP''\in [\sP<\sP']$.
Let
\begin{equation}
\label{eq:UpperPartitionR4ASDNormal}
\sO^{\asd}(\Theta,[\sP<\sP'],\delta)
=
\left(\coprod_{\sP''\in [\sP<\sP']}\sO^{\asd}(\Theta,\sP'',\delta)
\right)
/\fS(\sP)
\subset
\nu(\Theta,[\sP<\sP'],\delta),
\end{equation}
where $\nu(\Theta,[\sP<\sP'],\delta)$
is defined in \eqref{eq:SymmetricProductNormalBundlesOfHigherStratum}.
Then,

\begin{lem}
\label{lem:SplicedEndsEmbeddingR4u}
Continue the assumptions and hypotheses of Lemma
\ref{lem:SplicedEndsEmbeddingR4d}.
Then the restriction of the map
$\rho^{\Theta,u}_{\sP,[\sP']}$
 to $\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$
is an open embedding into $\sO^{\asd}(\Theta,[\sP<\sP'],\delta)$.
\end{lem}

\begin{proof}
The lemma follows immediately from the definition
of  $\rho^{\Theta,u}_{\sP,[\sP']}$ because,
being defined largely by the exponential map $e(Z_\ka,\sP)$
(see \eqref{eq:R4TrivialSplicingUpwardsOverlapMap}),
the map $\rho^{\Theta,u}_{\sP,[\sP']}$ does not change the connections
in the domain $\sO(\Theta,\sP,[\sP'],\delta)$.
\end{proof}


The following lemma shows the existence of suitably small,
$\SO(3)\times\SO(4)$ invariant
neighborhoods of $T(\Theta,\sP,\delta)$.


\begin{lem}
\label{lem:SeparatingNgh}
For any subspace $V$ of $\bar{\sB}^s_{\ka}$, let
$\cl(V)$ denote the closure of $V$ in the Uhlenbeck topology.
If $V$ is any  set in
$\bar{\sB}^s_{\ka}$ with satisfying
$$
\cl(V) \cap \bga'_{\Theta,\sP}\left(
T(\Theta,\sP,\delta)\right)=\emptyset,
$$
then there is an $\SO(3)\times\SO(4)\times \fS(\sP)$ invariant
neighborhood, $\sO^{\asd}_1(\Theta,\sP,\delta)$, of $T(\Theta,\sP,\delta)$ in
$\sO^{\asd}(\Theta,\sP,\delta)$ such that the intersection
$$
\cl(V) \cap \bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right)
$$
is empty and the inclusion,
$$
\cl(V) \cap \cl
  \left(
    \bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right)
  \right)
\subset \cl(V) \cap \cl
  \left(
     \bga'_{\Theta,\sP}\left( T(\Theta,\sP,\delta)\right)
  \right),
$$
is satisfied.
\end{lem}

\begin{proof}
For $\bA\in \sO^{\asd}(\Theta,\sP,\delta)$ given by $\bA=\left(
(x_P)_{P\in\sP},([A_P,F^s_P])_{P\in\sP}\right)$, let
$\la_P(\bA)=\la(A_P)$. For $\bx=(x_P)_{P\in\sP}\in
\Delta^\circ(Z_\ka,\sP)$, define
$$
V(\bx,\sP):= \{\left( \bx, (A_P)_{P\in\sP}
  \right)
  \in (\bga'_{\Theta,\sP})^{-1}\left(\cl( V) \right)\}
$$
and $\bla_{V}:\Delta^\circ(Z_\ka,\sP)\to (0,\infty)$ by
$$
\bla_{V}(\bx) = \min_{\bA\in
V(\bx,\sP)}\sum_{P\in\sP}\la_P^2(\bA).
$$
By assumption $\bla_{V}$ is non-zero on $\Delta^\circ(Z_\ka(\delta),\sP)$.
Let $f:\Delta^\circ(Z_\ka(\delta),\sP)\to (0,\eps)$ be a continuous
function with $f(\bx)< \half \bla_{V}(\bx)$, and
\begin{equation}
\label{eq:fBoundedByDistance}
f\left( (x_P)_{P\in\sP}\right) < \half \min_{P_1\neq P_2\in\sP}
|x_{P_1}- x_{P_2}|.
\end{equation}
Then we define the neighborhood $\sO^{\asd}_1(\Theta,\sP,\delta)$ by
$$
\sO^{\asd}_1(\Theta,\sP,\delta):= \{\bA=\left
(\bx,([A_P,F^s_P])_{P\in\sP}\right)\in\sO^{\asd}(\Theta,\sP,\delta):
  \sum_{P\in\sP} \la_P^2(\bA)< f(\bx)\}.
$$
This set has the desired invariance and its image does not
intersect $\cl(V)$.

If $\{\bA(\alpha)\}_{\alpha=1}^{\infty}\subset\sO^{\asd}_1(\Theta,\sP,\delta)$,
with
$$
\bA(\alpha)=[ \bx(\alpha), ([A_P(\alpha),F^s_P(\alpha)])_{P\in\sP}],
$$
where $\bx(\alpha)\in \Delta^\circ(Z_\ka,\sP)$ is the splicing points
of $\bA(\alpha)$ and if
$$
\lim_{\alpha\to\infty} \bga'_{\Theta,\sP}\left( \bA(\alpha)\right)
\in \cl(V),
$$
then
$$
\lim_{\alpha\to\infty} \bx(\alpha)\notin \Delta^\circ(Z_\ka,\sP).
$$
The inequality \eqref{eq:fBoundedByDistance} then implies that
$$
\lim_{\alpha\to\infty} f(\bx(\alpha))=0,
$$
so the condition
$$
\sum_{P\in\sP} \la_P^2(\bA(\alpha))< f(\bx(\alpha))
$$
in the definition of $\sO^{\asd}_1(\Theta,\sP,\delta)$ implies that
for all $P\in\sP$
$$
\lim_{\alpha\to\infty} \la (A_P(\alpha))=0,
$$
so $\lim_{\alpha\to\infty}A_P(\alpha)=[\Theta,c_{|P|}]$,
giving the final conclusion of the lemma.
\end{proof}

We now construct the subspaces $\sO^{\asd}_1(\Theta,\sP,\delta)$
referred to in Proposition \ref{prop:ExistenceOfSplicedEnd}.
We will construct these subspaces to be $\fS_\ka$ invariant
in the sense that if $\sP''\in [\sP]$, then the open
subspaces $\sO^{\asd}_1(\Theta,\sP,\delta)$ and $\sO^{\asd}_1(\Theta,\sP'',\delta)$
are related by the
are identified by the natural
action of $\fS_\ka$ on $\coprod_{\sP_i\in[\sP_1]}\nu(\Theta,\sP_i)$.
Thus, defining $\sO^{\asd}_1(\RR^4,\sP')$ for one $\sP'\in [\sP']$
suffices to define the space
\begin{equation}
\label{eq:DefineSymmetricQuotientOfR4UpperASDNgh}
\sO^{\asd}_1(\Theta,[\sP<\sP'])
=
\left(
\coprod_{\sP''\in [\sP<\sP']} \sO^{\asd}_1(\Theta,\sP'',\delta)
\right)\fS(\sP)
\subset
\sO^{\asd}(\Theta,[\sP<\sP'],\delta),
\end{equation}
where $\sO^{\asd}(\Theta,[\sP<\sP'],\delta)$ is defined in
\eqref{eq:UpperPartitionR4ASDNormal}.

\begin{lem}
\label{lem:ConstructCollar}
Assume $\barM^{s,\natural}_{spl,\ka'}(\delta)$
has been constructed satisfying the properties enumerated in
Theorem \ref{thm:ExistenceOfSplicedEndsModuli} for all
$\ka'<\ka$. Then, to every partition $\sP$ of $N_\ka$ there is an
open neighborhood $\sO^{\asd}_1(\Theta,\sP,\delta)$ of $T(\Theta,\sP,\delta)$ in
$\sO^{\asd}(\Theta,\sP,\delta)$, such that
\begin{enumerate}
\item
These neighborhoods are closed  under the
$\SO(3)\times\SO(4)$ action given in Lemma
\ref{eq:SO3SO4Actions},
\item
If $\sP_1$ and $\sP_2$ are conjugate under the action of
$\fS_\ka$, then the neighborhoods $\sO^{\asd}_1(\Theta,\sP_1,\delta)$
and $\sO^{\asd}_1(\Theta,\sP_2,\delta)$  are identified by the natural
action of $\fS_\ka$ on $\coprod_{\sP_i\in[\sP_1]}\nu(\Theta,\sP_i)$,
\item
If $\sP\not < \sP'$ and $\sP'\not <\sP$, then
$$
\bga'_{\Theta,\sP}(\sO^{\asd}_1(\Theta,\sP,\delta))
  \cap
\bga'_{\Theta,\sP'}(\sO^{\asd}_1(\Theta,\sP',\delta))=\emptyset,
$$
\item
If $\sP< \sP'$ then there is an open neighborhood
$\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$ of the space
$T(\Theta,\sP,[\sP'],\delta)$ in
$\sO^{\asd} (\Theta,\sP,[\sP'],\delta)$ such that
\begin{equation}
\label{eq:CompleteOverlapsR4}
\begin{aligned}
\bga'_{\Theta,\sP} & \left( \sO^{\asd}_1(\Theta,\sP,\delta)\right) \cap
\bga'_{\Theta,\sP'}\left( \sO^{\asd}_1(\RR^4,\sP')\right)
\\
&\subseteq \bga'_{\Theta,\sP'}\left(
\rho^{\Theta,u}_{\sP,[\sP']}(\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta))\right)
\\
&=
\bga'_{\Theta,\sP}\left(
\rho^{\Theta,d}_{\sP,[\sP']}(\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta))\right)
\end{aligned}
\end{equation}
\end{enumerate}
\end{lem}

\begin{proof}
By induction, assume the open sets $\sO^{\asd}_1(\Theta,\sP,\delta)$ have been
defined for all $[\sP]<\sP'$.
Note that the conclusion still holds if we
shrink the sets $\sO^{\asd}_1(\Theta,\sP,\delta)$.
We now construct $\sO^{\asd}_1(\Theta,\sP',\delta)$.

It suffices to construct $\sO^{\asd}_1(\Theta,\sP',\delta)$ for
a single $\sP'\in [\sP']$ because for another
element, $\sP''\in [\sP']$, one can define $\sO^{\asd}_1(\Theta,\sP'',\delta)$
to be the image of $\sO^{\asd}_1(\Theta,\sP',\delta)$ under the action of
$\fS_\ka$.


If $\sP\not <\sP'$ and $\sP'\not < \sP$, then
we  assume $\sO^{\asd}_1(\Theta,\sP,\delta)$ has already been constructed.
If $\sO^{\asd}_1(\Theta,\sP,\delta)$ has not already been constructed, then
this step will appear later when the induction reaches $\sP$.
The relation $\sP\not <\sP'$ and $\sP'\not < \sP$ implies that
\begin{align}
\label{eq:UnrelatedTrivIntersection1} \cl
\left(\bga'_{\Theta,\sP}\left( T(\Theta,\sP,\delta)\right)\right)
\cap \bga'_{\Theta,\sP'}\left( T(\Theta,\sP',\delta)\right) & =
\emptyset, \\
\label{eq:UnrelatedTrivIntersection2} \bga'_{\Theta,\sP}\left(
T(\Theta,\sP,\delta)\right)) \cap \cl\left(
\bga'_{\Theta,\sP'}\left( T(\Theta,\sP',\delta)\right)\right) &=
\emptyset .
\end{align}
Lemma \ref{lem:SeparatingNgh} and
\eqref{eq:UnrelatedTrivIntersection1} imply there is an
$\SO(3)\times\SO(4)$ invariant
neighborhood $\sO^{\asd}_1(\Theta,\sP',\delta)$ such that
$$
\cl
 \left(  \bga'_{\Theta,\sP}\left( T(\Theta,\sP,\delta)\right)\right)
\cap \bga'_{\Theta,\sP'}\left( \sO^{\asd}_1(\Theta,\sP',\delta)\right) =\emptyset.
$$
Lemma \ref{lem:SeparatingNgh} also implies that
\begin{align}
\notag \cl& \left(
      \bga'_{\Theta,\sP}\left( T(\Theta,\sP,\delta)\right)
    \right)
\cap \cl \left(
      \bga'_{\Theta,\sP'}\left( \sO^{\asd}_1(\Theta,\sP',\delta)\right)
    \right)
\\
\label{eq:UnrelatedClosureContainment} &\subseteq \cl \left(
      \bga'_{\Theta,\sP}\left( T(\Theta,\sP,\delta)\right)
    \right)
\cap \cl \left(
      \bga'_{\Theta,\sP'}\left( T(\Theta,\sP',\delta)\right)
    \right).
\end{align}
Equations \eqref{eq:UnrelatedClosureContainment} and
\eqref{eq:UnrelatedTrivIntersection2} imply
\begin{align*}
 \bga'_{\Theta,\sP}& \left( T(\Theta,\sP,\delta)\right)
\cap \cl \left(
      \bga'_{\Theta,\sP'}\left( \sO^{\asd}_1(\Theta,\sP',\delta)\right)
    \right) \\
& \subset
 \bga'_{\Theta,\sP}  \left( T(\Theta,\sP,\delta)\right)
\cap \cl \left(
      \bga'_{\Theta,\sP'}\left( T(\Theta,\sP',\delta)\right)
    \right)=\emptyset.
\end{align*}
Thus,  Lemma \ref{lem:SeparatingNgh} implies there is a
(smaller) $\SO(3)\times\SO(4)$ invariant
neighborhood $\sO^{\asd}_2(\RR^4,\sP)
\subset \sO^{\asd}_1(\Theta,\sP,\delta)$ of $T(\Theta,\sP,\delta)$ with
$$
 \bga'_{\Theta,\sP}\left( \sO^{\asd}_2(\RR^4,\sP)\right)
\cap \cl \left(
      \bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right)
    \right)=\emptyset
$$
as desired.  Replace the open neighborhood $\sO^{\asd}_1(\Theta,\sP,\delta)$
with $\sO^{\asd}_2(\RR^4,\sP)$.
For all other $\sP''\in[\sP]$, replace $\sO^{\asd}_1(\Theta,\sP'',\delta)$
with the image of $\sO^{\asd}_2(\RR^4,\sP)$  under the action of
$\fS_\ka$.  This completes the
case of $\sP\not <\sP'$ and $\sP'\not < \sP$ in the induction.

If $[\sP]<\sP'$, by conjugating $\sP$ if necessary and using the
invariance of the neighborhoods $\sO^{\asd}_1(\Theta,\sP,\delta)$ under the
action of $\fS_{\ka}$, we can assume $\sP<\sP'$.
By Lemma \ref{lem:ScaleLimit}, the closure of
$$
[\Theta] \times \left( \Si(Z_\ka(\delta),\sP') - \Imag
(e(Z_\ka,\sP)) \right),
$$
(where $e(Z_\ka,\sP)$ is the exponential map defined in
\eqref{eq:R4Exponential})
does not intersect $\bga'_{\Theta,\sP}(T(\Theta,\sP,\delta))$.
Thus, applying Lemma \ref{lem:SeparatingNgh} and shrinking
$\sO^{\asd}_1(\Theta,\sP,\delta)$, we can assume
$$
\cl \left(
\bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right)
    \right)
\cap
\left(
[\Theta] \times \left( \Si(Z_\ka,\sP'') - \Imag
(e(Z_\ka,\sP)) \right)
\right)
=\emptyset
$$
for all $\sP''\in [\sP<\sP']$
and thus
\begin{equation}
\label{eq:ClosureLowerOpenLessTrivOverlap}
\begin{aligned}
{}&\cl \left(
\bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right)
    \right)
\cap
\left(
\left(
\cup_{\sP''\in [\sP<\sP']}
    \bga'_{\Theta,\sP'}( T(\Theta,\sP'',\delta))
\right)/\fS(\sP)\right)
\\
&\quad
\Subset [\Theta] \times
e(Z_\ka(\delta),\sP)\left(\sO(Z_\ka(\delta),[\sP<\sP'])\right),
\end{aligned}
\end{equation}
where $\sO(Z_\ka(\delta),[\sP<\sP'])$ is defined in
Lemma \ref{lem:SymmProdNeigh}.
Let $T(\Theta,\sP,[\sP'],\delta)$
and
$\sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)$
be the sets defined in \eqref{eq:TrivialStrataOverlapSymmQuotient}.
The equality and inclusion
\begin{align*}
[\Theta] & \times e(Z_\ka,\sP)\left(\sO(Z_\ka(\delta),[\sP<\sP'])\right)
=
\bga'_{\Theta,[\sP<\sP']}
  \left(
    \rho^{\Theta,u}_{\sP,[\sP']}\left( T(\Theta,\sP,[\sP'],\delta)\right)
  \right)
\\
& \subset \bga'_{\Theta,[\sP<\sP']}
  \left(
    \rho^{\Theta,u}_{\sP,[\sP']}\left( \sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)\right)
  \right),
\end{align*}
together with \eqref{eq:ClosureLowerOpenLessTrivOverlap}
then imply that
\begin{equation}
\label{eq:LowerR4SplicingMinusOverlap}
\cl \left( \bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right)
    \right)
\less \bga'_{\Theta,\sP'}
  \left(
    \rho^{\Theta,u}_{\sP,\sP'}\left( \sO^{\asd}_1(\Theta,\sP,\sP',\delta)\right)
  \right)
\end{equation}
is disjoint from $\bga'_{\Theta,\sP''}(T(\Theta,\sP'',\delta))$ for all
$\sP''\in [\sP<\sP']$.
Lemma \ref{lem:SeparatingNgh} gives an
$\SO(3)\times\SO(4)\times\fS(\sP)$ invariant neighborhood
$\sO^{\asd}_1(\Theta,[\sP<\sP'],\delta)$
of
$$
\left(\coprod_{\sP''\in [\sP<\sP']}T(\Theta,\sP'',\delta)\right)/\fS(\sP)
\quad\text{in}\quad
\left(\coprod_{\sP''\in [\sP<\sP']}\sO^{\asd}(\Theta,\sP'',\delta)\right)/\fS(\sP)
$$
whose image under $\bga'_{\Theta,[\sP<\sP']}$ is
disjoint from the last difference
\eqref{eq:LowerR4SplicingMinusOverlap}.
This disjointess implies
\begin{align*}
{}&
\bga'_{\Theta,\sP}\left( \sO^{\asd}_1(\Theta,\sP,\delta)\right) \cap
\bga'_{\Theta,\sP''}\left( \sO^{\asd}_1(\Theta,\sP'',\delta)\right)
\\
{}&\quad\subset
\bga'_{\Theta,[\sP<\sP']}
  \left(
    \rho^{\Theta,u}_{\sP,[\sP']}\left( \sO^{\asd}_1(\Theta,\sP,[\sP'],\delta)\right)
  \right),
\end{align*}
for all
$\sP''\in [\sP<\sP']$ and so
as required in the conclusion of the lemma.
\end{proof}


\begin{proof}[Proof of Proposition \ref{prop:ExistenceOfSplicedEnd}]
For each partition $\sP$ of $N_\ka$, let $\sO^{\asd}_1(\Theta,\sP,\delta)$
be the $\SO(3)\times\SO(4)$ invariant subspace of $\sO^{\asd}(\Theta,\sP,\delta)$
constructed in Lemma \ref{lem:ConstructCollar}.
For any two partitions $\sP$ and $\sP'$ the images
$\bga'_{\Theta,\sP}(\sO^{\asd}_1(\Theta,\sP,\delta))$
and
$\bga'_{\Theta,\sP'}(\sO^{\asd}_1(\Theta,\sP',\delta))$
are either disjoint or, by the last item in Lemma \ref{lem:ConstructCollar},
are given by:
$$
\bga'_{\Theta,\sP}
\left(
(\rho^{\Theta,d}_{\sP,[\sP']})^{-1}(\sO^{\asd}_1(\Theta,\sP,\delta)
\right).
$$
Thus, the overlaps are open subspaces and the union of the images
$W_\ka$ has the desired properties.  The estimates on $F^+_A$ in
the proposition follow from the arguments in \cite{FLKM1}, details
will appear in \cite{FL3,FL4}.
\end{proof}

\subsection{Tubular neighborhoods of the spliced end moduli space}
\label{subsec:TMOfSplicedEnd}
We now introduce the tubular neighborhood structure of the spliced
end $W_\ka$ defined in the previous section.  This structure
will satisfy the conditions discussed in the beginning of
\S \ref{sec:diagTM}.
In this lemma, we introduce the projection map.

\begin{lem}
\label{lem:TMProjectionOnSplicedEndR4}
For each partition $\sP$ of $N_\ka$ with
$|\sP|>1$, let $\sU(\Theta,\sP)\subset W_\ka$
be the image of the open subspace $\sO^{\asd}_1(\Theta,\sP,\delta)$
defined in Lemma \ref{lem:ConstructCollar}.
There is an $\SO(3)\times\SO(4)$ equivariant map
\begin{equation}
\label{eq:SplicedEndR4Projection}
\pi(\Theta,\sP):
\sU(\Theta,\sP) \to \{[\Theta]\}\times\Si(Z_\ka(\delta),\sP),
\end{equation}
satisfying
\begin{enumerate}
\item
If $\sP'\in [\sP]$, then
$\sU(\Theta,\sP)=\sU(\Theta,\sP')$,
$\Si(Z_\ka(\delta),\sP)=\Si(Z_\ka(\delta),\sP')$, and
$\pi(\Theta,\sP)=\pi(\Theta,\sP')$.
\item
On the overlap $\sU(\Theta,\sP)\cap\sU(\Theta,\sP')$,
\begin{equation}
\label{eq:R4SplicedEndProjectionTM}
\pi(\Theta,\sP)\circ \pi(\Theta,\sP')=\pi(\Theta,\sP)
\end{equation}
\end{enumerate}
\end{lem}

\begin{proof}
The map $\pi(\Theta,\sP)$ is defined to be the
composition of the projection
$$
\sO^{\asd}_1(\Theta,\sP,\delta)\to\Si(Z_\ka,\sP),
$$
with the inverse of the splicing map $\bga'_{\Theta,\sP}$.
The equality \eqref{eq:R4SplicedEndProjectionTM} follows
from \eqref{eq:CompleteOverlapsR4}.
\end{proof}

The obvious identification
\begin{equation}
\label{eq:TrivStratIdent}
\{[\Theta]\}\times Z_\ka/\fS_\ka
\cong
Z_\ka/\fS_\ka,
\end{equation}
allows us to define the projection maps $\pi(Z_\ka,\sP)$
and functions $t(Z_\ka)$ defined in
\S \ref{subsec:R4Diags} as maps and functions on
the trivial strata of $W_\ka$.
We identify the restriction of the projection
$\pi(\Theta,\sP)$ with such a map in the following.

\begin{lem}
\label{lem:R4TrivialProjectionIdentification}
Under the identification \eqref{eq:TrivStratIdent},
the restriction of the projection map $\pi(\Theta,\sP)$
of Lemma \ref{lem:TMProjectionOnSplicedEndR4} to the
intersection
$$
\left(\{[\Theta]\}\times Z_\ka/\fS_\ka\right)
\cap
\sU(\Theta,\sP)
$$
is equal to the map $\pi(Z_\ka,\sP):U(Z_\ka,\sP)/\fS(\sP)\to
\Delta^\circ(Z_\ka,\sP)=\Si(Z_\ka,\sP)/\fS(\sP)$
defined in \eqref{eq:R4SplicedEndProjectionTM}.
\end{lem}

\begin{proof}
The lemma follows from noting that the restriction of the
splicing map $\bga'_{\Theta,\sP}$ to points where the connections being spliced
in are trivial is exactly the map $e(Z_\ka,\sP)$ of
\eqref{eq:NormalOfR4Diagonal}.
\end{proof}


Before defining the tubular distance functions,
we define a perturbation of the scale function
which is constant along the fibers of the
projections $\pi(\Theta,\sP)$.  This scale
shall be used in the construction of the tubular
distance function and in the definition of the link of
the reducible $\PU(2)$ monopoles.

\begin{lem}
\label{lem:TMTubularDistanceR4}
There are continuous maps
\begin{equation}
\label{eq:TMScaleR4}
\tilde\la_\ka: \barM^{s,\natural}_{spl,\ka}(\delta)\to [0,2\delta),
\end{equation}
and
\begin{equation}
t_r(\Theta,\sP): \sU(\Theta,\sP)\to [0,1],
\quad\text{and}\quad
t(\Theta,\sP): \sU(\Theta,\sP)\to [0,1].
\end{equation}
which are smooth on each stratum, $\SO(3)\times\SO(4)$
invariant, and satisfy for some neighborhoods
$\sU'(\Theta,\sP)\Subset\sU(\Theta,\sP)$ of
$\{[\Theta]\}\times\Si(Z_\ka(\delta),\sP)$
\begin{enumerate}
\item
The function $\tilde\la_\ka$ is equal to the scale function $\la$ on the
complement of $W_\ka$,
\item
Under the identification \eqref{eq:TrivStratIdent},
the restriction of $\tilde\la_\ka$ to the trivial strata
in \eqref{eq:TrivStratIdent} is equal to the function
$t(Z_\ka)$ defined in Lemma \ref{lem:R4TMScale},
\item
For all partitions $\sP$ of $N_\ka$ with $|\sP|>1$,
$\tilde\la_\ka=\tilde\la_\ka\circ\pi(\Theta,\sP)$ on
$\sU'(\Theta,\sP)$,
\item
If $\sP'\in [\sP]$ then,
$t_r(\Theta,\sP)=t(\Theta,\sP')$ and
$t(\Theta,\sP)=\tilde t(\Theta,\sP')$,
\item
On the overlap $\sU'(\Theta,\sP)\cap\sU'(\Theta,\sP')$,
\begin{equation}
\label{eq:R4SplicedEndTubularTM}
t_r(\Theta,\sP)\circ \pi(\Theta,\sP')=t_r(\Theta,\sP),
\quad\text{and}\quad
t(\Theta,\sP)\circ \pi(\Theta,\sP')=t(\Theta,\sP)
\end{equation}
\item
$t_r(\Theta,\sP)^{-1}(0)
=t(\Theta,\sP)^{-1}(0)
=\{\Theta\}\times\Si(Z_\ka(\delta),\sP)$,
\item
There are open neighborhoods
$$
U_1(\Theta,\sP)
\Subset
\sU_2(\Theta,\sP)
\Subset\sU(\Theta,\sP)
$$
of $\{\Theta\}\times\Si(Z_\ka,\sP)$, such that
\begin{equation}
\label{eq:TubularNeighborhoodsByFunction}
\begin{aligned}
\sU_1(\Theta,\sP)&=t_r(\Theta,\sP)^{-1}([0,\thalf)),
\\
\sU(\Theta,\sP) - \sU_2(\Theta,\sP)
&= t_r(\Theta,\sP)^{-1}(\{1\})
\end{aligned}
\end{equation}
\end{enumerate}
\end{lem}

\begin{proof}
The function $\tilde\la_\ka$ is constructed using induction
on $\ka$.  Thus, we assume the functions
$\tilde\la_{\ka'}$, $t_r(\Theta,\sP)$, and $t(\Theta,\sP)$
have been constructed for all $\ka'<\ka$ and all partitions
$\sP$ of $N_{\ka'}$.

Begin the induction with $\tilde\la_1=\la_1$.
First, observe that the restriction
of $\tilde\la_\ka$ to the trivial strata in $W_\ka$
appearing in \eqref{eq:TrivStratIdent} is
determined by item (2).  This is consistent with
item (3) by virtue of the identification of
$\pi(\Theta,\sP)$ in Lemma \ref{lem:R4TrivialProjectionIdentification}
and the equality in item (2) of Lemma \ref{lem:R4TMScale}.
Then construct $\tilde\la_\ka$ following the
pattern of the construction of the function
$t(Z_\ka)$ in Lemma \ref{lem:R4TMScale}
using $t_r(\Theta,\sP)$ in place of $t_r(Z_P,\sP)$.

For each partition $\sP$ of $N_\ka$,
the function $t(\Theta,\sP)$ is then defined by
\begin{equation}
\label{eq:DefineTubular1}
t(\Theta,\sP)=\sum_{P\in\sP}|P|\tilde\la_P^2,
\end{equation}
where the functions $\tilde\la_P$ are defined on
$\sO^{\asd}(\Theta,\sP,\delta)$ by the new scale
function $\tilde\la_{|P|}$ on the $S^4$ connection
being spliced in at $x_P$.

The equality \eqref{eq:R4SplicedEndProjectionTM} holds
for $t(\Theta,\sP)$ by the same argument used
to establish \eqref{eq:XTMTubularDistance}, using
the observation that for $\sP<\sP'$, the projection
$\pi(\Theta,\sP')$ on $\sU(\Theta,\sP)\cap\sU(\Theta,\sP')$
is given by the restriction of the projection
$$
\begin{CD}
\Delta^\circ(Z_\ka(\delta),\sP) \times
\prod_{P\in\sP}
\left(
    \Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)
    \times
    \prod_{P'\in\sP_P'} \barM^{s,\natural}_{spl,|P'|}(\delta_{P'})
\right)
\\
@VVV
\\
\Delta^\circ(Z_\ka(\delta),\sP)
\times
\prod_{P\in\sP}
\left(
    \Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)
    \times
    \prod_{P'\in\sP_P'} \{[\Theta,c_{P'}]\}
\right)
\end{CD}
$$
to the appropriate subspaces
(see \eqref{eq:CommutingSplicingQuotient} and
\eqref{eq:SplicedModuliOverlapR4}).



The function $t_r(\Theta,\sP)$ is then a
rescaling of the function $t(\Theta,\sP)$ in the sense of
Lemma \ref{lem:XTMTubularDistance}.  That is, $t_r(\Theta,\sP)$ is
defined by multiplying $t(\Theta,\sP)$ by a non-vanishing,
smooth function pulled back from $T(\Theta,\sP,\delta)$
by the map $\pi(\Theta,\sP)$.  The property
\eqref{eq:R4SplicedEndProjectionTM} of $\pi(\Theta,\sP)$ implies
that the pullback of a function $f$ by $\pi(\Theta,\sP)$ satisfies
$$
(f\circ\pi(\Theta,\sP))\circ \pi(\Theta,\sP')
=
f\circ \pi(\Theta,\sP)
$$
so defining $t_r(\Theta,\sP)$ by such a rescaling of
$t(\Theta,\sP)$ implies that if $t(\Theta,\sP)$
has the property \eqref{eq:R4SplicedEndProjectionTM} then
$t_r(\Theta,\sP)$ does as well.
\end{proof}


\subsection{The isotopy of the spliced end}
\label{subsec:Collar}
The final step in the proof of Theorem \ref{thm:ExistenceOfSplicedEndsModuli}
is to use the isotopy provided by the gluing and centering maps to
piece together the spliced end $W_\ka$ with the actual moduli space
$\barM^{s,\natural}_{\ka}(\delta)$.


We first construct the isotopy of $W_\ka$ in $\bar\sB^s_{\ka}(\delta)$.

\begin{lem}
\label{lem:SplicedEndIsotopy}
There is continuous, smoothly-stratified map
\begin{equation}
\label{eq:Isotopy}
R: (-\infty,1] \times W_\ka \to \bar\sB^s_\ka(\delta),
\end{equation}
such that
\begin{enumerate}
\item
For all $t\in (-\infty,\thalf]$
and all $[A,F^s,\bx]\in W_\ka$,
$R(t,[A,F^s,\bx])=[A,F^s,\bx]$,
\item
For all $t\in [\tthreequarter,1]$
and all $[A,F^s,\bx]\in W_\ka$,
$R(t,[A,F^s,\bx])\in\barM^s_\ka(\delta)$,
\item
For all $[A,F^s,\bx]\in W_\ka$,
$R(1,[A,F^s,\bx])\in \barM^{s,\natural}_\ka(\delta)$,
\item
For all $t\in (-\infty,1]$, the map $R(t,\cdot):W_\ka\to\bar\sB^s_\ka(\delta)$
is $\SO(3)\times\SO(4)$ equivariant.
\end{enumerate}
\end{lem}


\begin{proof}
The bound on the $L^{\sharp,2}$-norm of the $F^+_A$ for all
$[A,\bx]\in W_\ka$ implies (see \cite[Proposition 7.6]{FLKM1})
that there is a continuous, smoothly-stratified,
$\SO(3)\times\SO(4)$ equivariant embedding,
$G:W_\ka\to\barM^{s,\natural}(\delta)$
\begin{equation}
\label{eq:S4GluingMap}
G([A,F^s,\bx])=[A+a(A),F^s,\bx].
\end{equation}
For $t\in [\thalf,\tthreequarter]$,
we define
\begin{equation}
\label{eq:GluingIsotopy}
R(t,[A,F^s,\bx])=[A+4(t-\thalf)a(A),F^s,\bx].
\end{equation}
The remainder of the isotopy, for $t\in [\tthreequarter,1]$
is defined by the mass-centering map. That is, pulling any
$[A,F^s,\bx]$ back by an appropriate conformal diffeomorphism
ensures it is mass-centered (see \cite[\S 3.2.1]{FKLM}).
This defines an natural isotopy, parameterized by
$t\in [\tthreequarter,1]$,
of the image of $W_\ka$ under $R(\thalf,\cdot)$ into the space of
mass-centered connections.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:ExistenceOfSplicedEndsModuli}]
The function $1-t_r(\Theta,\sP)$ is compactly supported on
$\sU(\Theta,\sP)$ and thus defines an
$\SO(3)\times\SO(4)$-invariant function on $W_\ka$.
Observe that by \eqref{eq:TubularNeighborhoodsByFunction},
the map
\begin{equation}
\label{eq:IsotopyParameter}
t=1-\sum_{\sP} (1-t_r(\Theta,\sP)).
\end{equation}
satisfies
\begin{align*}
t(\cup_{\sP} \sU_1(\Theta,\sP)) &\subset (-\infty,\thalf],
\\
t\left(
\cap_{\sP}(W_\ka -\sU_2(\Theta,\sP))
\right)&=1.
\end{align*}
That is, if $[A,F^s,\bx]\in \sU_1(\Theta,\sP)$ for some $\sP$
then $t([A,F^s,\bx])<\thalf$ while if
$[A,F^s,\bx]\notin\sU_2(\Theta,\sP)$ for all $\sP$, then
$t([A,F^s,\bx])=1$.  Alternately, $t<\half$ on the
``interior'' of $W_\ka$, near the trivial strata while
$t$ is equal to one the ``exterior'', away from the trivial strata.
Define $R(W_\ka)$ to be the image of $W_\ka$ under the map $R(t(\cdot),\cdot)$:
$$
R(W_\ka)
=
\{R(t([A,F^s,\bx]),[A,F^s,\bx]): [A,F^s,\bx]\in W_\ka\}.
$$
Then if we define the spliced end moduli space by
$$
\barM^{s,\natural}_{spl,\ka}(\delta)
=
\left(
\barM^{s,\natural}_{\ka}(\delta)
-
R(1,W_\ka)
\right)
\cup
R(W_\ka),
$$
it will have the properties specified in Theorem
\ref{thm:ExistenceOfSplicedEndsModuli}.
\end{proof}


%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Article: INTERSECTION
%File: globalsplicingdata.tex
%Last update: October 22nd (Tom)
%Spellcheck:  February 18th, 2002 (Tom)
%Todo:

\section{The space of global splicing data}
\label{sec:GlobalSplicingData}
In this section, we construct the space of global splicing data.
This space will be the union of spaces of splicing data,
$\sO(\ft,\sP)$, associated to diagonals $\Si(X^\ell,\sP)$,
as $\sP$ varies over partitions of $N_{\ell}$.  (Of course,
we will do this equivariantly with respect to the symmetric group.)
To each partition $\sP$, we will define a
{\em crude splicing map\/}
which  is identical to the splicing map
defined in \cite{FL3} except
\begin{enumerate}
\item
The connections on $S^4$ being spliced in
are elements of the
spliced ends moduli space rather than the moduli space
of anti-self-dual connections on $S^4$,
\item
The background pair, $(A_0,\Phi_0)$, are ``flattened''
(i.e. the connection $A_0$ is replaced with one which is
flat while the section $\Phi_0$ is multiplied by a vanishing cut-off
function)
on balls of fixed radius around the splicing point rather
than balls whose radius vanishes as the connection on $S^4$
becomes more concentrated.
\end{enumerate}
This last difference is necessary to mimic the result which
holds when splicing connections to the trivial connection on
$\RR^4$ (Proposition \ref{prop:CommutingSplicingSymmQuotient}),
allowing us to control the overlap of the images of the crude
splicing map as we did when constructing the spliced end $W_\ka$
by defining a space of overlap data and overlap maps.
With this control of the overlaps of the images of the crude splicing
maps, we can define
the space of global splicing data is then the union of the images
of the crude splicing maps.

After constructing the domain of the splicing map in
\S \ref{subsec:SplicingData}, we show how to flatten
pairs as described above in \S \ref{subsec:FlatteningPairs}
which defines the crude splicing map
in \S \ref{subsec:CrudeSplicingMap}.
With the understanding of the overlaps of the images
of the crude splicing maps developed in \ref{subsec:OverlapSpacesModuli},
we construct the space of global splicing data in
\S \ref{subsec:GlobalSplicingData}.
In \S \ref{subsec:TMStr}, we construct
the partial Thom-Mather structure which will be used in
the construction of the link $\bar\bL_{\ft,\fs}$.
In \S \ref{subsec:GlobalSplicingMap},
we produce a global splicing map whose image will
contain the reducible monopoles.
Finally, in \S \ref{subsec:ProjectionToSym},
we construct a projection map from the space
of global splicing data to $\Sym^\ell(X)$.


\subsection{The splicing data}
\label{subsec:SplicingData}
We begin by recalling the domains of the splicing maps from
\cite{FL3}.

\subsubsection{The background pairs}
Let $\tilde M_{\fs}\subset\tsC_{\ft_0}$ be the image of
the embedding of the solutions of the perturbation of
the Seiberg-Witten equations considered in \cite{FL2a}.
The gauge group quotient, $M_{\fs}=\tilde M_{\fs}/\sG_{\fs}$
embeds in $\sC_{\ft_0}$.
A virtual normal
bundle $N_{\ft_0,\fs}\to M_{\fs}$ with an embedding
of a $\delta$-disk subbundle
$N_{\ft_0,\fs}(\delta)\subset N_{\ft_0,\fs}\to \sC_{\ft_0}$,
was defined in \cite{FL2a}.
We will abbreviate $N_{\ft_0,\fs}(\delta)$ by $N(\delta)$.
If $\tilde N\to\tilde M_{\fs}$ is the pullback of $N_{\ft_0,\fs}$
by the projection $\tilde M_{\fs}\to M_{\fs}$, we showed
in \cite{FLLevelOne} that there is an embedding
$\tilde N(\delta)\to\tsC_{\ft_0}$ covering the embedding of
$N(\delta)\to \sC_{\ft_0}$.

\subsubsection{The metrics}
\label{subsubsec:FiberBundleMetrics}
For each partition $\sP$ of $N_\ell$, let $g_{\sP}$ be the smooth
family of metrics on $X$ parameterized by $\Delta^\circ(X^\ell,\sP)$
and
let $\tilde \sU(X^\ell,\sP)\subset X^\ell$ be the tubular neighborhood
of $\Delta^\circ(X^\ell,\sP)$
constructed in Lemma \ref{lem:FlatteningMetrics2}.
Recall that $\pi(X^\ell,\sP):\tilde\sU(X^\ell,\sP)\to\Delta^\circ(X^\ell,\sP)$
is the projection map of this tubular neighborhoods.
We list here the relevant properties:
\begin{enumerate}
\item
For all $\by\in\Delta^\circ(X^\ell,\sP)$, the metric
$g_{\sP,\by}$ is flat on the support of
$\tilde\sU(X^\ell,\sP)\cap \pi(X^\ell,\sP)^{-1}(\by)$,
\item
For $\sP<sP'$ partitions of $N_\ell$ and for
$\by'\in \tilde\sU(X^\ell,\sP)\cap\Delta^\circ(X^\ell,\sP')$
with $\pi(X^\ell,\sP)(\by')=\by$, the equality
$g_{\sP,\by}=g_{\sP,\by'}$ holds,
\item
The metrics are $\fS_\ell$ invariant in the sense that
$g_{\sP,\by}=g_{\si(\sP),\si(\by)}$ for all $\si\in\fS_\ell$.
\end{enumerate}

\subsubsection{The frame bundles}
For $\sP$ be a partition of $N_{\ell}$,
let
$$
\Fr(TX^\ell,\sP,g_\sP)\to \Delta^\circ(X^\ell,\sP),
$$
be the fiber bundle defined before \eqref{eq:DiagonalNormalAssocBundle}.
Similarly, for $\fg_0=\fg_{V_0}$ we define
$$
\Fr(\fg_0,\sP)\to\Delta^\circ(X^\ell,\sP),
$$
as
\begin{equation}
\label{eq:DefineDiagonalOfBackgroundFrameBundle}
\begin{aligned}
\Fr(\fg_0,\sP)
&=
\{(F^{\fg}_1,\dots,F^{\fg}_\ell)\in \prod_{i=1}^\ell \Fr(\fg_0):
\\
&\quad
F^{\fg}_i=F^{\fg}_j\ \text{if and only if there is $P\in\sP$ and $i,j\in P$}\}.
\end{aligned}
\end{equation}
Then, the gluing data bundle is:
\begin{equation}
\label{eq:DefineGluingDataBundle}
\Fr(\ft,\fs,\sP,g_{\sP})=
\Fr(TX,\sP,g_{\sP})\times_{\Delta^\circ(X^\ell,\sP)}\Fr(\fg_{\ft_0},\sP).
\end{equation}
Recall that we denote elements of $\Delta^\circ(X^\ell,\sP)$
as $(y_P)_{P\in\sP}$ where $y_P\in X$.
We will similarly denote elements of the bundle
$\Fr(\ft,\fs,\sP,g_{\sP})$ lying
over $(y_P)_{P\in\sP}$ as $(F^T_P,F^{\fg}_P)_{P\in \sP}$
where $F^T_P\in \Fr(TX)|_{y_P}$ and $F^{\fg}_P\in\Fr(\fg_{\ft_0})|_{y_P}$.
We also write $(F^{T,\fg}_P)_{P\in\sP}$ for such an element of
$\Fr(\ft,\fs,\sP,g_{\sP})$.

The structure groups of $\Fr(X,\sP)$ over
$\Delta^\circ(X^\ell,\sP)$ and over $\Si(X^\ell,\sP)$ are
$\tG(\sP)$ and $G(\sP)$ respectively where
\begin{equation}
\label{eq:DefineGluingDataBundleStructureGroup}
\begin{aligned}
\tG(\sP)& = \{\left((R_1,M_1),\dots,(R_\ell,M_\ell)\right)\in
\left( \SO(4)\times\SO(3)\right)^\ell:
\\
&\quad \text{for every $i,j\in P\in\sP$ $R_i=R_j$ and
$M_i=M_j$}\},
\\
G(\sP)&= \tG(\sP) \ltimes \fS(\sP).
\end{aligned}
\end{equation}
Observe that for $\sP<\sP'$, there is an inclusion,
\begin{equation}
\label{eq:StructureGroupInclusion}
\tilde G(\sP)\to \tilde G(\sP').
\end{equation}
We can write elements of $\tG(\sP)$ as $(R_P,M_P)_{P\in\sP}$.
With this notation the action of $G(\sP)$ on
$$
\prod_{P\in\sP} \barM^{s,\natural}_{spl,|P|}(\delta)
$$
is given by the factor $(R_P,M_P)$ acting by the
standard $\SO(3)\times\SO(4)$ action on
$\barM^{s,\natural}_{spl,|P|}(\delta)$ and the
natural permutation action of $\fS(\sP)$.



\subsubsection{The space of splicing data}

The splicing map will be defined on a subspace
$$
\tilde N(\delta)\times_{\sG_{\fs}}
\sO(\ft,\fs,\sP)
$$
where
\begin{equation}
\sO(\ft,\fs,\sP)
\subset
\Gl(\ft,\fs,\sP,g_{\sP})=
\Fr(\ft,\fs,\sP,g_{\sP})
\times_{G(\sP)}
\prod_{P\in\sP} \barM^{s,\natural}_{spl,|P|}(\delta_P),
\end{equation}
is an open neighborhood of
\begin{equation}
\label{eq:DefineTrivialStratumXConePoints}
\Si(\ft,\fs,\sP,g_{\sP})
=
\{
\left(
(F^{T,\fg}_P)_{P\in\sP},([\Theta,F^s_P,c_{|P|}])_{P\in\sP}
\right) \in\Gl(\ft,\fs,\sP,g_{\sP})\}
\end{equation}
where $c_{|P|}$ is the cone point of $\Sym^{|P|,\natural}(\RR^4)$.
The notation in \eqref{eq:DefineTrivialStratumXConePoints}
is motivated by the observation that because $\SO(3)\times\SO(4)$
acts trivially on $[\Theta,F^s_P,c_{|P|}]$, there is
an identification,
\begin{equation}
\label{eq:IdentifyConePointStratum}
\Si(\ft,\fs,\sP,g_{\sP})
\cong
\Si(X^\ell,\sP).
\end{equation}
We also define a larger subspace of $\Gl(\ft,\fs,\sP,g_{\sP})$ by:
\begin{equation}
\label{eq:DefineTrivialStratumX}
T(\ft,\fs,\sP,g_{\sP})
=
\{
\left(
(F^{T,\fg}_P)_{P\in\sP},([\Theta,F^s_P,\bv_P])_{P\in\sP}
\right) \in\Gl(\ft,\fs,\sP,g_{\sP})\}.
\end{equation}



One requirement in the definition of $\sO(\ft,\fs,\sP)$ will be that
for any point,
$$
\bA=
\left((A_0,\Phi_0)
(F^T_P,F^{\fg}_P)_{P\in\sP},([A_P,F^s_P,\bv_P])_{P\in\sP}
\right)
\in\sO(\ft,\fs,\sP)
$$
where $F^T_P$ and $F^{\fg}_P$ lie over $y_P\in X$
that
\begin{equation}
8\la([A_P,F^s_P,\bv_P]) < \min_{P'\in\sP,P'\neq P} d(y_P,y_P')
\end{equation}
for all $P\in\sP$.  Finally, we define
\begin{equation}
\label{eq:DefineSplicingDataToDiagonalProjection}
\pi_{\Si}: \tilde N(\delta)\times_{\sG_{\fs}}\Gl(\ft,\fs,\sP,g_{\sP})
 \to \Si(X^\ell,\sP)
\end{equation}
to be the obvious projection.  The $S^1$ action on $\tilde N_{\ft_0,\fs}$
defined in \ref{sec:prelim} defines an $S^1$ action on
$$
\tilde N(\delta)\times_{\sG_{\fs}}\Gl(\ft,\fs,\sP,g_{\sP})
$$
and we will see that all the constructions of this section
are equivariant with respect to this $S^1$ action.
See \cite{FLLevelOne} for different ways of presenting this $S^1$
action.

\subsection{Flattening pairs}
\label{subsec:FlatteningPairs}
The commutativity of the diagram \eqref{eq:CommutingSplicingQuotient}
for the spliced ends moduli space
depended on the flatness of the metric on $\RR^4$ and on
the flatness of the trivial connection $\Theta$.
The locally flat metrics $g_{\sP}$ described in
\S \ref{subsubsec:FiberBundleMetrics} have this property.
We now introduce a method for achieving the same kind of
local flatness for the background connection $A_0$.
This method also yields a way of cutting off the background section
consistently from one stratum to another.

\begin{defn}
\label{defn:SeparatingFunctions}
A collection of smooth functions $s_P:\Delta^\circ(X^\ell,\sP)\to (0,1)$,
indexed by $P\in\sP$ are a {\em separating family\/} if
\begin{enumerate}
\item
The functions are $\fS(\sP)$ invariant in the sense that
$s_{P}=s_{\si(P)}$  and $s_{P}=s_P\circ\si$, for all $\si\in\fS(\sP)$,
\item
For all $\by=(y_P)_{P\in\sP}\in\Delta^\circ(X^\ell,\sP)$
and for all $P\neq P'\in\sP$, the balls $B(y_P,4s_P(\by)^{1/2})$
and $B(y_{P'},4s_{P'}(\by)^{1/2})$ are disjoint.
\end{enumerate}
\end{defn}

\begin{defn}
Let $\{s_P\}$ be a separating family of smooth functions on
$\Delta^\circ(X^\ell,\sP)$.
A connection $\hat A_0$ is {\em flat with respect to $\{s_P\}$\/}
at $\by=(y_P)_{P\in\sP}\in\Delta^\circ(X^\ell,\sP)$
if the connection $\hat A_0$ is flat on
\begin{equation}
\label{eq:SeparatingFunctionBalls1}
\cup_{P\in\sP}\ B(y_P,2s_P(\by)^{1/2}).
\end{equation}
A pair $(A,\Phi)\in\sC_{\ft_0}$ is {\em flat with respect to $\{s_P\}$\/}
if $\hat A$ is flat with respect to $\{s_P\}$ and $\Phi$ vanishes
on
\begin{equation}
\label{eq:SeparatingFunctionBalls2}
\cup_{P\in\sP}\ B(y_P,4s_P(\by)^{1/3}).
\end{equation}
\end{defn}

The following lemma is the analogue of the flattening construction for
metrics given in Lemma \ref{lem:FlatteningMetrics}.

\begin{lem}
\label{lem:FlatteningPairs}
Let $\sP$ be a partition of $N_\ell$.
Let $\{s_P\}$ be a separating family of smooth functions on
$\Delta^\circ(X^\ell,\sP)$. Then,
there is an $S^1$-equivariant map,
\begin{equation}
\label{eq:DefineFlatteningMap}
\bTheta_{\sP}':
\tN(\delta)\times \Delta^\circ(X^\ell,\sP)
\to
\tsC_{\ft_0}
\end{equation}
which is $\sG_{\fs}$-equivariant and descends to the quotient
$$
N(\delta)\times \Delta^\circ(X^\ell,\sP)
\to
\sC_{\ft_0}
$$
In addition, for all $\by=(y_P)_{P\in\sP}\in\Delta^\circ(X^\ell,\sP)$,
if
$(A',\Phi')=\bTheta'_{\sP}((A_0,\Phi_0),\by)$ then
\begin{enumerate}
\item
The connection $\hat A'$ is flat on
$$
\cup_{P\in\sP} B(y_P,2s_P(\by)^{1/2}),
$$
and equal to $A_0$ on the complement of
\begin{equation}
\label{eq:ConnToBeFlat}
\cup_{P\in\sP} B(y_P,4s_P(\by)^{1/2})
\end{equation}
\item
The section $\Phi'$ vanishes on
$$
\cup_{P\in\sP} B(y_P,4s_P(\by)^{1/3})
$$
and is equal to $\Phi$ on the complement of
\begin{equation}
\label{eq:SectionToVanish}
\cup_{P\in\sP} B(y_P,8s_P(\by)^{1/3}).
\end{equation}
\end{enumerate}
If the connection $\hat A_0$ is already flat on
the space \eqref{eq:ConnToBeFlat}, then $A'=A_0$.
If the section $\Phi$ already vanishes on
the space \eqref{eq:SectionToVanish}, then $\Phi'=\Phi_0$.
\end{lem}

\begin{proof}
The map $\bTheta_{\sP'}$ is
initially defined as a map,
$$
\bTheta_{\sP}':\tilde N(\delta)\times\Fr(\ft,\fs,\sP,g_{\sP})
\to
\tsC_{\ft_0}
$$
as described in \cite[\S 3.2]{FL3}.  Specifically, the frames and radial
parallel translation define a trivial connection on
the space \eqref{eq:ConnToBeFlat}.  If $\hat A_0$ is flat on
the space \eqref{eq:ConnToBeFlat}, this trivial connection is equal
to $\hat A_0$.  Then, use a cut-off function to interpolate
between the trivial connection and
$\hat A_0$ on the annuli around the points $y_P$, defining a new
connection $\hat A'$ which is flat on the inner balls and equal to
$\hat A_0$ outside the space \eqref{eq:ConnToBeFlat}.
Observe that the resulting connection $\hat A'$ is independent of the choice of
the frames used as it is equal to the connection obtained by splicing in
the trivial connection which has stabilizer $\SO(3)\times\SO(4)$.
Thus, the map $\bTheta_{\sP}'$ descends to the domain stated in the lemma.

The section $\Phi'$ is defined by multiplying $\Phi_0$ by a cut-off function
equal to on the space \eqref{eq:SectionToVanish} and vanishing on
the balls of radius $4s_P(\by)^{1/3}$.
\end{proof}

To define the crude splicing maps in such a way that an
analogue of Proposition \ref{prop:CommutingSplicingSymmQuotient}
controls the overlaps of their images,
we must redefine the flattening maps in such a way
that for $\by'\in\Delta^\circ(X^\ell,\sP')$ near
$\by\in\Delta^\circ(X^\ell,\sP)$ the flattened background pairs
$\bTheta_{\sP'}(A_0,\Phi_0,\by')=\bTheta_{\sP}(A_0,\Phi_0,\by)$.
(Compare the second property of the metrics $g_{\sP}$ appearing
in \S \ref{subsubsec:FiberBundleMetrics}.)
To this end, we introduce a refinement of the flattening
map of Lemma \ref{lem:FlatteningPairs}.

\begin{lem}
\label{lem:CompatibleFlattening}
Continue the assumptions of Lemma \ref{lem:FlatteningPairs}.
Then,
for every partition $\sP$ of $N_\ell$
there are separating functions $\{\tilde s_P\}$ and
is an $S^1$-equivariant map
\begin{equation}
\label{eq:CompatibleFlattening}
\bTheta_{\sP}: \tilde N(\delta)\times \Delta^\circ(X^\ell,\sP)
\to
\tsC_{\ft_0},
\end{equation}
which is $\fS(\sP)$-invariant,
$\sG_{\fs}$-equivariant and descends to the quotient
$$
N(\delta)\times \Delta^\circ(X^\ell,\sP)
\to
\sC_{\ft_0}
$$
such that if $(A',\Phi')=\bTheta_{\sP}(A_0,\Phi_0,\by)$
where $\by=(y_P)_{P\in\sP}\in\Delta^\circ(X^\ell,\sP)$
then
\begin{enumerate}
\item
The connection $\hat A'$ is flat on
$$
\cup_{P\in\sP} B(y_P,2\tilde s_P(\by)^{1/2}),
$$
\item
The section $\Phi'$ vanishes on
$$
\cup_{P\in\sP} B(y_P,4\tilde s_P(\by)^{1/3}).
$$
\end{enumerate}
Furthermore, if $\sP<\sP'$ and the tubular neighborhood
$\sU(X^\ell,\sP)$ is suitably small,
if $\by'\in\sU(X^\ell,\sP)\cap\Delta^\circ(X^\ell,\sP')$
and
$
\pi(X^\ell,\sP)(\by')=\by\in\Delta^\circ(X^\ell,\sP),
$
then
\begin{equation}
\label{eq:ConsistentFlattening}
\bTheta_{\sP}(A_0,\Phi_0,\by)=\bTheta_{\sP'}(A_0,\Phi_0,\by').
\end{equation}
\end{lem}

\begin{proof}
The proof is identical to the construction of the consistent
flat families of metrics in Lemma \ref{lem:FlatteningMetrics2}.
One constructs $\bTheta_{\sP}$ by induction on $\sP$.

For $\sP_0=\{N_\ell\}$ the
crudest partition of $N_\ell$,
define $\bTheta_{\sP_0}=\bTheta_{\sP_0}'$ (as constructed in
Lemma \ref{lem:FlatteningPairs}).

Assume $\bTheta_{\sP}$ has been constructed for all $\sP<\sP'$.
To construct $\bTheta_{\sP'}$, we first define
$$
\bTheta_{\sP,\sP'}:\tilde N(\delta)\times
\left(\sU(X^\ell,\sP\cap \Delta^\circ(X^\ell,\sP')\right)
\to
\tsC_{\ft_0},
$$
by
$$
\bTheta_{\sP',\sP}(A_0,\Phi_0,\by')
=\bTheta_{\sP}(A_0,\Phi_0,\pi(X^\ell,\sP)(\by')).
$$
Observe that if $\sP_1<\sP$ and $\sP_2<\sP$ and
$\sU(X^\ell,\sP_1)\cap\sU(X^\ell,\sP_2)$ is non-empty,
then we can assume $\sP_1<\sP_2<\sP$.  The Thom-Mather
condition on the projections $\pi(X^\ell,\sP)$,
\eqref{eq:DiagonalsTM1}:
$$
\pi(X^\ell,\sP_1)\circ\pi(X^\ell,\sP_2)=\pi(X^\ell,\sP_1),
$$
and the inductive assumption imply that for
$\by'\in\sU(X^\ell,\sP_1)\cap\sU(X^\ell,\sP_2)\cap\Delta^\circ(X^\ell,\sP')$,
\begin{align*}
\bTheta_{\sP',\sP_2}(A_0,\Phi_0,\by')
&=
\bTheta_{\sP_2}(A_0,\Phi_0,\pi(X^\ell,\sP_2)(\by'))
\\
&=
\bTheta_{\sP_1}(A_0,\Phi_0,\pi(X^\ell,\sP_1)\pi(X^\ell,\sP_2)(\by'))
\quad\text{By inductive hypothesis}
\\
&=
\bTheta_{\sP_1}(A_0,\Phi_0,\pi(X^\ell,\sP_1)(\by'))
\quad\text{By Thom-Mather property \eqref{eq:DiagonalsTM1}}
\\
&=
\bTheta_{\sP',\sP_1}(A_0,\Phi_0,\by').
\end{align*}
By the preceding equality, we can define
$$
\bTheta_{\sP',\sU}(A_0,\Phi_0,\by')
=
\bTheta_{\sP',\sP}(A_0,\Phi_0,\by')
\quad\text{for $\by'\in\sU(X^\ell,\sP)$}
$$
on $\tilde N(\delta)\times \sU(\sP')$ where
\begin{equation}
\label{eq:EndOfDiagonal}
\sU(\sP')=
\cup_{\sP<\sP'}\sU(X^\ell,\sP)\cap\Delta^\circ(X^\ell,\sP').
\end{equation}
Then, we define
$$
\bTheta_{\sP',1}:\tilde N(\delta)\times
\Delta^\circ(X^\ell,\sP')
\to
\tsC_{\ft_0},
$$
by
$$
\bTheta_{\sP',1}(A_0,\Phi_0,\by')
=
(1-\bchi(\by'))(A_0,\Phi_0)
+
\bchi(\by')\bTheta_{\sP',\sU}(A_0,\Phi_0,\by')
$$
where $\bchi:\Delta^\circ(X^\ell,\sP)\to [0,1]$ is an
$\fS(\sP)$ invariant function supported in
\eqref{eq:EndOfDiagonal} and equal to one on a neighborhood
$\sU_2$ of the end of $\Delta^\circ(X^\ell,\sP)$
(hence $\sU_2$ is a proper subspace of $\sU(\sP')$).
By shrinking the tubular neighborhoods $\sU(X^\ell,\sP)$, we can
find a family of separating functions $\{\tilde s_{P'}\}_{P'\in\sP'}$,
such that for $\by'=(y'_{P'})_{P'\in\sP'}
\in\sU(X^\ell,\sP)\cap\Delta^\circ(X^\ell,\sP')$ with
$\pi(X^\ell,\sP)(\by')=\by=(y_P)_{P\in\sP}$
\begin{equation}
\label{eq:BallInclusions}
\begin{aligned}
{}&B(y'_{P'},8\tilde s_{P'}(\by')^{1/3})
\Subset
B(y_P,4\tilde s_{P}(\by)^{1/3})
\\&
\quad\text{and}\quad
B(y'_{P'},4\tilde s_{P'}(\by')^{1/2})
\Subset
B(y_P,2\tilde s_{P}(\by)^{1/2})
\end{aligned}
\end{equation}
for all $P'\in\sP'_P$.  By these inclusions,
by the inductive assumption, and by the final statement
of Lemma \ref{lem:FlatteningPairs}, we have the equality,
for $\by'\in \sU_2$,
$$
\bTheta_{\sP'}'
\left( \bTheta_{\sP,1}(A_0,\Phi_0,\by'),\by'\right)
=
\bTheta_{\sP,1}(A_0,\Phi_0,\by').
$$
That is, for $\by'\in\sU_2$,
the pair $\bTheta_{\sP,1}(A_0,\Phi_0,\by')$ is already flat
on the relevant balls and thus the flattening construction of
Lemma \ref{lem:FlatteningPairs} does not change the pair.
Hence, if we define
$$
\bTheta_{\sP'}(A_0,\Phi_0,\by')
=\bTheta_{\sP'}'(\bTheta_{\sP,1}(A_0,\Phi_0,\by'),\by'),
$$
this flattening map will satisfy \eqref{eq:ConsistentFlattening},
completing the construction of the map $\bTheta_{\sP'}$
and thus completing the induction.
\end{proof}

\subsection{The crude splicing map}
\label{subsec:CrudeSplicingMap}
The crude splicing map:
\begin{equation}
\label{eq:DefineCrudeSplicing}
\bga''_{\ft,\fs,\sP}:
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP}) \to \bar\sC_{\ft}
\end{equation}
is $S^1$-equivariant and defined on the domain
\begin{equation}
\label{eq:DefineCrudeSplicingDomain}
\begin{aligned}
{}&
\sO(\ft,\fs,\sP,g_{\sP})
\\
{}&\quad=
\{(F^{T,\fg}_P,[A_P,F^s_P,\bv_P])_{P\in\sP}
\in \Gl(\ft,\fs,\sP,g_{\sP}):
8\la([A_P,F^s_P,\bv_P])^{1/3}\le s_P(F^{T,\fg}_P)\},
\end{aligned}
\end{equation}
where $\{s_P\}$ is the pullback
of the family of separating functions $\{\tilde s_P\}$ constructed
in Lemma \ref{lem:CompatibleFlattening}   by the projection
$\Gl(\ft,\fs,\sP,g_{\sP})\to\Si(X^\ell,\sP)$.

The crude splicing map
is defined exactly as is the splicing map in
\cite[\S 3.2]{FL3} with the following exceptions.


First, the connections on $S^4$ being spliced in are elements
of the spliced ends moduli space $\barM^{s,\natural}_{spl,|P|}$
constructed in \S \ref{sec:SplicedEnd} rather than elements
of the anti-self-dual moduli space $\barM^{s,\natural}_{|P|}$.

Second, the metric on $X$ used to identify balls in $X$ with balls around
the north pole of $S^4$ is fixed for the standard splicing map.
In the crude splicing map, the metric varies with the splicing points
$\by\in\Delta^\circ(X^\ell,\sP)$, varying in a smooth family of metrics
$g_{\sP}$.

Third, in the definition of the splicing map in
\cite[\S 3.2]{FL3}, the background connection $A_0$
is flattened on the balls
$$
B(y_P,2\la_P^{1/2})
$$
while the background section is multiplied by a cut-off function
vanishing on
$$
B(y_P,4\la_P^{1/3}),
$$
where $\la_P$ is the scale of the connection on $S^4$ being spliced
in at $x_P\in X$.
The crude splicing map is defined instead by replacing the background
pair $(A_0,\Phi_0)$ with the background pair
$\bTheta_{\sP}(A_0,\Phi_0,\by)$ constructed in Lemma
\ref{lem:CompatibleFlattening}.  Thus, the background pair
is flattened on balls whose radius does not depend on
the scale of the connections on $S^4$.

The definition of the crude splicing map immediately implies
the following observation about the crude splicing map.

\begin{lem}
\label{lem:CrudeSplicingOnTrivials}
Let $\Si(\ft,\fs,\sP,g_{\sP})$ be as defined in
\eqref{eq:DefineTrivialStratumXConePoints}.
Let $\id_\Si:\Si(\ft,\fs,\sP,g_{\sP})\cong\Si(X^\ell,\sP)$
be the identification in \eqref{eq:IdentifyConePointStratum}.
Then
the restriction of the crude splicing map $\bga''_{\ft,\fs,\sP}$ to
$$
\tilde N(\delta)\times_{\sG_{\fs}}\Si(\ft,\fs,\sP,g_{\sP})
=
N(\delta)\times\Si(\ft,\fs,\sP,g_{\sP})
$$
takes values in
$$
\sC_{\ft_0}\times \Si(X^\ell,\sP)
\subset \bar\sC_{\ft}
$$
and for any $\bF\in \Si(\ft,\fs,\sP,g_{\sP})$
and $(A_0,\Phi_0)\in\tilde N_{\ft_0,\fs}$,
$$
\bga''_{\ft,\fs,\sP}((A_0,\Phi_0),\bF)
=
\left(\bTheta_{\sP}(A_0,\Phi_0,\id_\Si(\bF)),\id_\Si(\bF)\right).
$$
\end{lem}

\begin{rmk}
\label{rmk:LocalSplicingMap}
In contrast to the result of Lemma \ref{lem:CrudeSplicingOnTrivials},
the standard splicing map would satisfy
$$
\bga'_{X,\sP}((A_0,\Phi_0),\bF)
=\left([ A_0,\Phi_0],\id_\Si(\bF)\right),
$$
that is, the background pair are not flattened.
\end{rmk}


\subsection{The overlap spaces}
\label{subsec:OverlapSpacesModuli}
We now introduce spaces of overlap data
$\Gl(\ft,\fs,\sP,[\sP'])$, attempting to mimic the construction of the
spliced end in \S \ref{sec:SplicedEnd}.


\subsubsection{The overlap space}
For $\sP<\sP'$ partitions of $N_\ell$ and $g_{\sP}$ a family of
smooth metrics on $X$ parameterized by $\Delta^\circ(X^\ell,\sP)$,
we define the space of overlap data by:
\begin{equation}
\label{eq:OverlapGluingDataSpace}
\begin{aligned}
{}&
\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\\
&\quad=
\Fr(\ft,\fs,\sP,g_{\sP})\times_{G(\sP)}
\coprod_{\sP''\in[\sP<\sP']}
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P)
\times\prod_{Q\in\sP''_P} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
\right).
\end{aligned}
\end{equation}
Recall that we denote elements of
$\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P)$ as
$(v_Q)_{Q\in\sP''_P}$ where $v_Q\in\RR^4$.
We then denote elements of $\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})$ by
\begin{equation}
\label{eq:NotationForXOverlapSplicingData}
\bA=
\left(
(F^{T,\fg}_P)_{P\in\sP},
\left( (v_Q)_{Q\in\sP''_P}, ([A_Q,F^s_Q,\bv_Q])_{Q\in\sP''_P}\right)_{P\in\sP}
\right)
\in
\Gl(\ft,\fs,\sP,[\sP'],g_{\sP}),
\end{equation}
where $(F^{T,\fg}_P)_{P\in\sP}\in \Fr(\ft,\fs,\sP,g_{\sP})$,
$(v_Q)_{Q\in\sP''_P}\in \Delta^\circ(Z_{|P|},\sP''_P)$,
and
$[A_Q,F^s_Q,\bv_Q]\in \barM^{s,\natural}_{spl,|Q|}(\delta)$.

The projection $\Fr(\ft,\fs,\sP,g_{\sP})\to\Si(X^\ell,\sP)$
defines a projection,
\begin{equation}
\label{eq:ProjectionGluingOverlapToDiagonalInX}
\pi_{\Si}:
\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
\Si(X^\ell,\sP).
\end{equation}
Because $[\Theta,F^s,c_{|Q|}]\in\barM^{s,\natural}_{spl,|Q|}$
is a fixed point of the action of $\SO(3)\times\SO(4)$,
the subspace
\begin{equation}
\label{eq:DefineTrivialInOverlap}
\begin{aligned}
{}&
\Si(\ft,\fs,\sP,[\sP'],g_{\sP})
\\
&\quad
=
\{
\left(
(F^{T,\fg}_P)_{P\in\sP},
\left(
(v_Q)_{Q\in\sP''_P}, ([\Theta,F^s_Q,c_{|Q|}])_{Q\in\sP''_P}\right)_{P\in\sP}
\right) \in \Gl(\ft,\fs,\sP,[\sP'],g_{\sP})\}
\end{aligned}
\end{equation}
is identified with the normal bundle
$\nu(X^\ell,\sP,[\sP'])$
\begin{equation}
\label{eq:IdentifyNormal}
k_{\Si,\sP,[\sP']}:
\Si(\ft,\fs,\sP,[\sP'],g_{\sP})
\cong
\nu(X^\ell,\sP,[\sP'])
\end{equation}
as, the reader may recall, $\nu(X^\ell,\sP,[\sP'])$ is defined
by
$$
\Fr(TX^\ell,\sP,g_{\sP})
\times_{G_T(\sP)}\coprod_{\sP''\in [\sP<\sP']}
\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P).
$$
There is thus a projection,
\begin{equation}
\label{eq:ProjectionGluingOverlapToNormalInX}
\pi_{\Si,\sP,[\sP']}: \Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
\Si(\ft,\fs,\sP,[\sP'],g_{\sP}),
\end{equation}
given by
\begin{align*}
{}&
\pi_{T}
\left(
(F^T_P,F^{\fg}_P)_{P\in\sP},
\left(
(v_Q)_{Q\in\sP''_P}, ([A_Q,F^s_Q,\bx_Q])_{Q\in\sP''_P}\right)_{P\in\sP}
\right)
\\&\quad=
\left(
(F^T_P,F^{\fg}_P)_{P\in\sP},
\left(
(v_Q)_{Q\in\sP''_P}, ([\Theta,F^s_Q,c_{|Q|}])_{Q\in\sP''_P}\right)_{P\in\sP}
\right).
\end{align*}
We will then write
\begin{equation}
\pi_{\nu(\Si)}: \Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
\nu(X^\ell,\sP,[\sP']),
\quad
\pi_{\nu(\Si)}=k_{\Si,\sP,[\sP']}\circ\pi_{\Si,\sP,[\sP']},
\end{equation}
for the projection to the normal bundle.

\subsubsection{The upwards overlap map}
To work simultaneously with all the diagonals $\sP''\in[\sP<\sP']$,
we introduce the gluing space,
$$
\Gl(\ft,\fs,[\sP<\sP'],g_{\sP'})
=
\left(
\coprod_{\sP''\in[\sP<\sP'],g_{\sP''}}
\Fr(\ft,\fs,\sP'',g_{\sP''})\times_{\tilde G(\sP'')}
\prod_{Q\in\sP''}\barM^{s,\natural}_{spl,|Q|}
\right)/\fS(\sP).
$$
Note that by the $\fS(\sP)$ invariance of the families
of metrics $g_{\sP''}$ (see \S \ref{subsubsec:FiberBundleMetrics}),
the space $\Gl(\ft,\fs,[\sP<\sP'],g_{\sP'})$ depends only
on the choice of one family of metrics $g_{\sP'}$.
By comparing the orbits under the symmetric group actions,
one can see there is an identification
$$
\Gl(\ft,\fs,\sP'',g_{\sP''}) \simeq \Gl(\ft,\fs,[\sP<\sP'],g_{\sP'}),
$$
for any $\sP''\in[\sP<\sP']$.
Then, for each background connection $\hat A_0$ on $\fg_0$,
we will define an upwards overlap map,
\begin{equation}
\label{eq:XUpwardsTransitionMap}
\rho^{\ft,\fs,u}_{\sP,[\sP']}:
\sO(\ft,\fs,\sP,[\sP'],g_{\sP})
\subset
\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
\Gl(\ft,\fs,[\sP<\sP'],g_{\sP'}).
\end{equation}
In words, the map $\rho^{\ft,\fs,u}_{\sP,[\sP']}$ will be defined
by keeping the $S^4$ connections,
and parallel translating the frames $(F^T_P,F^{\fg}_P)$
from $y_P\in X$ to each $x_Q\in X$, where
$(x_Q)_{Q\in\sP''}$ is the image of $\bA$ under the
composition $e(X^\ell,\sP)\circ\pi_{\nu(\Si)}$.

We assume that the domain $\sO(\ft,\fs,\sP,[\sP'],g_{\sP})$
of $\rho^{\ft,\fs,u}_{\sP,[\sP']}$ satisfies
\begin{equation}
\label{eq:XUpwardsTransitionDomain1}
\sO(\ft,\fs,\sP,[\sP'],g_{\sP})
\subset
\pi_{\nu(\Si)}^{-1}
\left(
\sO(X^\ell,\sP,[\sP'])\right),
\end{equation}
where $\sO(X^\ell,\sP,[\sP'])\subset\nu(X^\ell,\sP,[\sP'])$ is the subspace
mapped to the tubular neighborhood
$\sU(X^\ell,\sP)$ in Lemma \ref{lem:CompatibleFlattening}.
Then, the map
$e(X^\ell,\sP)\circ\pi_{\nu(\Si)}$ is
defined on $\sO(X^\ell,\sP,[\sP'])$ and takes values in
$\sU(X^\ell,[\sP<\sP'])$, the intersection of
$\sU(X^\ell,\sP)$ with the diagonals $\Si(X^\ell,\sP'')$
for $\sP''\in [\sP<\sP']$.

Assume $x$ is a point in
a geodesic ball around $y\in X$.
Then for any connection $A$ on a bundle over $X$
and $F$ a frame of the bundle lying over $y$,
let $T^A_{x,y}(F)$ denote parallel translation (parallel
with respect to the connection $A$) of the frame $F$ from
$x$ to $y$ along the radial geodesic.
We will use $T^{\LC}_{x,y}$ to denote parallel translation
with respect to the Levi-Civita connection.

We now define the upwards overlap map.
Assume $\bA\in\sO(\ft,\fs,\sP,[\sP'])$ satisfies
$$
\pi_{\Si}(\bA)=\by=(y_P)_{P\in\sP}
\quad\text{and}\quad
e(X^\ell,\sP)\circ \pi_{\nu(\Si)}(\bA)= (x_Q)_{Q\in\sP''}.
$$
Then,
\begin{equation}
\label{eq:UnparamUpwardsTransition}
\begin{aligned}
\rho^{\ft,\fs,u}_{\sP,[\sP']}(\bA)
&=
\rho^{\ft,\fs,u}_{\sP,[\sP']}
\left(
(F^T_P,F^{\fg}_P)_{P\in\sP},
\left(
(v_Q)_{Q\in\sP''_P}, ([A_Q,F^s_Q,\bx_Q])_{Q\in\sP''_P}\right)_{P\in\sP}
\right)
\\
&=
\left(
(T^{\LC}_{x_Q,y_P}(F^T_P),T^{\hat A_0}_{x_Q,y_P}(F^{\fg}_P))_{Q\in\sP''},
([A_Q,F^s_Q,\bx_Q])_{Q\in\sP''_P}
\right),
\end{aligned}
\end{equation}
where the indices $P$ and $Q$
appearing in the parallel translations $T^{\LC}_{x_Q,y_P}$ and
$T^A_{x_Q,y_P}$ satisfy $Q\subseteq P$.
The Levi-Civita connection used to define $T^{\LC}_{x_Q,y_P}$ is
the metric $g_{\sP,\by}$.  The map $\rho^{\ft,\fs,u}_{\sP,[\sP']}$ thus
depends on the background connection $\hat A_0$ used to define
$T^{\hat A_0}_{x_Q,y_P}$.

We define the subspace,
\begin{equation}
\label{eq:DefineTrivialOverlapX}
\begin{aligned}
{}&T(\ft,\fs,\sP,[\sP'])
\\
{}&\quad=
\{
\left(
(F^T_P,F^{\fg}_P)_{P\in\sP},
\left(
(v_Q)_{Q\in\sP''_P}, [\Theta,F^s_Q,\bx_Q])_{Q\in\sP''_P}\right)_{P\in\sP}
\right)
\in \Gl(\ft,\fs,\sP,[\sP'])
\}.
\end{aligned}
\end{equation}
That is, $T(\ft,\fs,\sP,[\sP'])$ is the subspace of
$\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})$ where all the $S^4$ connections
are trivial.  The proof of the following is trivial.

\begin{lem}
\label{lem:UpwardsTransition}
Let $\sP<\sP'$ be partitions of $N_\ell$.
Assume the open subspace
$\sO(\ft,\fs,\sP,[\sP'],g_{\sP})$ of
$\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})$
satisfies the condition \eqref{eq:XUpwardsTransitionDomain1}
Then, for each connection $\hat A_0$ on $\fg_0$,
the map
$$
\rho^{\ft,\fs,u}_{\sP,[\sP']}:
\sO(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
\Gl(\ft,\fs,[\sP<\sP'],g_{\sP'}),
$$
defines an open embedding of
$\sO(\ft,\fs,\sP,[\sP'],g_{\sP})$ into $\Gl(\ft,\fs,[\sP<\sP'],g_{\sP'})$.
\end{lem}

The upwards overlap map $\rho^{\ft,\fs,u}_{\sP,[\sP']}$ depends on
the choice of a connection $\hat A_0$ on $\fg_0$.
For a background pair $(A_0,\Phi_0)\in\tsC_{\ft_0}$,
we will write $\rho^{\ft,\fs,u}_{\sP,[\sP']}(A_0,\Phi_0)$ for the
upwards overlap map defined by the connection $\hat A_0$.
Then, we define
\begin{equation}
\label{eq:DefineParameterizedUpwardsTransitionMap}
\rho^{N,\ft,\fs,u}_{[\sP<\sP']}:
\tilde N(\delta)\times_{\sG_{\fs}}
\sO(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
\tilde N(\delta)\times_{\sG_{\fs}}
\sO(\ft,\fs,[\sP<\sP'],g_{\sP'})
\end{equation}
by, for $(A_0,\Phi_0)\in \tilde N_{\ft_0,\fs}$ and
$\bF\in \sO(\ft,\fs,\sP,[\sP'],g_{\sP})$,
$$
\rho^{N,\ft,\fs,u}_{[\sP<\sP']}\left( (A_0,\Phi_0),\bF\right)
=
\left( (A_0,\Phi_0),
\rho^{\ft,\fs,u}_{\sP,[\sP']}(\bTheta_{\sP}(A_0,\Phi_0,\pi_{\Si}(\bF))(\bF))
\right).
$$
In words, $\rho^{N,\ft,\fs,u}_{[\sP<\sP']}$ leaves the background pair
alone and the restriction of $\rho^{N,\ft,\fs,u}_{[\sP<\sP']}$
to
$$
\sG_{\fs}(A_0,\Phi_0)\times_{\sG_{\fs}}
\sO(\ft,\fs,\sP,[\sP'],g_{\sP})|_{\pi_{\Si}^{-1}(\by)}
$$
is equal to the upwards overlap map $\rho^{\ft,\fs,u}_{\sP,[\sP']}$
given by the background connection $\hat A'$ where
$(A',\Phi')=\bTheta_{\sP}(A_0,\Phi_0,\by)$.

\subsubsection{The downwards overlap map}
The downwards overlap map is a map,
$$
\rho^{\ft,\fs,d}_{\sP,\sP'}:
\sO_d(\ft,\fs,\sP,[\sP'],g_{\sP})\subset
\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
\Gl(\ft,\fs,\sP,g_{\sP}),
$$
defined analogously to the map $\rho^{\Theta,d}_{\sP,\sP'}$.
We wish to define, for $\bA\in \Gl(\ft,\fs,\sP,[\sP'],g_{\sP})$,
\begin{equation}
\label{eq:XDownwardTransition}
\begin{aligned}
\rho^{\ft,\fs,d}_{\sP,[\sP']}(\bA)
&=
\rho^{\ft,\fs,d}_{\sP,[\sP']}
\left(
(F^{T,\fg}_P)_{P\in\sP},
\left(
(v_Q)_{Q\in\sP''_P}, ([A_Q,F^s_Q,\bx_Q])_{Q\in\sP''_P}\right)_{P\in\sP}
\right)
\\
&=
\left(
(F^{T,\fg}_P)_{P\in\sP},
\left(
\bga'_{\RR^4,\sP''_P}
\left( (v_Q)_{Q\in\sP''_P}, ([A_Q,F^s_Q,\bx_Q])_{Q\in\sP''_P} \right)
\right)
\right).
\end{aligned}
\end{equation}
In words, the map $\rho^{\ft,\fs,d}_{\sP,[\sP']}$ is defined by
applying the splicing map $\bga'_{\Theta,\sP''_P}$
of \eqref{eq:R4TrivialSplicing} to the data in
\begin{equation}
\label{eq:SplicingDataFactor}
\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P)\times
\prod_{Q\in\sP''_P}\barM^{s,\natural}_{spl,|Q|}(\delta_Q).
\end{equation}
Thus, for the map $\rho^{\ft,\fs,d}_{\sP,[\sP']}$ to be defined,
we must restrict $\rho^{\ft,\fs,d}_{\sP,[\sP']}$ to the points
where the data in \eqref{eq:SplicingDataFactor} lie
in the domain of the splicing map $\bga'_{\Theta,\sP''_P}$.
More formally,
we must assume that domain of
$\rho^{\ft,\fs,d}_{\sP,[\sP']}$ satisfies
\begin{equation}
\label{eq:XUpwardsTransitionDeomainRequirement}
\begin{aligned}
{}&
\sO_d(\ft,\fs,\sP,[\sP'],g_{\sP})
\\
&\quad\subseteq
\{
\left(
(F^T_P,F^{\fg}_P)_{P\in\sP},
\left(
(v_Q)_{Q\in\sP''_P}, ([A_Q,F^s_Q,\bx_Q])_{Q\in\sP''_P}\right)_{P\in\sP}
\right) \in \Gl(\ft,\fs,\sP,[\sP'],g_{\sP}):
\\
&\qquad
\left( (v_Q)_{Q\in\sP''_P}, ([A_Q,F^s_Q,\bx_Q])_{Q\in\sP''_P}\right)
\in \sO^{\asd}_1(\RR^4,\sP''_P)
\quad\text{for all $P\in\sP$}\},
\end{aligned}
\end{equation}
where $\sO^{\asd}_1(\Theta,\sP''_P,\delta_P)$ is the open set defined in
Lemma \ref{lem:ConstructCollar}.

\begin{lem}
\label{lem:XUpwardsTransition}
Let $\sO_d(\ft,\fs,\sP,[\sP'],g_{\sP})\subseteq \Gl(\ft,\fs,\sP,[\sP'],g_{\sP})$
be any open neighborhood of $\Si(\ft,\fs,\sP,[\sP'],g_{\sP})$
satisfying \eqref{eq:XUpwardsTransitionDeomainRequirement}.
Then the restriction  of  $\rho^{\ft,\fs,d}_{\sP,[\sP']}$
to $\sO_d(\ft,\fs,\sP,[\sP'],g_{\sP})$
is an open embedding whose image is an open
neighborhood of $\Si(\ft,\fs,\sP,g_{\sP})$ in $\Gl(\ft,\fs,\sP,g_{\sP})$.
\end{lem}

\subsubsection{The splicing equality}
We now show how the constructions of the
overlap maps and crude splicing maps lead to
a diagram similar to the diagram
\eqref{eq:CommutingSplicingQuotient} used to construct
the spliced end $W_\ka$.

For $\sP''\in [\sP<\sP']$,
let $\sO(\ft,\fs,\sP'',g_{\sP''})\subset \Gl(\ft,\fs,\sP'',g_{\sP''})$
be a $\fS(\sP)$-invariant collection of open subspaces on which
the crude splicing maps $\bga''_{X,\sP''}$ are defined.
Define
\begin{equation}
\label{eq:DefineSymmQuotientOfCrudeSplicingDomains}
\sO(\ft,\fs,[\sP<\sP'],g_{\sP'})
=
\left(\coprod_{\sP''\in [\sP<\sP']} \sO(\ft,\fs,\sP'',g_{\sP''})\right)/\fS(\sP)
\subset
\Gl(\ft,\fs,[\sP<\sP'],g_{\sP'}).
\end{equation}
Then,
by the $\fS(\sP)$-invariance of the crude splicing
maps $\bga''_{\ft,\fs,\sP''}$ for $\sP''\in [\sP<\sP']$, the
maps$\bga''_{\ft,\fs,\sP''}$  define a
crude splicing map
\begin{equation}
\label{eq:UpperStratumCrudeSplicing}
\bga_{\ft,\fs,[\sP<\sP']}'':
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,[\sP<\sP'],g_{\sP'})
\to
\bar\sC_{\ft}.
\end{equation}
Given subspaces, $\sO(\ft,\fs,\sP,g_{\sP})$ and an
$\fS(\sP)$-invariant collection of subspaces
$\sO(\ft,\fs,\sP'',g_{\sP''})$ for $\sP''\in [\sP<\sP']$,
we define an open subspace,
\begin{equation}
\label{eq:DefineOverlapDomain}
\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})
\subset
\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\end{equation}
to be the points satisfying the conditions
\eqref{eq:XUpwardsTransitionDomain1}
and
\eqref{eq:XUpwardsTransitionDeomainRequirement}
and, in addition, satisfying
$$
\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})
\subseteq
(\rho^{\ft,\fs,u}_{\sP,[\sP']})^{-1}
\left(
\sO(\ft,[\sP<\sP'])
\right)
\cap
(\rho^{\ft,\fs,d}_{\sP,[\sP']})^{-1}
\left(
\sO(\ft,\fs,\sP,g_{\sP})
\right).
$$
Then we have,

\begin{prop}
\label{prop:XOverlapControl}
Let $\sP<\sP'$ be partitions of $N_\ell$.
Assume the families of metrics $g_{\sP''}$ satisfy the conditions
in \S \ref{subsubsec:FiberBundleMetrics}.
Let $\sO(\ft,\fs,\sP,g_{\sP}\subset\Gl(\ft,\fs,\sP,g_{\sP})$ be an open
subspace on which the crude splicing
map is defined.
Let $\{\sO(\ft,\fs,\sP'',g_{\sP''})\}_{\sP''\in [\sP<\sP']}$
be an $\fS(\sP)$-invariant collection of open subspaces on
which the crude splicing maps, $\bga''_{\ft,\fs,\sP''}$ are defined
and let $\sO(\ft,\fs,[\sP<\sP'],g_{\sP'})$ be the
resulting space defined by \eqref{eq:DefineSymmQuotientOfCrudeSplicingDomains}.
Write $\id_N$ for the identity map on $\tilde N=\tilde N_{\ft_0,\fs}$.
Then, if $\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})$ is the subspace
defined in \eqref{eq:DefineOverlapDomain},
the maps in the diagram
\begin{equation}
\label{eq:GlobalSplicingCD}
\begin{CD}
\tilde N(\delta)\times_{\sG_{\fs}}\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})
@> \rho^{N,\ft,\fs,u}_{\sP,[\sP']} >>
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,[\sP<\sP'],g_{\sP'})
\\
@V \id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']} VV
@V \bga''_{\ft,\fs,[\sP<\sP']} VV
\\
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})
@> \bga''_{\ft,\fs,\sP} >> \bar\sC_{\ft}
\end{CD}
\end{equation}
are $S^1$-equivariant and the diagram commutes.
\end{prop}

\begin{proof}
The conditions in the definition of the
subspace $\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})$ imply that
the compositions
$$
\bga''_{\ft,\fs,[\sP<\sP']}\circ \rho^{N,\ft,\fs,u}_{\sP,[\sP']}
\quad\text{and}\quad
\bga''_{\ft,\fs,\sP}\circ(\id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']})
$$
are defined on
$\tilde N(\delta)\times_{\sG_{\fs}}\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})$.

The $S^1$-equivariance follows from the $S^1$-equivariance
of the crude splicing maps and the observation that
the overlap maps in \eqref{eq:GlobalSplicingCD} are
$S^1$-equivariant because the $S^1$ action is simply scalar
multiplication on the section in $\tilde N_{\ft_0,\fs}$
and the overlap maps do not change the background section.

To see that the diagram \eqref{eq:GlobalSplicingCD} commutes,
begin by restricting the diagram to
$$
\sG_{\fs}(A_0,\Phi_0)\times_{\sG_{\fs}}
\sO(\ft,\fs,\sP,[\sP'],g_{\sP})|_{\pi_{\Si}^{-1}(\by)}
\subset
\tN\times_{\sG_{\fs}}
\sO(\ft,\fs,\sP,[\sP'],g_{\sP})
$$
for $(A_0,\Phi_0)\in \tN$ and $\by\in\Delta^\circ(X^\ell,\sP)$.
The definition of the overlap maps imply that
the crude splicing maps are defined by splicing in
$S^4$ connections at $\by$ for $\bga''_{\ft,\fs,\sP}$ and
at some $\by'$ for $\bga''_{X,[\sP<\sP'}$ where
$\pi(X^\ell,\sP)(\by')=\by$.  Then, the consistency of the families
of metrics $g_{\sP}$ and $g_{\sP'}$ imply
$g_{\sP,\by}=g_{\sP',\by'}$.  Similarly, the consistency
condition of the flattening maps in \eqref{eq:ConsistentFlattening}
imply that $(A'_0,\Phi'_0)
=\bTheta_{\sP}(A_0,\Phi_0,\by)=\bTheta_{\sP'}(A_0,\Phi_0,\by')$.
This proves that the sections obtained by the two
compositions in the diagram
\eqref{eq:GlobalSplicingCD} are equal to $\Phi'_0$.
The connections obtained by the two
compositions in the diagram
\eqref{eq:GlobalSplicingCD} are  defined by splicing connections on
$S^4$ to the background connection $A_0'$.


The property \eqref{eq:XUpwardsTransitionDomain1} of
$\sO(\ft,\fs,\sP,[\sP'],g_{\sP})$ implies that $\by'$ lies
in the suitably small tubular neighborhood $\sU(X^\ell,\sP)$ defined
in Lemma \ref{lem:CompatibleFlattening}.
Thus, from \eqref{eq:BallInclusions} the inclusions
$$
B(y'_{P'},4\tilde s_{P'}(\by')^{1/2})
\Subset
B(y_P,2\tilde s_{P}(\by)^{1/2})
$$
hold for the balls where the splicing takes place.
Then because the connection $\hat A_0'$ and
metric $g_{\sP,\by}$ are flat on each ball
$B(y_P,2\tilde s_{P}(\by)^{1/2})$, the proof of the
proposition reduces to
Lemma \ref{lem:CommutingSplicing}.
\end{proof}

\subsection{Constructing the space of global splicing data}
\label{subsec:GlobalSplicingData}
The construction of the space of global splicing data
follows from  Proposition \ref{prop:XOverlapControl}
and the arguments on shrinking the neighborhoods
$\sO(\ft,\fs,\sP,g_{\sP})$ given in the construction of the
spliced end $W_\ka$ in \S \ref{subsec:SplicedEnd}.

\begin{thm}
\label{thm:GlobalSplicingDataOverlaps}
For every partition $\sP$ of $N_\ell$,
there is a neighborhood $\sO(\ft,\fs,\sP,g_{\sP})$ of
$\Si(\ft,\fs,\sP,g_{\sP})$ in $\Gl(\ft,\fs,\sP,g_{\sP})$ such that
the intersection of the images
$$
\bga''_{\ft,\fs,\sP}(\sO(\ft,\fs,\sP,g_{\sP}))\cap\bga''_{\sP'}(\sO(\ft,\fs,\sP',g_{\sP'}))
$$
is empty if $\sP\not < \sP'$ and $\sP'\not <\sP$
and is contained in
\begin{align*}
{}&
\bga''_{\ft,\fs,\sP}
\left(\id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']}
\left(
\tilde N\times_{\sG_{\fs}}\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})
\right)
\right)
\\
&\quad=
\bga''_{[\sP<\sP']}
\left(
\rho^{N,d}_{\sP,[\sP']}
\left(
\tilde N\times_{\sG_{\fs}}\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})
\right)
\right)
\end{align*}
if $\sP<\sP'$, where $\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})$
is the subspace appearing in
Proposition \ref{prop:XOverlapControl}.
\end{thm}

\begin{proof}
The proof is identical to the arguments appearing in
Lemma \ref{lem:ConstructCollar} for
the construction of $W_\ka$.
\end{proof}

We then define the {\em space of global splicing data\/}
to be
\begin{equation}
\label{eq:DefineGlobalGluingDataSpace}
\bar\sM^{\vir}_{\ft,\fs}
=
\sqcup_{\sP} \ \left( \tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})\right)/\sim
\end{equation}
where $\sO(\ft,\fs,\sP,g_{\sP})$ is the space appearing in
Theorem \ref{thm:GlobalSplicingDataOverlaps}
and points in
$$
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})
\quad\text{and}\quad
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP',g_{\sP'})
$$
are identified by the relation $\sim$ if their images under the crude splicing
maps are equal.
Thus, the space $\bar\sM^{\vir}_{\ft,\fs}$ is the homotopy pushout of the
diagram \eqref{eq:GlobalSplicingCD} in the sense of the following lemma.

\begin{lem}
\label{lem:IdentifyPointsInOverlap}
If $\sP<\sP'$ are partitions of $N_\ell$,
$(A_0,\Phi_0,\bF)\in
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})$
and
$(A_0',\Phi_0',\bF')\in
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP',g_{\sP'})$,
then
$$
(A_0,\Phi_0,\bF)
\sim
(A_0',\Phi_0',\bF')
$$
if and only if $(A_0,\Phi_0)=(A_0',\Phi_0')$ and
there is
$(A_0,\Phi_0,\bA)\in
\tilde N\times_{\sG_{\fs}}\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})$
satisfying
\begin{equation}
\label{eq:OverlapEquality1}
\begin{aligned}
(A_0,\Phi_0,\bF)
&=
(\id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']})(A_0,\Phi_0,\bF_0)
\\
(A_0',\Phi_0',\bF')
&=
\rho^{N,d}_{\sP,[\sP']}(A_0,\Phi_0,\bF_0).
\end{aligned}
\end{equation}
\end{lem}

Because the overlaps of the images of $\bga''_{\ft,\fs,\sP}$ are controlled
by the diagram \eqref{eq:GlobalSplicingCD}
and because the maps $\id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']}$ and
$\rho^{N,\ft,\fs,u}_{\sP,[\sP']}$ are smoothly-stratified embeddings,
we have the following.

\begin{cor}
The space $\bar\sM^{\vir}_{\ft,\fs}$ is a smoothly-stratified space
admitting an $S^1$-equivariant fibration,
\begin{equation}
\label{eq:GlobalProjectionToN}
\pi_N: \bar\sM^{\vir}_{\ft,\fs} \to N_{\ft_0,\fs}(\delta),
\end{equation}
and an $S^1$-equivariant embedding
\begin{equation}
\label{eq:GlobalCrudeSplicingMap}
\bga''_{\sM}:\bar\sM^{\vir}_{\ft,\fs}\to \bar\sC_{\ft},
\end{equation}
such that the restriction of $\bga''_{\sM}$ to
$\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})$ is equal
to $\bga''_{\ft,\fs,\sP}$.
\end{cor}


\subsection{Thom-Mather structures on the space of global splicing data}
\label{subsec:TMStr}
On each open subspace
$$
\sU(\ft,\fs,\sP)=
\tN(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})
\subset
\barM^{\vir}_{\ft,\fs},
$$
there is a projection map
\begin{equation}
\label{eq:DefineXTubularProj}
\pi(\ft,\fs,\sP):
\sU(\ft,\fs,\sP)
\to
N(\delta)\times\Si(\ft,\fs,\sP,g_{\sP})
\end{equation}
defined by $\id_N\times \pi_\Si$, where $\pi_\Si$ is the restriction
of the projection defined in \eqref{eq:DefineSplicingDataToDiagonalProjection}
to $\sO(\ft,\fs,\sP,g_{\sP})$ and $\id_N$ is the identity on
$\tilde N(\delta)$.

To prove the relations \eqref{eq:TMProperties} holds for the projection
maps $\pi(\ft,\fs,\sP)$ we make the following observations
on the overlap maps.
First, note that the projection maps,
$$
\pi(\Theta,\sP'_P):
\sU(\Theta,\sP'_P)\subset \barM^{s,\natural}_{spl,|P|}(\delta_P)
\to
\{[\Theta]\}\times \Si(Z_P(\delta_P),\sP'_P)
$$
defined in \eqref{eq:SplicedEndR4Projection} are
$\SO(3)\times\SO(4)$ equivariant.  Together with
the identify map $\id_N$ on $\tN(\delta)$, the maps
$\pi(\Theta,\sP''_P)$ (for $\sP''\in [\sP<\sP']$) define
a map, $\id_N\times p_{\sP,[\sP']}$,
\begin{equation}
\label{eq:DefineUpperProjectionInLowerNgh}
\begin{CD}
\Fr(\ft,\fs,\sP,g_{\sP})
\times_{\tilde G(\sP)}
\prod_{P\in\sP}
\coprod_{\sP''\in[\sP<\sP']}
\sU(\Theta,\sP''_P)
\\
@V \id_N\times p_{\sP,[\sP']} VV
\\
\Fr(\ft,\fs,\sP,g_{\sP})
\times_{\tilde G(\sP)}
\prod_{P\in\sP}
\coprod_{\sP''\in[\sP<\sP']}
\{[\Theta]\}\times \Si(Z_P(\delta_P),\sP''_P)
\end{CD}
\end{equation}
Note that because
\begin{align*}
{}&\coprod_{\sP''\in[\sP<\sP']}
\sU(\Theta,\sP''_P)
\subset
\barM^{s,\natural}_{spl,|P|}(\delta_P),
\quad
\text{and}
\quad
\\
{}&
\coprod_{\sP''\in[\sP<\sP']}
\{[\Theta]\}\times \Si(Z_P(\delta_P),\sP''_P)
\subset
\barM^{s,\natural}_{spl,|P|}(\delta_P)
\end{align*}
the map $\id_N\times p_{\sP,\sP'}$ is defined on
a subspace of
$\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})$.
Observe that the relation
\begin{equation}
\label{eq:ProjectionRelationOnLowerSet}
(\id_N\times \pi_\Si)\circ
(\id_N\times p_{\sP,\sP'})
=
(\id_N\times \pi_\Si)
\end{equation}
holds where $\pi_\Si$ is the projection given
in \eqref{eq:DefineSplicingDataToDiagonalProjection}
because $p_{\sP,\sP'}$ respects the fibers of
$\Gl(\ft,\fs,\sP,g_{\sP})$.


Then for $\sP<\sP'$, the following lemma describes the
projection map $\pi(\ft,\fs,\sP')$ on
the image of the overlap map $\rho^{\ft,\fs,d}_{\sP,[\sP']}$.

\begin{lem}
\label{lem:XProjectionOnOverlaps}
Let $\sP<\sP'$ be partitions of $N_\ell$.
For $\pi_{\Si,\sP,[\sP']}$ the projection
defined in \eqref{eq:ProjectionGluingOverlapToNormalInX},
let
$$
\id_N\times \pi_{\Si,\sP,[\sP']}:
\tilde N(\delta)\times_{\sG_{\fs}}\Gl(\ft,\fs,\sP,[\sP'],g_{\sP})
\to
N(\delta)\times\Si(\ft,\fs,\sP,[\sP'],g_{\sP})
$$
be the product of $\pi_{\Si,\sP,[\sP']}$ with the identity
map on $\tilde N(\delta)$.
Then, the relations
\begin{equation}
\label{eq:XProjectionRelationsOnOverlap}
\begin{aligned}
\pi(\ft,\fs,\sP')\circ \rho^{N,\ft,\fs,u}_{\sP,[\sP']}
&=
\rho^{N,\ft,\fs,u}_{\sP,[\sP']}\circ (\id_N\times \pi_{\Si,\sP,[\sP']})
\\
(\id_N\times p_{\sP,[\sP']})\circ
( \id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']})
&=
(\id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']})\circ
(\id_N\times \pi_{\Si,\sP,[\sP']})
\end{aligned}
\end{equation}
hold on $\tN\times_{\sG_{\fs}}\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})$.
\end{lem}

We now prove the first Thom-Mather relation between
the projections.

\begin{lem}
\label{lem:TM1ForGlobalSplicingData}
Let $\sP<\sP'$ be partitions of $N_\ell$.
Then, restricted to the intersection
$$
\sU(\ft,\fs,\sP)\cap \sU(\ft,\fs,\sP'),
$$
the equality
\begin{equation}
\label{eq:TM1ForGlobalSplicing}
\pi(\ft,\fs,\sP)\circ\pi(\ft,\fs,\sP')=\pi(\ft,\fs,\sP)
\end{equation}
holds.
\end{lem}

\begin{proof}
By Theorem \ref{thm:GlobalSplicingDataOverlaps},
the intersection $\sU(\ft,\fs,\sP)\cap\sU(\ft,\fs,\sP')$
is contained in the image of the overlap maps
in the diagram \eqref{eq:GlobalSplicingCD}.
Therefore every point $\bA'$ in the intersection is
given by
$$
\bA'=
(\id_N\times \rho^{\ft,\fs,d}_{\sP,[\sP']})(\bA)
=
\rho^{N,\ft,\fs,u}_{\sP,[\sP']}(\bA).
$$
for some $\bA\in \tN\times_{\sG_{\fs}}\sO_1(\ft,\fs,\sP,[\sP'],g_{\sP})$.
Then
\begin{align*}
\pi(\ft,\fs,\sP')(\bA')
&=
\pi(\ft,\fs,\sP')\circ \rho^{N,\ft,\fs,u}_{\sP,[\sP']}(\bA)
\\
&=
\rho^{N,\ft,\fs,u}_{\sP,[\sP']}\circ (\id_N\times \pi_{\Si,\sP,[\sP']})(\bA)
\quad\text{By equation \eqref{eq:XProjectionRelationsOnOverlap}}
\\
&=
(\id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']})\circ
(\id_N\times \pi_{\Si,\sP,[\sP']})(\bA)
\quad\text{By the definition of $\sim$}
\\
&=
(\id_N\times p_{\sP,[\sP']})\circ
( \id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']})(\bA)
\quad\text{By equation \eqref{eq:XProjectionRelationsOnOverlap}}
\end{align*}
The preceding equality implies
\begin{align*}
{}&
(\pi(\ft,\fs,\sP)\circ \pi(\ft,\fs,\sP'))(\bA')
\\
&\quad
=
(\pi(\ft,\fs,\sP)\circ
(\id_N\times p_{\sP,[\sP']})\circ
( \id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']}))(\bA)
\\
&\quad
=
((\id_N\times\pi_\Si)\circ
(\id_N\times p_{\sP,[\sP']})\circ
( \id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']}))(\bA)
\quad\text{By the definition following \eqref{eq:DefineXTubularProj}}
\\
&\quad=
(\id_N\times\pi_\Si)
\circ
( \id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']}))(\bA)
\quad\text{By equation \eqref{eq:ProjectionRelationOnLowerSet}}
\\
&\quad =
(\pi(\ft,\fs,\sP)\circ ( \id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']}))(\bA)
\quad\text{By the definition following \eqref{eq:DefineXTubularProj}}
\\
&\quad =
(\pi(\ft,\fs,\sP)(\bA'),
\end{align*}
thus proving \eqref{eq:TM1ForGlobalSplicing}.
\end{proof}

We now define a tubular distance function,
$$
t(\ft,\fs,\sP): \sO(\ft,\fs,\sP)\to [0,1)
$$
which is invariant under the actions of $\sG_{\fs}$ and $S^1$
on $\Gl(\ft,\fs,\sP,g_{\sP})$ and which satisfies:
$$
t(\ft,\fs,\sP)^{-1}(0)=\Si(\ft,\fs,\sP,g_{\sP}).
$$
The natural choice is
\begin{equation}
\label{eq:DefineTubularDistanceFunction}
t(\ft,\fs,\sP)=\sum_{P\in\sP} |P|\tilde\la_P^2,
\end{equation}
where $\tilde \la_P$ is the modified scale parameter
defined in \eqref{eq:TMScaleR4}.  We shall also
use a rescaling of this map.

We note that all maps defined on subspaces of
$\Sym^\ell(X)$ define maps on the subspace
$N(\delta)\times\Sym^\ell(X)$ by pullback.

\begin{lem}
\label{lem:GlobalSplicingTubularNgh}
There are tubular distance functions,
\begin{equation}
t(\ft,\fs,\sP):\sU(\ft,\fs,\sP)\to [0,1),
\quad\text{and}\quad
t_r(\ft,\fs,\sP):\sU(\ft,\fs,\sP)\to [0,1)
\end{equation}
satisfying
\begin{enumerate}
\item
$t(\ft,\fs,\sP)^{-1}(0)
=
t_r(\ft,\fs,\sP)^{-1}(0)
=
N(\delta)\times\Si(X^\ell,\sP)$,
\item
The restriction of $t(\ft,\fs,\sP)$ to
the subspace
$$
\left(N(\delta)\times \sU''(X^\ell,\sP,g_{\sP})\right)\cap U(\ft,\fs,\sP)
$$
is equal to the pullback of the map $t(X^\ell,\sP)$ defined
in \eqref{eq:XTubularDistFunc} under the obvious projection,
\item
On the intersection $\sU(\ft,\fs,\sP)\cap\sU(\ft,\fs,\sP')$
the relations,
\begin{equation}
\label{eq:TMForTubularReds}
t(\ft,\fs,\sP)\circ \pi(\ft,\fs,\sP')
=
t(\ft,\fs,\sP)
\quad\text{and}\quad
t_r(\ft,\fs,\sP)\circ \pi(\ft,\fs,\sP')
=
t_r(\ft,\fs,\sP)
\end{equation}
hold.
\item
There are open neighborhoods
$$
U_1(\ft,\fs,\sP)
\Subset
\sU_2(\ft,\fs,\sP)
\Subset\sU(\ft,\fs,\sP)
$$
of $N(\delta)\times \Si(X^\ell,\sP)$, such that
\begin{equation}
\label{eq:TubularNeighborhoodsOfRedByFunction}
\begin{aligned}
\sU_1(\ft,\fs,\sP)&=t_r(\ft,\fs,\sP)^{-1}([0,\thalf)),
\\
\sU(\ft,\fs,\sP) - \sU_2(\ft,\fs,\sP)
&= t_r(\ft,\fs,\sP)^{-1}(\{1\})
\end{aligned}
\end{equation}
\end{enumerate}
\end{lem}

\begin{proof}
The first and second items follow immediately from the definition.
To prove the equality \eqref{eq:TMForTubularReds} for the function
$t(\ft,\fs,\sP)$ defined in \eqref{eq:DefineTubularDistanceFunction},
we begin by recalling that the modified scale function
$\tilde \la_P$ satisfied
$$
\tilde\la_P\circ \pi(\Theta,\sP''_P)=\tilde\la_P.
$$
Then the definition of the projection
$\id_N\times p_{\sP,[\sP']}$ in
\eqref{eq:DefineUpperProjectionInLowerNgh}
implies
$$
t(\ft,\fs,\sP)\circ (\id_N\times p_{\sP,[\sP']})
=
t(\ft,\fs,\sP).
$$
The equality \eqref{eq:TMForTubularReds} then
follows by the argument in Lemma \ref{lem:TM1ForGlobalSplicingData}.

The function $t_r(\ft,\fs,\sP)$ is defined by a rescaling of
$t(\ft,\fs,\sP)$ (i.e. multiplying by a function pulled
back from $N(\delta)\times \Si(X^\ell,\sP)$) which does
not affect the property \eqref{eq:TMForTubularReds}.
The third item then follows by the argument given for
item (6) in Lemma \ref{lem:TMTubularDistanceR4}.
\end{proof}


\subsection{The global splicing map}
\label{subsec:GlobalSplicingMap}
The global crude splicing map $\bga''_{\sM}$ of
\eqref{eq:GlobalCrudeSplicingMap} is unsatisfactory
because its image does not include $M_{\fs}\times\Sym^\ell(X)$
(see Lemma \ref{lem:CrudeSplicingOnTrivials} and
Remark \ref{rmk:LocalSplicingMap}).  To remedy this,
we define an isotopy of the embedding $\bga''_{\sM}$.

First, observe that if $\beta:[0,1]\to [0,1]$ is a smooth
function with $\beta([0,\tquarter])=1$ and
$\beta([\thalf,1])=0$, then $\beta\circ t_r(\ft,\fs,\sP)$
is supported in  $\sU_1(\ft,\fs,\sP)$  and
vanishes on $\sU(\ft,\fs,\sP)-\sU_2(\ft,\fs,\sP)$
where the spaces $\sU_i(\ft,\fs,\sP)$ are given in
appearing in Lemma \ref{lem:GlobalSplicingTubularNgh}.
Hence, the function
$$
\beta_\sP=\beta\circ t_r(\ft,\fs,\sP)
$$
can be considered a global function on $\bar\sM^{\vir}_{\ft,\fs}$
satisfying:
\begin{equation}
\label{eq:SupportOfCutOffs}
\bar\sM^{\vir}_{\ft,\fs}-\sU_2(\ft,\fs,\sP)
\subset \beta_{\sP}^{-1}(0)
\quad\text{and}\quad
\sU_1(\ft,\fs,\sP)
\subset \beta_{\sP}^{-1}(1).
\end{equation}


Pick one representative $\sP$ of each conjugacy class $[\sP]$
and enumerate these representatives in such a way that
$$
\Si(X^\ell,\sP_i) \subset \cl_{\Sym^\ell(X)} \Si(X^\ell,\sP_j)
$$
implies $i<j$.  Assume
$\sP_1=\{N_\ell\}$ is the crudest partition.
We define a finite sequence of embeddings,
$$
\bga''_{\sM,i}:\bar\sM^{\vir}_{\ft,\fs} \to \bar\sC_{\ft},
$$
by, if $\bga'_{\ft,\fs,\sP}$ is the standard splicing map
defined on $\sU(\ft,\fs,\sP)$,
\begin{equation}
\label{eq:IsotopyOfSplicingMap}
\begin{aligned}
\bga''_{\sM,0}&=\bga''_{\sM},
\\
\bga''_{\sM,i}&=\beta_{\sP}(\cdot)\bga'_{\ft,\fs,\sP}(\cdot)
+(1-\beta_\sP)(\cdot)\bga''_{\sM,i-1}(\cdot).
\end{aligned}
\end{equation}
Observe that although $\bga'_{\ft,\fs,\sP}$ is only defined
on the subspace $\sU(\ft,\fs,\sP)\subset\bar\sM^{\vir}_{\ft,\fs}$,
the convex combination of the embeddings $\bga'_{\ft,\fs,\sP}$
and $\bga''_{\sM,i-1}$ is defined on all of $\bar\sM^{ir}_{\ft,\fs}$
because $\beta_{\sP}$ vanishes on the complement of
$\sU(\ft,\fs,\sP)$.
By \eqref{eq:SupportOfCutOffs},
$\bga''_{\sM,i}$ is equal to the standard splicing map
$\bga'_{\ft,\fs,\sP}$ on $\sU_1(\ft,\fs,\sP)$ and equal
to $\bga''_{\sM,i-1}$ on $\bar\sM^{\vir}_{\ft,\fs}-\sU_2(\ft,\fs,\sP)$.
Assume that $\sP_r=N_\ell$ is the finest partition.
We then define
\begin{equation}
\label{eq:DefineGlobalSplice}
\bga'_{\sM}=\bga''_{\sM,r},
\end{equation}
It is straightforward to verify that
\begin{enumerate}
\item
The map $\bga'_{\sM}$ is a continuous $S^1$-equivariant
embedding and is smooth on each stratum,
\item
The restriction of $\bga'_{\sM}$ to $N(\delta)\times\Sym^\ell(X)$
is equal to the product of the embedding $N(\delta)\to\sC_{\ft_0}$
with the identity on $\Sym^\ell(X)$.
\end{enumerate}


\subsection{Projection to $\Sym^\ell(X)$}
\label{subsec:ProjectionToSym}
We now define a projection map,
\begin{equation}
\label{eq:GlobalProjectionToX}
\pi_X: \bar\sM^{\vir}_{\ft,\fs} \to \Sym^\ell(X),
\end{equation}
which will be used in describing cohomology classes
on $\bar\sM^{\vir}_{\ft,\fs}$.

We will denote
the projection $\pi_\Si$ of
\eqref{eq:DefineSplicingDataToDiagonalProjection} by
$$
\pi_\Si(\ft,\fs,\sP):\sU(\ft,\fs,\sP)\to \Si(X^\ell,\sP),
$$
to avoid confusion between the projection maps of the different
strata.  These local projections maps do not agree on
the intersections $\sU(\ft,\fs,\sP)\cap\sU(\ft,\fs,\sP')$
but can be related by the following.

\begin{lem}
\label{eq:LocalXProjectionsRelation}
If $\sP<\sP'$ are partitions of $N_\ell$, then
\begin{equation}
\pi(X^\ell,\sP)\circ \pi_\Si(\ft,\fs,\sP')=\pi_\Si(\ft,\fs,\sP)
\end{equation}
on $\sU(\ft,\fs,\sP)\cap \sU(\ft,\fs,\sP')$.
\end{lem}

\begin{proof}
The lemma follows immediately from Lemma \ref{lem:IdentifyPointsInOverlap}
and the definitions of the projection maps.
\end{proof}

For $\sP$ a partition of $N_\ell$, let
$$
r_{\sP}: \sU(X^\ell,\sP)\times [0,1] \to \sU(X^\ell,\sP),
$$
be the deformation retraction of $\sU(X^\ell,\sP)$ to
$\Si(X^\ell,\sP)$ defined by identifying $\sU(X^\ell,\sP)$
with the normal bundle $\nu(X^\ell,\sP)$ by the
exponential map $e(X^\ell,\sP,g_{\sP})$
and fiber multiplication by the parameter $t\in [0,1]$.  Thus,
$r_{\sP}(\cdot,1)$ is the identity map and
$r_{\sP}(\cdot,0)=\pi(X^\ell,\sP,g_{\sP})$.
Because $r_{\sP'}(\cdot,1)$ is the identity, we
can define
$$
r_{\sP}\left( \pi_\Si(\ft,\fs,\sP')(\cdot),1-\beta_{\sP}(\cdot)\right):
\sU(\ft,\fs,\sP') \to \Si(X^\ell,\sP')\cup \Si(X^\ell,\sP).
$$
Observe that on  $\sU_1(\ft,\fs,\sP)\cap\sU(\ft,\fs,\sP')$ the
preceding map is equal to $\pi(X^\ell,\sP)\circ\pi_\Si(\ft,\fs,\sP')$
which by Lemma \ref{eq:LocalXProjectionsRelation} is equal to
$\pi_\Si(\ft,\fs,\sP)$.
We now prove the existence of the global projection $\pi_X$
on an open subspace of $\barM^{\vir}_{\ft,\fs}$.

\begin{lem}
\label{lem:GlobalProjectionToX}
There is an open neighborhood $\sU_1\subset \bar\sM^{\vir}_{\ft,\fs}$
of $N(\delta)\times\Sym^\ell(X)$ and a projection map,
$$
\pi_X:\sU_1\to \Sym^\ell(X),
$$
such that the restriction of $\pi_X$ to $\sU_1\cap\sU(\ft,\fs,\sP)$
is homotopic to $\pi_\Si(\ft,\fs,\sP)$.
\end{lem}

\begin{proof}
We construct the map
$$
\tilde\pi_\Si(\ft,\fs,\sP'):\sU(\ft,\fs,\sP') \to \cl \Si(X^\ell,\sP')
$$
inductively so that
\begin{enumerate}
\item
The map $\tilde\pi_\Si(\ft,\fs,\sP')$ is homotopic to
$\pi_\Si(\ft,\fs,\sP)$,
\item
For all $\sP<\sP'$, on $\sU_1(\ft,\fs,\sP)\cap\sU_1(\ft,\fs,\sP)$,
we have $\tilde\pi_\Si(\ft,\fs,\sP')=\tilde \pi_\Si(\ft,\fs,\sP)$.
\end{enumerate}
The open neighborhood $\sU_1$ refered to in the statement of the lemma
will be the union of $\sU_1(\ft,\fs,\sP')$ and the map
$\pi_X$ will be defined by $\tilde\pi_\Si(\ft,\fs,\sP)$ on each
subspace $\sU_1(\ft,\fs,\sP)$.

Let $\sP_0=\{N_\ell\}$ be the crudest partition of $N_\ell$.
Let $\tilde\pi_\Si(\ft,\fs,\sP_0)=\pi_\Si(\ft,\fs,\sP_0)$.

Assume $\tilde\pi_\Si(\ft,\fs,\sP)$ has been constructed for all
$\sP<\sP'$.  The projection $\tilde\pi_\Si(\ft,\fs,\sP)$
is constructed by
induction on $\sP'$ with $\sP<\sP'$ using the deformation
$r_{\sP'}$ and the parameter given by the function $t_r(\ft,\fs,\sP')$
to homotope between $\pi_\Si(\ft,\fs,\sP)$ and a new map
$\tilde\pi_\Si(\ft,\fs,\sP')$ equal to $\tilde\pi(\ft,\fs,\sP')$.
The inductive assumption ensures there is no conflict for different $\sP'$.
Let $\tilde\pi_\Si(\ft,\fs,\sP)$ be the end of this string
of homotopies.
\end{proof}

%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Article: INTERSECTION
%File: splicedendobstruction.tex
%Last update: February 14th, 2002 (Tom)
%Spellcheck:  February 18th, 2002 (Tom)
%Todo:


\section{The spliced-end obstruction bundle}
\label{sec:SplicedEndsObstruction}
The gluing map, described in Theorem \ref{thm:Gluing},
does not map all points in $\bar\sM^{\vir}_{\ft,\fs}$ to the
moduli space $\bar\sM_{\ft}$.  Instead, there is a section
(the {\em obstruction section\/}) of a
pseudo-vector bundle (the {\em obstruction bundle\/})
over $\bar\sM^{\vir}_{\ft,\fs}$ such that the intersection
of $\sM_{\ft}$ with the image of the gluing map is given
by the image of the zero-locus of the obstruction section.

The obstruction bundle has two components: the background obstruction
and the instanton obstruction.  In this section, we define
a pseudo-bundle, the {\em spliced-end obstruction\/} bundle,
$\bar\Upsilon^i_{spl,\ka}\to\barM^{s,\natural}_{spl,\ka}(\delta)$
which will be used to define the instanton obstruction over
$\bar\sM^{\vir}_{\ft,\fs}$.

The restriction of $\bar\Upsilon^i_{spl,\ka}$
to the top stratum of $\barM^{s,\natural}_{spl,\ka}$,
$\Upsilon^i_{spl,\ka}\to M^{s,\natural}_{spl,\ka}$
is a finite-dimensional subbundle
of the infinite-dimensional vector bundle,
\begin{equation}
\label{eq:S4InfiniteDimObstrBundle}
\fV_\ka=
\sA^\natural_\ka(2\delta)\times\Fr(E_\ka)|_s\times_{\sG_\ka} L^2_{k-1}(V^-_\ka)
\to
\sB^s_\ka(2\delta),
\end{equation}
restricted to $M^{s,\natural}_{spl,\ka}(\delta)$,
where $V^\pm_\ka=\bS^\pm\otimes E_\ka$ for the standard spin
structure $(\fS^\pm,\rho)$ on $S^4$.  The pseudo-bundle
$\bar\Upsilon^i_{spl,\ka}$ will be a sub-bundle of the space
\begin{equation}
\label{eq:ExtendedS4InfiniteDimObstrBundle}
\bar\fV_\ka=\sqcup_{\ell=0}^\ka \fV_{\ka-\ell}\times\Sym^\ell(X),
\end{equation}
but this extension will only be
a pseudo-bundle rather than as a vector bundle as the rank
of the extension will decrease on the lower strata.

The bundle $\Upsilon^i_{spl,\ka}$ is
an approximation to the index bundle $\Ind(\bD_\ka)$
associated to the Dirac operator $D_A:\Ga(V^-_\ka)\to\Ga(V^+_\ka)$
(as defined in \S \ref{subsec:EquivDiracBundle}) but an approximation
that works more conveniently with the splicing construction
of the space of global splicing data.  To use the
$\bar\Upsilon^i_{spl,\ka}$ to define an obstruction bundle over the
space of splicing data, the $\bar\Upsilon^i_{spl,\ka}$ must
be equivariant with respect to a $\Spinu(4)$ action described
in \S \ref{subsec:SpinuAction}, just as the
spliced ends moduli space is closed under the
$\SO(3)\times\SO(4)$ action.
The construction of the $\bar\Upsilon^i_{spl,\ka}$
is analogous to the that of the spliced-ends moduli space.
Using induction, we require the restriction of
$\bar\Upsilon^i_{spl,\ka}$ to the image of the splicing map
$\bga'_{\Theta,\sP}$ to be given by the image of a splicing map
for sections of \eqref{eq:ExtendedS4InfiniteDimObstrBundle}.
An inductive argument, analogous to that used in constructing
the subspace $W_\ka$ in \S \ref{subsec:SplicedEnd}.
Then, following the argument in \S \ref{subsec:Collar},
the resulting bundle over $W_\ka$ is isotoped to one which
is equal to $\Ind(\bD_\ka^*)$ away from the trivial strata.
These last arguments, because they are so similar to those
in \S \ref{sec:SplicedEnd}
are carried out in a compressed fashion in \S
\ref{subsec:ObstructionSplicing}; the properties of
$\bar\Upsilon^i_{spl,\ka}$ are enumerated in
Proposition \ref{prop:ExistenceOfSplicedEndsIndex}.

\subsection{The equivariant Dirac index bundle}
\label{subsec:EquivDiracBundle}
We begin by defining the bundle $\Ind(\bD^*_\ka)$.
Let $\fs_{S^4}=(\rho,\bS^+,\bS^-)$ be the standard spin structure
on $S^4$ with respect to the round metric.
Let $E_{\ka}\to S^4$ be the rank-two, complex Hermitian bundle
with $c_2(E_{\ka})=\ka$.  For $[A]\in M^{s,\natural}_{\ka}(\delta)$
we define the Dirac operator
\begin{equation}
\label{eq:DefineS4Dirac}
D_A: L^2_k( \bS^+\otimes E_{\ka}) \to L^2_{k-1}(\bS^-\otimes E_{\ka}),
\end{equation}
to be the Dirac operator defined by the Levi-Civita connection
on $\fs_{S^4}$ and $A$ on $E_{\ka}$.  The spliced-end obstruction
bundle is an approximation to the bundle
$$
\ind(\bD^*_{\ka}) \to M^{s,\natural}_{\ka}(\delta),
$$
defined by the cokernels of the Dirac operators:
\begin{equation}
\label{eq:DefineInstantonDiracBundle}
\ind(\bD^*_{\ka})=
\{ (A,F^s,\Psi)\in \tM^{s,\natural}_{\ka}\times \Fr(E_{\ka})|_s
\times L^2_{k-1}(\bS^-\otimes E_{\ka}): D_A^*\Psi=0
\}/\sG_{\ka}.
\end{equation}
Because of the positive scalar curvature on $S^4$,
the Weitzenbock estimates in \cite{FL1} imply that the cokernel
of $D_A^*$ vanishes for $[A]\in M^{s,\natural}_\ka(\delta)$ and
thus $\ind(\bD^*_\ka)$ actually forms a vector bundle.
An index theorem computation shows that $\ind(\bD^*_\ka)$
is a complex, rank $\ka$ vector bundle.  We will also use the
notation $\ind(\bD^*_{\ka})$ to denote the bundle over
$\barM^{s,\natural}_{\ka}(\delta)$ given by,
$$
\{[A,F^s,\bx,\Psi]: [A,F^s,\bx]\in \barM^{s,\natural}_{\ka}(\delta)
\quad\text{and}\quad D_A\Psi=0\}.
$$
Because the index of the Dirac operator depends on $\ka$,
this extension of $\ind(\bD^*_\ka)$ is only a pseudo-vector bundle,
that is, the rank of the fiber depends on the stratum.

\subsection{The action of $\Spinu(4)$}
\label{subsec:SpinuAction}
The group
$$
\Spinu(4)
=
\left( \U(2)\times \Spinc(4)\right)/S^1
$$
is defined in \cite[\S 3.2]{FLLevelOne}
as are a pair of surjective homomorphisms
$$
\Ad^u_{\SO(3)}:\Spinu(4)\to\SO(3),
\quad
\Ad^u_{\SO(4)}:\Spinu(4)\to\SO(4)
$$
(see \cite[Equation (3.13)]{FLLevelOne}).
The bundle $\fV_\ka\to\sB^s_\ka$ is an equivariant $\Spinu(4)$
bundle with respect to the action of $\Spinu(4)$ on $\sB^s_\ka$
defined by the homomorphism
$$
\Spinu(4)\to\SO(3)\times\SO(4),
\quad
A\mapsto (\Ad(A),\Ad(A)).
$$
The bundle $\Ind(\bD^*_\ka)$ is invariant under this action and
thus is also an $\Spinu(4)$ equivariant bundle.




\subsection{The splicing map}
\label{subsec:ObstructionSplicing}
The spliced-ends obstruction bundle will be identical to
the index bundle $\Ind(\bD^*_\ka)$ on the complement
of the spliced ends of $\barM^{s,\natural}_{spl,\ka}(\delta)$.
On the spliced end, $W_\ka$, the spliced-ends obstruction
bundle will be given by the image of a splicing map
which we now introduce.

Recall that the end of $\barM^{s,\natural}_{spl,\ka}(\delta)$
near the trivial stratum,
$$
\{[\Theta]\}\times\Si(Z_\ka(\delta),\sP)
$$
where $\sP$ is a partition of $N_\ka$ with $|\sP|>1$,
was given by the image of a splicing map
$$
\bga'_{\Theta,\sP}:
\sO^{\asd}(\Theta,\sP)\subset
\Delta^\circ(Z_\ka(\delta),\sP)\times_{\fS(\sP)}
\prod_{P\in\sP} \barM^{s,\natural}_{spl,|P|}(\delta_P)
\to
\barM^{s,\natural}_{spl,\ka}(\delta).
$$
We will define the spliced-ends obstruction bundle inductively.


An element of the space $\bar\fV_\ka$ from
\eqref{eq:ExtendedS4InfiniteDimObstrBundle}
can be written as
$[A,F^s,\bx,\Psi]$ where $[A,F^s,\bx]\in\bar\sB^{s}_{\ka}(2\delta)$,
$\bx\in\Sym^\ell(\RR^4)$,
and $\Psi\in L^2_{k-1}(\bS^-\otimes E_{\ka-\ell})$.
Then, for each partition
$\sP$ of $N_\ka$ we define an embedding, covering
the splicing map $\bga_{\Theta,\sP}$ by,
\begin{equation}
\label{eq:DefineS4SectionSplicingToTrivial}
\begin{aligned}
{}&
\varphi_{\Theta,\sP}':
\Delta^\circ(Z_\ka(\delta),\sP)
\times_{\fS(\sP)}
\prod_{P\in\sP} \bar\fV_{|P|}
\to
\bar\fV_\ka,
\\
{}&\quad
\varphi_{\Theta,\sP}'\left( [x_P,[A_P,F^s_P,v_P,\Psi_P]]_{P\in\sP}\right)
=
\left(\bga'_{\Theta,\sP}([x_P,[A_P,F^s_P,v_P]_{P\in\sP}),
\sum_{P\in\sP} \chi_{x_P,\la_P} c_{x_P,1}^*\Psi_P
\right).
\end{aligned}
\end{equation}
Here $\la_P=\la([A_P,F^s_P,\bx_P])$
and $\chi_{x,\la}$ is a cut-off function
supported on
$B(x,\tquarter \la^{1/3})$ and equal to one on
$B(x,\teighth \la^{1/2})$.
Observe that the map \eqref{eq:DefineS4SectionSplicingToTrivial}
is equivariant with respect to the $\Spinu(4)$ action if
$\Spinu(4)$ acts diagonally on the domain, by the projection $\Ad^u_{\SO(4)}$
to $\SO(4)$ on the factor $\Delta^\circ(Z_\ka(\delta),\sP)$.
With this vocabulary established, we can now state the proposition
detailing the properties of the spliced-end obstruction bundle.

\begin{prop}
\label{prop:ExistenceOfSplicedEndsIndex}
There is a pseudo-bundle
\begin{equation}
\label{eq:SplicedEndsIndexBundle}
\bar\Upsilon^i_{spl,\ka}\to \barM^{s,\natural}_{spl,\ka}(\delta)
\end{equation}
which is a $\Spinu(4)$-invariant subbundle of the space
$\bar\fV_\ka\to\bar\sB^{s,\natural}_\ka(2\delta)$
and satisfies
\begin{enumerate}
\item
The restriction of $\bar\Upsilon^i_{spl,\ka}$ to
the top stratum, $M^{s,\natural}_{spl,\ka}(\delta)$,
is a complex, rank $\ka$ vector bundle,
\item
For any partition $\sP$ of $\ka$ with $|\sP|>1$,
the restriction of $\bar\Upsilon^i_{spl,\ka}$ to the
neighborhood of $\{[\Theta]\}\times \Si(Z_\ka(\delta),\sP)$
given in \eqref{eq:EqualsSplicingImage} is given
by the image of
\begin{equation}
\label{eq:SplicedEndIndexBundle}
\Delta^\circ(Z_\ka(\delta),\sP)\times_{\fS(\sP)}
\prod_{P\in\sP}\bar\Upsilon^i_{spl,|P|},
\end{equation}
under the splicing map $\varphi'_{\Theta,\sP}$,
\item
The restriction of $\bar\Upsilon^i_{spl,\ka}$
to the complement of $W_\ka$ is given by the
restriction of  the bundle $\Ind(\bD^*_\ka)$
defined in \eqref{eq:DefineInstantonDiracBundle} to
the complement of $W_\ka$,
\item
For any $[A',F^s,\bx]\in\barM^{s,\natural}_{spl,\ka}(\delta)$,
$L^2$-orthogonal projection defines an isomorphism,
$$
\Upsilon^i_{spl,\ka}|_{[A,F^s,\bx]} \cong
\Coker D_{A_t}
$$
for all $t\in [0,1]$
where $A_t=A'+ta$ and $A=A'+a$ is the value of the gluing map
at $A'$.
\end{enumerate}
\end{prop}

The construction of the bundle $\bar\Upsilon^i_{spl,\ka}$
uses induction on $\ka$ and is parallel to the construction of
the spliced-ends moduli space.  Thus, the first step
(done in \S \ref{subsubsec:SplicedEndsObstrOverlap})
is show that the images of the splicing maps $\varphi'_{\Theta,\sP}$
form a vector bundle over $W_\ka$.
We then use the final property
in the proposition property that $L^2$-orthogonal projection gives
an isomorphism to define the isotopy between the image of the splicing
maps over $W_\ka$ and the bundle $\Ind(\bD^*_\ka)$.

\subsubsection{The overlap bundles}
\label{subsubsec:SplicedEndsObstrOverlap}
If the restriction of the
spliced end index bundle to the image of the splicing map
$\bga'_{\Theta,\sP}$
is to be given by the image
of the splicing map $\varphi'_{\Theta,\sP}$ restricted to
the subspace \eqref{eq:SplicedEndIndexBundle}, then
we need to describe the overlaps of the images of
the splicing maps $\varphi'_{\Theta,\sP}$ and
$\varphi'_{\Theta,\sP'}$ where $\sP<\sP'$.
This is done by constructing a space of overlap
data analogous to that appearing in
Proposition \ref{prop:CommutingSplicingSymmQuotient}.

Begin by initially considering the vector pseudo-bundle
$\tilde\Upsilon(\Theta,\sP,\sP')$ defined by restricting
the vector bundle
\begin{equation}
\label{eq:DefineOverlapInstantonObstr}
\begin{CD}
\Delta^\circ(Z_\ka(\delta),\sP)
\times
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)
\times\prod_{Q\in\sP'_P} \bar\Upsilon^i_{spl,|Q|}
\right)
\\
@VVV
\\
\Delta^\circ(Z_\ka(\delta),\sP)
\times
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)
\times\prod_{Q\in\sP'_P} \bar M^{s,\natural}_{spl,|Q|}(\delta_Q)
\right)
\end{CD}
\end{equation}
to the intersection of its base with the
space $\tilde\sO(\Theta,\sP,\sP',\delta)$
defined in Lemma \ref{lem:OverlappingTrivial}.
By induction on $\ka$, the splicing maps
$\varphi_{\Theta,\sP'_P}$ define maps
$$
\varphi_{\Theta,\sP'_P}:
\Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)
\times\prod_{Q\in\sP'_P} \Upsilon^i_{spl,|Q|}
\to
\Upsilon^i_{spl,|P|},
$$
covering the splicing map $\bga'_{\Theta,\sP'_P}$.
Hence, there is a map
$$
\rho^{\Psi,d}_{\sP,\sP'}:
\tilde\Upsilon(\Theta,\sP,\sP')
\to
\Upsilon(\Theta,\sP)=
\Delta^\circ(Z_{\ka}(\delta),\sP)\times\prod_{P\in\sP}\Upsilon^i_{spl,|P|}
$$
defined by
$$
\rho^{\Psi,d}_{\sP,\sP'}
=
\id_\Delta\times\prod_{P\in\sP}\varphi'_{\Theta,\sP'_P},
$$
where $\id_\Delta$ is the identity map on
$\Delta^\circ(Z_{\ka}(\delta),\sP)$.
By the definition of the overlap
map $\rho^{\Theta,d}_{\sP,\sP'}$
in \eqref{eq:DefineUpwardTransition}, the
map $\rho^{\Psi,d}_{\sP,\sP'}$ fits into the vector bundle diagram
\begin{equation}
\label{eq:SplicedObstrUpwards}
\begin{CD}
\tilde\Upsilon(\Theta,\sP,\sP')
@> \rho^{\Psi,d}_{\sP,\sP'} >>
\Upsilon(\Theta,\sP)
\\
@VVV @VVV
\\
\sO^{\asd}(\Theta,\sP,\sP',\delta)
@> \rho^{\Theta,d}_{\sP,\sP'} >>
\sO^{\asd}(\Theta,\sP)
\end{CD}
\end{equation}
The map $\rho^{\Psi,d}_{\sP,\sP'}$ is not injective because
of the action of the symmetric group, but this will not
affect our construction.

We also define a map
\begin{equation}
\label{eq:UpwardsTransitionObstruction}
\rho^{\Psi,u}_{\sP,\sP'}:
\tilde\Upsilon(\Theta,\sP,\sP')
\to
\Upsilon(\Theta,\sP')
\end{equation}
which fits into the vector bundle diagram
\begin{equation}
\label{eq:SplicedObstrDownwards}
\begin{CD}
\tilde\Upsilon(\Theta,\sP,\sP')
@> \rho^{\Psi,u}_{\sP,\sP'} >>
\Upsilon(\Theta,\sP')
\\
@VVV @VVV
\\
\sO^{\asd}(\Theta,\sP,\sP',\delta)
@> \rho^{\Theta,u}_{\sP,\sP'} >>
\sO^{\asd}(\Theta,\sP')
\end{CD}
\end{equation}
Then,
the following lemma, analogous to
Lemma \ref{lem:CommutingSplicing}, is the key
to the construction of the bundle $\Upsilon^i_{spl,\ka}$.

\begin{lem}
\label{lem:OverlapObstructionCommuting}
Let $\sP<\sP'$ be partitions of $N_\ka$ with
$|\sP|>1$.  Let $\tilde\Upsilon(\Theta,\sP,[\sP'])$
be the space defined in \eqref{eq:DefineOverlapInstantonObstr}.
Then, the following diagram of vector bundles,
\begin{equation}
\label{eq:OverlapObstructionCommuting}
\begin{CD}
\tilde\Upsilon(\Theta,\sP,\sP')
@> \rho^{\Psi,d}_{\sP,\sP'} >>
\Upsilon(\Theta,\sP)
\\
@V \rho^{\Psi,u}_{\sP,\sP'} VV
@V \varphi'_{\Theta,\sP} VV
\\
\Upsilon(\Theta,\sP') @> \varphi'_{\Theta,\sP'} >>
\bar\fV_\ka
\end{CD}
\end{equation}
covers the diagram \eqref{eq:CommutingSplicing},
is $\Spinu(4)$ equivariant, and commutes.
\end{lem}

\begin{proof}
That the diagram \eqref{eq:OverlapObstructionCommuting}
covers the diagram \eqref{eq:CommutingSplicing} follows
from the definition of the maps
$\rho^{\Psi,d}_{\sP,\sP'}$ and $\rho^{\Psi,u}_{\sP,\sP'}$
(see the diagrams \eqref{eq:SplicedObstrUpwards} and
\eqref{eq:SplicedObstrDownwards}) and the fact that the
of the spicing maps $\varphi'_{\Theta,\sP}$ cover the
splicing maps $\bga'_{\Theta,\sP}$.

The $\Spinu(4)$ equivariance of the diagram follows
from that of the splicing maps $\varphi'_{\Theta,\sP}$.

The commutativity of the diagram
follows from the definition of the splicing maps
$\varphi'_{\Theta,\sP}$ and the property
\eqref{eq:BallContainment2}
which appears, in \eqref{eq:OverlapCondition3},
in the definition of the overlap space
$\tilde\sO(\Theta,\sP,\sP',\delta)$.


In detail, assume that the point in
$\tilde\Upsilon(\Theta,\sP,\sP')$ is given by data consisting of
points $x_P\in\Delta^\circ(Z_{\ka}(\delta),\sP)$
(for $P\in\sP$),
points $y_Q\in\Delta^\circ(Z_{|P|}(\delta_P),\sP'_P)$
(for $Q\in\sP'_P$),
and $[A_Q,F^s_Q,\bv_Q,\Psi_Q]\in\bar\Upsilon_{spl,|Q|}$.
Then, the first map in
the composition $\varphi'_{\Theta,\sP}\circ\rho^{\Psi,d}_{\sP,\sP'}$
entails defining sections $\Psi_P$ (for $P\in\sP$) through
the splicing maps $\varphi_{\Theta,\sP'}$ 
in the definition
of the map $\rho^{\Psi,d}_{\sP,\sP'}$ by
$$
\Psi_P=\sum_{Q\in\sP'_P} \chi_{y_Q,\la_Q}c_{y_Q,1}^*\Psi_Q,
$$
where $\la_Q=\la([A_Q,F^s_Q,\bv_Q])$.
Applying the splicing map $\varphi'_{\Theta,\sP}$ to the sections
$\Psi_P$ will give, for
$$
\la_P
=
\la\left( \bga'_{\Theta,\sP'_P}((y_Q,[A_Q,F^s_Q,\bv_Q])_{Q\in\sP'_P})
\right),
$$
the section
\begin{equation}
\label{eq:IteratedSplicing1}
\begin{aligned}
\sum_{P\in\sP}\chi_{x_P,\la_P}c_{x_P,1}^*\Psi_P
&=
\sum_{P\in\sP}\sum_{Q\in\sP'_P}
\chi_{x_P,\la_P}c_{x_P,1}^*\chi_{y_Q,\la_Q}c_{y_Q,1}^*\Psi_Q
\\
&=
\sum_{Q\in\sP'}
\chi_{x_P,\la_P}\chi_{x_P+y_Q,\la_Q}c^*_{x_Q+y_Q,1}\Psi_Q
\end{aligned}
\end{equation}
where the last equality follows from Lemma \ref{lem:ComposeCenterings}.
Recall that by the condition
\eqref{eq:OverlapCondition3},
the points $x_P$ and $y_Q$ in the definition of
the subspace $\tilde\sO(\Theta,\sP,\sP')$ will satisfy
(see condition \eqref{eq:BallContainment2}),
$$
B(x_P+y_Q,\tquarter\la_Q^{1/3})\Subset B(x_P,\teighth\la_P^{1/2}).
$$
Because $\chi_{x,\la}$ is supported on
$B(x,\tquarter \la^{1/3})$ and equal to one on
$B(x,\teighth \la^{1/2})$.
the preceding inclusion implies that
$\chi_{x_P,\la_P}$ is equal to one on the support of
$\chi_{x_P+y_Q,\la_Q}$, so the section in
\eqref{eq:IteratedSplicing1} is equal to
$$
\sum_{Q\in\sP'}
\chi_{x_P+y_Q,\la_Q}c^*_{x_Q+y_Q,1}\Psi_Q.
$$
This is exactly the section resulting from the applying the
map $\varphi'_{\Theta,\sP'}\circ\rho^{\Psi,u}_{\sP,\sP'}$ to
the same data, proving the diagram
\eqref{eq:OverlapObstructionCommuting} commutes.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:ExistenceOfSplicedEndsIndex}]
The proof of this proposition is largely identical to the
proof of Theorem \ref{thm:ExistenceOfSplicedEndsModuli} so
we omit many of the details.

The induction begins with setting $\Upsilon^i_{spl,1}$
equal to the bundle $\ind(\bD^*_1)$ on the top stratum of
$\barM^{s,\natural}_{spl,1}(\delta)=(0,\delta)\times\SO(3)$
(which is given by)
$$
(0,\delta)\times\SU(2)\times_{\pm 1}\CC \to (0,\delta)\times\SO(3).
$$
The bundle extends as a pseudo-vector bundle over the Uhlenbeck
compactification, $c(\SO(3))$ as
$$
c\left( \SU(2)\times_{\pm 1}\CC\right)
\to
c(\SO(3)).
$$
Assuming that $\bar\Upsilon^i_{spl,\ka'}$ has been constructed
for all $\ka'<\ka$,
we define a pseudo-vector bundle $\bar\Upsilon^i(W_\ka)\to W_\ka$
over the spliced end by the image of the splicing maps
$\varphi'_{\Theta,\sP}$ on the domains in
\eqref{eq:SplicedEndIndexBundle}.
This space is a pseudo-vector bundle because the overlap
of the images of the maps $\varphi'_{\Theta,\sP}$ is
controlled by Lemma \ref{lem:OverlapObstructionCommuting}.

The final property in Proposition \ref{prop:ExistenceOfSplicedEndsIndex}
on the surjectivity of $L^2$-orthogonal projection follows
from the results of
\cite[\S 8]{FL3} (see also \cite[Proposition 7.1.32]{DK}).
We can then define an isotopy $R_D$ of
$\bar\Upsilon^i(W_\ka)$ by, for $t\in [\thalf,\tthreequarter]$,
$$
R_D\left([A,F^s,\bx,\Psi],t\right)
=
\left(R([A,F^s,\bx,\Psi],t),
(1-4(t-\thalf))\Psi+ 4(t-\thalf)\Pi_{A+a(A)}\Psi
\right)
$$
where $R$ is the isotopy defined in
\eqref{eq:GluingIsotopy} and for
$G([A,F^s,\bx]=[A+a(A),F^s,\bx]$ is
the gluing map from \eqref{eq:S4GluingMap},
 $\Pi_{A+a(A)}$ denotes
$L^2$-orthogonal onto $\Coker D_{A+a(A)}$.
For for $t\in(-\infty,\thalf]$, the isotopy $R_D(\cdot,t)$
is equal to the identity and for
$t\in[\tthreequarter,1]$, the isotopy $R_D(\cdot,t)$
is defined by pulling back the section $\Psi$ by the centering map.

The bundle $\bar\Upsilon^i_{spl,\ka}$ is then defined by
the union of the image $\bar\Upsilon^i(W_\ka)$ under the map
$R_D(\cdot,t(\cdot))$ where $t$ is the function defined
in \eqref{eq:IsotopyParameter} with
$\Ind(\bD^*_\ka)$ restricted to the complement of
$$\left(
\barM^{s,\natural}_{\ka}(\delta)
-
R(1,W_\ka)
\right)
$$
as in the definition of $\barM^{s,\natural}_{spl,\ka}(\delta)$
in the proof of Theorem \ref{thm:ExistenceOfSplicedEndsModuli}.
\end{proof}

%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Article: INTERSECTION
%File: obstruction.tex
%Last update: November 28th, 2001 (Tom)
%Spellcheck:  February 18th, 2002 (Tom)
%Todo:


\section{The obstruction bundle}
\label{sec:obstr}
The gluing map is an embedding of $\bar\sM^{\vir}_{\ft,\fs}$
into $\bar\sC_{\ft}$ defined by a perturbation of the image
of the splicing map.  The intersection of the image of the
gluing map with $\bar\sM_{\ft}$ is defined by the zero-locus
of a finite-dimensional pseudo-bundle, the {\em obstruction bundle\/},
which is a smooth vector bundle on each stratum of $\bar\sM^{\vir}_{\ft,\fs}$.
In this section, we define this obstruction bundle.

The obstruction bundle is the direct sum of the Seiberg-Witten obstruction
bundle, defined in \S \ref{subsec:SWObstruction},
and the instanton obstruction pseudo-bundle,
defined in \S \ref{subsec:InstantonObstr}.
The gluing map, defined in Theorem \ref{thm:Gluing},
is homotopic to the global splicing map and the image of the gluing
map contains a neighborhood of $M_{\fs}\times\Sym^\ell(X)$ in
$\bar\sM_{\ft}$.  The intersection of the image of the gluing map
with $\bar\sM_{\ft}$ is given by the zero-locus of a section of
the obstruction bundles as discussed in \S \ref{subsec:GluingThm}.

\subsection{The infinite-dimensional obstruction bundle}
The infinite-dimensional obstruction bundle in which the
monopole map $\fS$ takes values is given by
\begin{equation}
\label{eq:InfiniteDimObstr}
\fV_{\ft}=
\tsC_{\ft}\times_{\sG_{\ft}}
L^2_{k-1}(\La^+\otimes \fg_{\ft}) \oplus L^2_{k-1}(V^-)
\to
\sC_{\ft}.
\end{equation}
The $S^1$ action on $\sC_{\ft}$ lifts to an $S^1$ action on
the bundle \eqref{eq:InfiniteDimObstr} given by scalar multiplication
on the element of $L^2_{k-1}(V^-)$.

We extend the above bundle to a pseudo-bundle over $\bar\sC_{\ft}$
by
\begin{equation}
\bar\fV_{\ft}=
\sqcup_{\ell} \
\left(\fV_{\ft(\ell)}\times\Sym^\ell(X)\right)
\to
\bar\sC_{\ft}=
\sqcup_{\ell}\ \left(\sC_{\ft(\ell)}\times\Sym^\ell(X)\right).
\end{equation}
The monopole map is a continuous map $\fS:\bar\sC_{\ft}\to\bar\fV_{\ft}$.

\subsection{The Seiberg-Witten obstruction}
\label{subsec:SWObstruction}
Recall from \cite{FL3} or \cite[\S 3.6.1]{FLLevelOne}
that the background or Seiberg-Witten component of the obstruction
is constructed by splicing in sections of the obstruction bundle
$$
\pi_N^*\Xi_{\ft,\fs}\to N_{\ft,\fs}(\delta),
$$
where $\pi_N:N_{\ft,\fs}(\delta)\to M_{\fs}$ and
$\Xi_{\ft,\fs}\cong M_{\fs}\times\CC^{r_\Xi}$ is a trivial bundle.
The embedding $N_{\ft,\fs}(\delta)\to\sC_{\ft(\ell)}$ is covered
by an vector bundle embedding:
$$
\begin{CD}
\pi_N^*\Xi_{\ft,\fs} @>>> \fV_{\ft(\ell)}
\\
@VVV @VVV
\\
N_{\ft,\fs}(\delta) @>>> \sC_{\ft(\ell)}
\end{CD}
$$
The above vector bundle map is $S^1$ equivariant with
respect to the actions described in \cite[\S 3.6.1]{FLLevelOne}
and is defined by the gauge group quotient of a vector bundle
embedding
$$
\pi_N^*\tXi_{\ft,\fs} \to \tsC_{\ft(\ell)}
\times
L^2_{k-1}(\La^+\otimes\fg_{\ft(\ell)}\oplus V^-_{\ft(\ell)}),
$$
where $\tXi_{\ft,\fs}\to\tM_{\fs}$ satisfies
$\tXi_{\ft,\fs}/\sG_{\fs}=\Xi_{\ft,\fs}$.

Define
$$
\Xi_\fs(\ft,\sP)=
\pi_N^*\tilde\Xi_{\ft,\fs}
\times_{\sG_{\fs}}
\sO(\ft,\fs,\sP,g_{\sP})
\subset
\pi_N^*\tilde\Xi_{\ft,\fs}
\times_{\sG_{\fs}}
\Gl(\ft,\fs,\sP).
$$
Then, the crude splicing construction
from \S \ref{subsec:CrudeSplicingMap}
(multiplying the obstruction sections
by the cut-off functions used to define the flattened section
$\Phi'$ in the definition of the flattening map $\bTheta_{\sP}$)
defines a vector bundle embedding,
$$
\varphi_{\fs}''(\sP):\Xi_{\fs}(\ft,\sP) \to \bar\fV_{\ft},
$$
covering the crude splicing map:
$$
\begin{CD}
\pi_N^*\tilde\Xi_{\ft,\fs}
\times_{\sG_{\fs}}
\sO(\ft,\fs,\sP,g_{\sP})
@> \varphi_{\fs}''(\sP) >>
\bar\fV_{\ft}
\\
@VVV @VVV
\\
\tilde N(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,g_{\sP})
@>\bga_{\ft,\fs,\sP}'' >> \bar\sC_{\ft}
\end{CD}
$$
The crude obstruction splicing maps $\varphi_{\fs}''(\sP')$ are
invariant under the action of the symmetric group and thus,
for $\sP<\sP'$ define a crude obstruction splicing map
$$
\varphi_{\fs}''([\sP<\sP'])
:
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,[\sP<\sP'])
\to
\bar\fV_{\ft}
$$
covering the crude splicing map $\bga''_{\ft,\fs,[\sP<\sP']}$
defined in \eqref{eq:UpperStratumCrudeSplicing}.

We can control the overlaps of the images of the embeddings $\varphi_{\fs}''(\sP)$
exactly as was done in the construction of the space of global splicing
data.  The overlap maps, $\rho^{N,\ft,\fs,u}_{\sP,[\sP']}$
and $\id_N\times\rho^{\ft,\fs,d}_{\sP,[\sP']}$, appearing in the diagram
\eqref{eq:GlobalSplicingCD} are covered by obvious bundle maps,
\begin{equation}
\label{eq:SWObstructionOverlap}
\begin{aligned}
{}&
\rho^{\Xi,\ft,\fs,u}_{\sP,[\sP']}:
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,[\sP'])
\to
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,[\sP<\sP'])
\\
{}&
\id_\Xi\times\rho^{\ft,\fs,d}_{\sP,[\sP']}:
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,[\sP'])
\to
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,\sP).
\end{aligned}
\end{equation}
Then, exactly as in the proof of Proposition
\ref{prop:XOverlapControl} (specifically the equality of
the section $\Phi_0'$), the diagram
\begin{equation}
\label{eq:SWObstructionOverlapDiagram}
\begin{CD}
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,\sP,[\sP'])
@> \rho^{\Xi,\ft,\fs,u}_{\sP,[\sP']} >>
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,[\sP<\sP'])
\\
@V \id_\Xi\times\rho^{\ft,\fs,d}_{\sP,[\sP']} VV
@V \varphi_{\fs}''([\sP<\sP']) VV
\\
\pi_N^*\tXi_{\fs}\times_{\sG_{\fs}}\sO(\ft,\fs,\sP)
@>\varphi_{\fs}''(\sP)>> \bar\fV_{\ft}
\end{CD}
\end{equation}
commutes and covers the diagram \eqref{eq:GlobalSplicingCD}.

We can therefore define a bundle
\begin{equation}
\bar\Upsilon^s_{\ft,\fs} \to \bar\sM^{\vir}_{\ft,\fs}
\end{equation}
as the union of the bundles $\Xi_s(\ft,\sP)$ as $\sP$ varies over
the partitions $\sP$ of $N_\ell$, patched together on
the overlaps by the diagram \eqref{eq:SWObstructionOverlapDiagram}.
The embeddings
$\varphi_s''(\sP)$ fit together to define a vector bundle
embedding
$$
\varphi''_s:\bar\Upsilon^s_{\ft,\fs} \to\bar\sM^{\vir}_{\ft,\fs}
$$
covering the embedding given by the crude splicing maps.

The embedding $\varphi''_s$ suffers from the same failings as
the crude splicing map:  it is
not equal to the identity.  We therefore define a new vector bundle
embedding following the isotopies of \eqref{eq:IsotopyOfSplicingMap}.
That is, let $\varphi'_s(\sP)$ be the standard slicing map
$$
\varphi'_s(\sP):\Xi_s(\ft,\fs,\sP) \to \bar\fV_{\ft}
$$
defined by using the cut-off functions vanishing on balls
$B(x_P,4\la_P^{1/3})$.  Define a sequence of vector bundle embeddings
$$
\varphi''_{s,j}:\bar\Upsilon^s_{\ft,\fs}\to\bar\fV_{\ft}
$$
exactly the embeddings $\bga''_{\sM,j}$ were defined in
\eqref{eq:IsotopyOfSplicingMap}, with the map
$\varphi'_s(\sP)$ in place of $\bga'_{\ft,\fs,\sP}$.
Define
\begin{equation}
\varphi'_s(\ft,\fs):\bar\Upsilon^s_{\ft,\fs}\to\bar\fV_{\ft}
\end{equation}
to be the end of the sequence of vector bundle embeddings
$\varphi''_{s,j}$.  It is a vector bundle embedding, covering
the global splicing map $\bga'_{\sM}$ of \eqref{eq:DefineGlobalSplice}.




\subsection{The instanton obstruction}
\label{subsec:InstantonObstr}
The other component of the obstruction is given by splicing in
the spliced-ends obstruction bundle over $\barM^{s,\natural}_{spl,\ka}(\delta)$.

\subsubsection{The frame bundles}
In \cite[\S 3.2]{FLLevelOne}, we defined a frame bundle
$$
\Fr_{\CCl(T^*X)}(V)\to \Fr(TX)\to X
$$
which is a $\U(2)$ bundle over $\Fr(TX)$ and a $\Spinu(4)$ bundle over
$X$.  The fiber over $F\in\Fr(TX)|_x$ is given by the Clifford module
isomorphisms from the standard Clifford module $\Delta\otimes\CC^2$
to $V|_x$ with respect to the
isomorphism $\RR^4\cong T^*X|_x$ defined by $F$.
The homomorphisms from $\Spinu(4)\to\SO(3)\times\SO(4)$
define a bundle map
$$
\Fr_{\CCl(T^*X)}(V) \to \Fr(\fg_{\ft})\times_X\Fr(TX).
$$
Analogously to the definition of the gluing data bundle
$\Fr(\ft,\fs,\sP,g_{\sP})$ in \eqref{eq:DefineGluingDataBundle},
we define
\begin{equation}
\Fr(V,\sP,g_{\sP}) \to \Fr(\ft,\fs,\sP,g_{\sP}),
\end{equation}
by
\begin{align*}
\Fr(V,\sP,g_{\sP})&=
\{(F^V_1,\dots,F^V_\ell)\in
\prod_{i=1}^\ell\Fr_{\CCl(T^*)}(V_{\ft(\ell)})|_{\Delta^\circ(X^\ell,\sP)}:
\\
&\qquad
F^V_i=F^V_j\quad\text{if and only there is $P\in\sP$ with $i,j\in P$}\}.
\end{align*}
The structure group of $\Fr(V,\sP,g_{\sP})\to \Delta^\circ(X^\ell,\sP)$
is
\begin{equation}
\begin{aligned}
\tilde G^V(\sP)&=\{(G_1,\dots,G_\ell)\in\prod_{i=1}^\ell\Spinu(4):
\\
&\qquad
G_i=G_j\quad\text{if there is $P\in\sP$ with $i,j\in P$}\}.
\end{aligned}
\end{equation}
The structure group of the bundle
$\Fr(V,\sP,g_{\sP})\to \Si(X^\ell,\sP)$ is
\begin{equation}
G^V(\sP)=\tilde G^V(\sP)\ltimes\fS(\sP).
\end{equation}
We then define the instanton obstruction bundle for the partition
$\sP$ by
\begin{equation}
\Ob(\ft,\fs,\sP)
=
\Fr(V,\sP,g_{\sP})
\times_{G^V(\sP)}
\prod_{P\in\sP}\bar\bD^{s,\natural}_{spl,|P|}.
\end{equation}

\subsubsection{The splicing map}
For each partition, $\sP$, of $N_\ell$, we define a splicing
map for the instanton obstruction,
\begin{equation}
\label{eq:InstantonObstrSplicingMap}
\varphi_{\ft,\fs,\sP}:
\tN(\delta)\times_{\sG_{\fs}}\Ob(\ft,\fs,\sP)
\to
\bar\fV_\ft,
\end{equation}
which will cover the crude splicing map $\bga''_{\ft,\fs,\sP}$.
Frames $(F^V_P)_{P\in\sP}\in\Fr(V,\sP)|_{\by}$,
where $\by=(y_P)_{P\in\sP}$ and a \spinu connection $A$
determine  embeddings
\begin{equation}
\label{eq:CliffordBundleIsomorphism}
\phi(A,F^V_P):V_{\ka_i}|_{B(0,8\la^{1/3})}
\cong
V_{\ft}|_{B(y_P,8\la^{1/3})}
\end{equation}
as described in \cite[\S 3.4.1]{FLLevelOne}.
If $\chi$ is a cut-off function supported on $B(0,8\la^{1/3})$
and $\Psi$ is a section of $V_{\ka_i}$, then
we write $\phi(A,F^V)(\chi\Psi)$ for the section of
$V_\ft|_{B(x,8\la^{1/3})}$ given by the isomorphism
\eqref{eq:CliffordBundleIsomorphism}.

To ensure the images of the different splicing maps
fit together coherently, we introduce a flattening map
on \spinu connections as was done for metrics and connections
in \S \ref{subsec:FlatteningPairs}.  That is, given a \spinu
connection $A_0$ on $V_{\ft(\ell)}$ we follow the construction
of Lemma \ref{lem:CompatibleFlattening} and for each
$\by\in \Si(X^\ell,\sP)$ define a locally flattened \spinu
connection $\bTheta^u(A,\by)$ on $V_{\ft(\ell)}$
satisfying
\begin{enumerate}
\item
For $\by=(y_P)_{P\in\sP}$,
$\bTheta^u(A_0,\by)$ is flat on
$$
\cup_{P\in\sP} B(y_p,4\tilde s_P(\by)^{1/3})
$$
\item
If $\sP<\sP'$ and $\by\in \Si(X^\ell,\sP)$ and
$\by'\in\Si(X^\ell,\sP')\cap \sU(X^\ell,\sP)$
satisfy $\pi(X^\ell,\sP)(\by')=\by$, then
$$
\bTheta^u(A_0,\by)=\bTheta^u(A_0,\by').
$$
\end{enumerate}
Henceforth, $\phi(A_0,F^V)$ will denote the isomorphism obtained by the
locally flattened connection $\bTheta^u(A_0,\by)$ where $F^V$ lies
over the support of $\by$.

For $(A_0,\Phi_0)\in N(\delta)\subset\sC_{\ft(\ell)}$
and a point in $\Ob(\ft,\fs,\sP)$
given by the data
$(F^V_P)_{P\in\sP}\in \Fr(V,\sP,g_{\sP})$,
$[A_P,F^s_P,\bv_P]\in\barM^{s,\natural}_{spl,|P|}(\delta_P)$
$[A_P,F^s_P,\bv_P,\Psi]\in\bar\Upsilon_{spl,|P|}$,
and
$$
\bga''_{\ft,\fs,\sP}((A_0,\Phi_0),F^V_P,[A_P,F^s_P,\bv_P])_{P\in\sP}
=(A'',\Phi'',\bx),
$$
the instanton obstruction splicing map is then defined by
\begin{equation}
\label{eq:DefineCrudeInstantonObstrSplicing}
\varphi_{\ft,\fs,\sP}
\left(
(A_0,\Phi_0),(F^V_P,[A_P,F^s_p,\bv_P,\Psi_P])_{P\in\sP}
\right)
=
\left(
A'',\Phi'',\bx,\sum_{P\in\sP}\phi(A_0,F^V_P)(\chi_{0,\la_P}\Psi_P)
\right)
\end{equation}
where the cut-off function $\chi_{0,\la_P}$ vanishes
on $B(0,\teighth\la^{1/3}_P)$
and is equal to one on $B(0, \tquarter \la^{1/2}_P)$.

\subsubsection{The overlap space and maps}
As is now standard, to control the overlap of the images
of the splicing maps $\varphi_{\ft,\fs,\sP}$ and
$\varphi_{\ft,\fs,\sP'}$, we introduce a space of
overlap data and overlap maps.
Analogously to the definition of the overlap space
$\Gl(\ft,\fs,\sP,[\sP'])$ in \eqref{eq:OverlapGluingDataSpace},
we define
$$
\Ob(\ft,\fs,\sP,[\sP'])
\to
\Gl(\ft,\fs,\sP,[\sP'])
$$
by
\begin{equation}
\begin{aligned}
{}&
\Ob(\ft,\fs,\sP,[\sP'])
\\
&\quad=
\Fr(V,\sP,g_{\sP})
\times_{G^V(\sP)}
\coprod_{\sP''\in[\sP<\sP']}
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P)
\times\prod_{Q\in\sP''_P} \bar\Upsilon_{spl,|Q|}
\right).
\end{aligned}
\end{equation}
We define overlap maps,
$$
\rho^{V,d}_{\sP,\sP'}:
\Ob(\ft,\fs,\sP,[\sP'])
\to
\Ob(\ft,\fs,\sP)
$$
covering the overlap maps $\rho^{\ft,\fs,d}_{\sP,[\sP']}$ by
applying the obstruction splicing map
$\varphi_{\Theta,\sP''_P}$
defined in \eqref{eq:DefineS4SectionSplicingToTrivial}
to the data given by the
factor
$$
\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P)
\times\prod_{Q\in\sP''_P} \bar\Upsilon_{spl,|Q|}
$$
in the definition of $\Ob(\ft,\fs,\sP,[\sP'])$.
More formally, the downwards overlap map is defined
by
\begin{equation}
\rho^{V,d}_{\sP,\sP'}=
\id_{\Fr(V)}\times
\coprod_{\sP''\in [\sP<\sP']}
\prod_{P\in\sP}\varphi_{\Theta,\sP''_P}
\end{equation}
where $\id_{\Fr(V)}$ is the identity map on
$\Fr(V,\sP,g_{\sP})$.

We define an upwards overlap map to the space,
$$
\Ob(\ft,\fs,[\sP<\sP'])
=
\left(
\coprod_{\sP''\in[\sP<\sP']}
\Ob(\ft,\fs,\sP'')
\right)/\fS(\sP)
$$
exactly as was done in
\eqref{eq:UnparamUpwardsTransition} and
\eqref{eq:DefineParameterizedUpwardsTransitionMap}.
Recall that the map
$\rho^{\ft,\fs,u}_{\sP,[\sP']}$ was defined by
a parallel translation of the frames in
the domain $\sO(\ft,\fs,\sP,[\sP'],g_{\sP})$
and leaving the $S^4$ connections unchanged.
Note that this parallel translation is done with
respect to the locally flattened connection $\bTheta^u(A_0,\by)$.
Thus, the map
\begin{equation}
\rho^{N,V,u}_{\sP,[\sP']}:
\tilde N(\delta)\times_{\sG_{\fs}}
\Ob(\ft,\fs,\sP,[\sP'])
\to
\tilde N(\delta)\times_{\sG_{\fs}}
\Ob(\ft,\fs,[\sP<\sP'])
\end{equation}
can be defined identically, using the flattened
\spinu connection $A_0$ to parallel translate the frames
of $\Fr_{\CCl(T^*X)}(V_{\ft(\ell)})$ and leaving the elements
of $\bar\Upsilon_{spl,|Q|}$ unchanged.

By the invariance of the obstruction splicing maps under
the action of the symmetric group, the splicing maps
$\varphi_{\ft,\fs,\sP''}$ for $\sP''\in [\sP<\sP']$
define a splicing map,
\begin{equation}
\varphi_{\ft,\fs,[\sP<\sP']}:
\tilde N(\delta)\times_{\sG_{\fs}}
\Ob(\ft,\fs,[\sP<\sP'])
\to
\bar\fV_{\ft}
\end{equation}
which covers the crude splicing map
$\bga''_{\ft,\fs,[\sP<\sP']}$ of \eqref{eq:UpperStratumCrudeSplicing}.
We then have the following relation between the splicing maps
$\varphi_{\ft,\fs,\sP}$ and $\varphi_{\ft,\fs,[\sP<\sP']}$.

\begin{lem}
\label{lem:InstantonObstrCommDiagr}
Let $\sP<\sP'$ be partitions of $N_\ell$.  Then the following diagram
commutes:
\begin{equation}
\label{eq:InstantonObstrCommDiagr}
\begin{CD}
\tilde N(\delta)\times_{\sG_{\fs}}
\Ob(\ft,\fs,\sP,[\sP'])
@>\rho^{N,V,u}_{\sP,[\sP']} >>
\tilde N(\delta)\times_{\sG_{\fs}}
\Ob(\ft,\fs,[\sP<\sP'])
\\
@V \id_N\times\rho^{V,d}_{\sP,[\sP']} VV
@V \varphi_{\ft,\fs,[\sP<\sP']} VV
\\
\tilde N(\delta)\times_{\sG_{\fs}}
\Ob(\ft,\fs,\sP)
@> \varphi_{\ft,\fs,\sP} >> \bar\fV_{\ft}
\end{CD}
\end{equation}
where $\id_N$ is the identity map on $\tilde N(\delta)$,
covering the diagram \eqref{eq:GlobalSplicingCD}.
\end{lem}

\begin{proof}
That the diagram \eqref{eq:InstantonObstrCommDiagr} covers
the diagram \eqref{eq:GlobalSplicingCD} follows immediately
from the definition of the overlap and obstruction splicing
maps.  Then, as in the proof of Proposition
\ref{prop:XOverlapControl}, assume that the splicing maps
$\varphi_{\ft,\fs,\sP}$ and $\varphi_{\ft,\fs,\sP'}$ are
defined at points $\by\in\Si(X^\ell,\sP)$ and
$\by'\in\Si(X^\ell,\sP')$ respectively where
$\pi(X^\ell,\sP)(\by')=\by$.  Thus, for
$P'\in\sP'$ and $P'\subseteq P\in\sP$, the inclusion
$$
B(y_{P'}, 8\tilde s(\by')^{1/3})
\Subset
B(y_P,4 \tilde s(\by)^{1/2})
$$
holds by \eqref{eq:BallInclusions}.
Because the metric and \spinu connection used in the splicing
argument are flat on the above balls,
the lemma then follows from the argument giving
Lemma \ref{lem:OverlapObstructionCommuting}.
\end{proof}

The commutativity of the diagram \eqref{eq:InstantonObstrCommDiagr}
and the definition of the space $\bar\sM^{\vir}_{\ft,\fs}$
as the union of the spaces
$$
\tN(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP)
$$
implies that we can define a vector pseudo-bundle
\begin{equation}
\bar\Upsilon^i_{\ft,\fs} \to \bar\sM^{\vir}_{\ft,\fs}
\end{equation}
as the union of the pseudo-bundles
$$
\tN(\delta)\times_{\sG_{\fs}}\Ob(\ft,\fs,\sP)
\to
\tN(\delta)\times_{\sG_{\fs}}\sO(\ft,\fs,\sP),
$$
with the overlaps identified by the diagram
\eqref{eq:InstantonObstrCommDiagr}.

The splicing maps $\varphi_{\ft,\fs,\sP}$ define
a vector bundle embedding of $\bar\Upsilon^i_{\ft,\fs}$
into $\bar\fV_{\ft}$ which covers the embedding of
$\bar\sM^{\vir}_{\ft,\fs}$ into $\bar\sC_{\ft}$ given
by the crude splicing maps.
Finally, we define an embedding
\begin{equation}
\varphi'_i:\bar\Upsilon^i_{\ft,\fs}\to\bar\fV_{\ft}
\end{equation}
covering the global splicing map $\bga'_{\sM}$ of
\eqref{eq:DefineGlobalSplice} by replacing the
triple $(A'',\Phi'',\bx)$ in the definition
\eqref{eq:DefineCrudeInstantonObstrSplicing} with the triple
$(A',\Phi',\bx)$ given by the global splicing map $\bga'_{\sM}$.


\subsection{The gluing theorem}
\label{subsec:GluingThm}
We are now in a position to state the relevant gluing theorem.


\begin{thm}
\label{thm:Gluing}
There is a continuous, $S^1$-equivariant embedding
\begin{equation}
\label{eq:GluingMap}
\bga_{\sM}:\bar\sM^{\vir}_{\ft,\fs} \to \bar\sC_{\ft}
\end{equation}
which is homotopic
through $S^1$-equivariant embeddings to the global splicing map $\bga'_{\sM}$,
smooth on each stratum of $\bar\sM^{\vir}_{\ft,\fs}$,
and
equal to the identity on $N(\delta)\times\Sym^\ell(X)$.
In addition, there are continuous, $S^1$-equivariant sections $\fo_s$ and $\fo_i$
of the bundles $\bar\Upsilon^s_{\ft,\fs}$ and
$\bar\Upsilon^i_{\ft,\fs}$ such that
\begin{enumerate}
\item
The section $\fo=\fo_s\oplus \fo_i$ of
$\bar\Upsilon^s_{\ft,\fs}\oplus \bar\Upsilon^i_{\ft,\fs}$
vanishes transversely on the top stratum of $\bar\sM^{\vir}_{\ft,\fs}$,
\item
The restriction of $\bga_{\sM}$ to the zero-locus
$\fo^{-1}(0)$ parameterizes a neighborhood of $M_{\fs}\times\Sym^\ell(X)$
in $\bar\sM_{\ft}$.
\end{enumerate}
\end{thm}

%end of file
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%Article: INTERSECTION
%File: cohom.tex
%Last update: February 15th 2002 (Tom)
%Spellcheck: March 5, 2002 (Paul)
%Todo:

\section{Cohomology}
\label{sec:Cohom}
In this section, we compute
the pullback of the cohomology classes by the global splicing map.
We define the cohomology classes and maps in terms of which these
pullbacks will be expressed in \S \ref{subsec:CohomDefn}.
We compute the pullbacks of the cohomology classes $\mu_p(\beta)$
and $\mu_c$ in \S \ref{subsec:MuClasses}  the Euler class of the Seiberg-Witten
component of the obstruction in \S \ref{subsec:SWObstrEuler}.
The computation of the Euler class of the instanton component of
the obstruction occupies \S \ref{subsec:InstObstrEulerLocal}, where
we compute the restriction of this Euler class to the top stratum
of the subspaces $\sU(\ft,\fs,\sP_i)\subset\bar\sM^{\vir}_{\ft,\fs}/S^1$
and \S \ref{subsec:InstObstrEulerGlobal} where discuss how these
local computations yield the global cohomology class.
Finally, in \S \ref{subsec:Extensions}, we define some extensions
of these cohomology classes from the top stratum of
$\bar\sM^{\vir}_{\ft,\fs}/S^1$ the complement of the reducible monopoles.

\subsection{Definitions}
\label{subsec:CohomDefn}

\subsubsection{Subspace and maps}
We begin by defining subspaces of $\bar\sM^{\vir}_{\ft,\fs}$.
First,
\begin{equation}
\label{eq:TopStratumOfVirtual}
\sM^{\vir}_{\ft,\fs}
\subset
\bar\sM^{\vir}_{\ft,\fs}
\end{equation}
will denote the top stratum which is mapped to $\sC_{\ft}$ by
the splicing and gluing maps $\bga'_{\sM}$ and $\bga_{\sM}$.
Let
\begin{equation}
\label{eq:TopStratumOfVirtualInclusion}
\iota:\sM^{\vir}_{\ft,\fs}\to\bar\sM^{\vir}_{\ft,\fs}
\end{equation}
be the inclusion.  Next, we define the complement of the reducible points,
\begin{equation}
\label{eq:RedComplementInVirtual}
\bar\sM^{\vir,*}_{\ft,\fs}
=
\bar\sM^{\vir}_{\ft,\fs} - M_{\fs}\times\Sym^\ell(X).
\end{equation}
The $S^1$ action is free on the subspace
$\bar\sM^{\vir,*}_{\ft,\fs}$.

The map
$$
\pi_N:\bar\sM^{\vir}_{\ft,\fs}\to N_{\ft_0,\fs}(\delta),
$$
was defined in \eqref{eq:GlobalProjectionToN}
and we define
\begin{equation}
\label{eq:ProjectionToSW}
\pi_{\fs}:\bar\sM^{\vir}_{\ft,\fs}\to M_{\fs}
\end{equation}
as the composition of $\pi_N$ with the projection
$N_{\ft_0,\fs}(\delta)\to M_{\fs}$.
Recall that the projection
$$
\pi_X:\bar\sM^{\vir}_{\ft,\fs}\to \Sym^\ell(X)
$$
was defined in \eqref{eq:GlobalProjectionToX}.
We will also write
\begin{equation}
\pi_{\fs,X}:\bar\sM^{\vir}_{\ft,\fs}
\to M_{\fs}\times\Sym^\ell(X)
\end{equation}
and
\begin{equation}
\pi_{N,X}:\bar\sM^{\vir}_{\ft,\fs}
\to N_{\ft_0,\fs}(\delta)\times\Sym^\ell(X)
\end{equation}
for the projections.



We use the same notation to denote the projections
from $\bar\sM^{\vir}_{\ft,\fs}/S^1$ (although note that
$\pi_N$ maps this quotient to $N_{\ft_0,\fs}(\delta)/S^1$).

\subsubsection{Cohomology classes}
We define some cohomology classes on $\bar\sM^{\vir,*}_{\ft,\fs}/S^1$.

\begin{defn}
Let $\nu\in H^2(\bar\sM^{vir,*}_{\ft,\fs}/S^1;\ZZ)$
be the first Chern class of the $S^1$ bundle,
\begin{equation}
\bar\sM^{vir,*}_{\ft,\fs}\to \bar\sM^{vir,*}_{\ft,\fs}/S^1.
\end{equation}
If
\begin{equation}
\label{eq:LineBundleForS1ZAction}
\LL_\nu=\bar\sM^{vir,*}_{\ft,\fs}\times_{S^1}\CC,
\end{equation}
to be the complex line bundle associated to the $S^1$ action on
$\bar\sM^{vir,*}_{\ft,\fs}$ so $c_1(\LL_\nu)=\nu$.
\end{defn}

\begin{defn}
For $\beta\in H_\bullet(X;\RR)$,
let $\mu_{\fs}(\beta)\in H^2(M_{\fs};\ZZ)$ be
the $\mu$-class defined in \S \ref{sec:prelim}.
We will use the same notation for the pullback, by the
projection $\pi_{\fs}$, of these classes to
$\bar\sM^{\vir}_{\ft,\fs}/S^1$.  Define
\begin{equation}
\LL_{\fs}
=
\tM_{\fs}\times_{\sG_{\fs}}X\times\CC
\to
M_{\fs}\times X.
\end{equation}
\end{defn}

\begin{defn}
If $\beta\in H^\bullet(X;\RR)$ and $\pi_i:X^\ell\to X$
is projection onto the $i$-th factor,
let $S^\ell(\beta)\in H^\bullet(\Sym^\ell(X);\RR)$ be the
cohomology class which pulls back to $\sum_i\pi_i^*\beta$
under the projection $X^\ell\to\Sym^\ell(X)$.
\end{defn}

The incidence locus,
\begin{equation}
\label{eq:Incidence}
\sV_\ell(\Delta)\subset \Sym^\ell(X)\times X
\end{equation}
is defined to be the points $(\bx,y)\in \Sym^\ell(X)\times X$ such
that $y$ is in the support of $\bx$.  Alternately, one can describe
it
$$
\sV_\ell(\Delta)
=
\left( \cup_i (\pi_i\times\id_X)^{-1}(\Delta)\right)/\fS_\ell,
$$
where $\pi_i:X^\ell\to X$ is projection onto the $i$-th factor
and $\Delta\subset X\times X$ is the diagonal.
Similarly, we define
\begin{equation}
\label{eq:IncidenceNormal}
\sV_\ell(\nu(\Delta))
=
\left( \cup_i (\pi_i\times\id_X)^{-1}(\nu(\Delta))\right)/\fS_\ell,
\end{equation}
where $\nu(\Delta)\subset X\times X$ is the normal bundle of $\Delta$.
Let
\begin{equation}
\label{eq:DefineDualOfIncidence}
\PD[\sV]\in H^4(\Sym^\ell(X)\times X;\ZZ),
\end{equation}
be the image of the Thom class of the disk bundle
$\sV_\ell(\nu(\Delta))\to \sV_\ell(\Delta)$ under the excision isomorphism
and the map
$$
H^4(\Sym^\ell(X)\times X,\Sym^\ell(X)\times X-\sV_\ell(\Delta);\ZZ)
\to
H^4(\Sym^\ell(X)\times X;\ZZ).
$$
From the equality on Poincare duality
given in \cite[Theorem 30.6]{GreenbergHarper},
we have

\begin{lem}
\label{lem:PDual}
For any $\beta\in H_\bullet(X;\RR)$,
$$
\PD[\sV]/\beta= S^\ell(\PD[\beta]).
$$
\end{lem}

\subsection{The $\mu$-classes}
\label{subsec:MuClasses}
We describe the pullbacks of the cohomology classes $\mu_c$ and
$\mu_p(\beta)$ by the gluing map in terms of the cohomology classes
introduced in \S \ref{subsec:CohomDefn}.

The computation of the pullback of $\mu_c$ is identical to that
appearing in \cite[Lemma 4.2]{FLLevelOne}.

\begin{lem}
\label{lem:PullbackOfMuC}
If $\mu_c\in H^2(\sC^{*,0}_{\ft}/S^1;\ZZ)$ is the cohomology
class defined in \eqref{eq:DefineMuC1},
$\bga_{\sM}$ is the gluing map of \eqref{eq:GluingMap},
and $\iota$ is the inclusion \eqref{eq:TopStratumOfVirtualInclusion}
then
\begin{equation}
\label{eq:MuCPullback}
\bga_{\sM}^*\mu_c=-\iota^*\nu.
\end{equation}
\end{lem}

To compute the pullback of the $\mu_p$ classes, we must study
the incidence locus $\sV_\ell(\Delta)$ more carefully.
Define
\begin{equation}
\label{eq:DefineIncidenceComplement}
\sO=
\sM^{\vir}_{\ft,\fs}/S^1 \times X
-
(\pi_X\times\id_X)^{-1} \sV_\ell(\nu(\Delta))
\end{equation}
We will consider compare the pullback of the universal bundle
$(\bga\times\id_X)^*\FF_{\ft}$, restricted to $\sO$, with
the same restriction of the bundle
\begin{equation}
\label{eq:VirtualUniversal}
\FF^{\vir}_{\ft,\fs}=
\cup_{\sP}\left(
\tN(\delta)\times\Gl(\ft,\fs,\sP)\times_{\sG_{\fs}\times S^1} \fg_{\ft}\right)
\end{equation}
which is defined analogously to the bundle in
\cite[Equation (4.7)]{FLLevelOne}.
The proof of \cite[Lemma 4.7]{FLLevelOne} then yields the
following expression for $p_1(\FF^{\vir}_{\ft,\fs})$.

\begin{lem}
\label{lem:PontrjaginOfVirtualUniv}
Assume the \spinu structure $\ft_\ell$ admits a splitting
$\ft_\ell=\fs\oplus \fs\otimes L$.
Let $\FF^{\vir}_{\ft,\fs}\to \sM^{\vir}_{\ft,\fs}/S^1\times X$
be the bundle defined in \eqref{eq:VirtualUniversal}.
Let $\pi_{X,2}:\sM^{\vir}_{\ft,\fs}/S^1\times X\to X$
be the projection onto the second factor.  Then,
\begin{equation}
\label{eq:PontrjaginOfVirtualUniv}
p_1(\FF^{\vir}_{\ft,\fs})
=
\iota^*\left(
(2\pi_{\fs}\times\id_X)^*c_1(\LL_\fs)-\nu+\pi_{X,2}^*c_1(L)
\right)^2.
\end{equation}
\end{lem}

The following comparison of
$\FF^{\vir}_{\ft,\fs}$ with
$(\bga_{\sM}\times\id_X)^*\FF_{\ft}$ is proven in
\cite[Lemma 4.3]{FLLevelOne}.

\begin{lem}
\label{lem:CompareUnivBundles}
Assume the \spinu structure $\ft_\ell$ admits a splitting
$\ft_\ell=\fs\oplus \fs\otimes L$.  Let
$\sO\subset \sM^{\vir}_{\ft,\fs}/S^1 \times X$
be the subspace defined in \eqref{eq:DefineIncidenceComplement}.
Then,
\begin{equation}
(\bga_{\sM}\times\id_X)^*\FF_{\ft}|_{\sO}
\cong
\FF^{\vir}_{\ft,\fs}|_{\sO}.
\end{equation}
\end{lem}
We specify the difference between the Pontrjagin classes
of the bundles in Lemma \ref{lem:CompareUnivBundles}
in the following.

\begin{lem}
\label{lem:CompareUnivPont}
Assume the \spinu structure $\ft_\ell$ admits a splitting
$\ft_\ell=\fs\oplus \fs\otimes L$.
Let $\PD[\sV]$ be the cohomology class defined in
\eqref{eq:DefineDualOfIncidence}.
\begin{equation}
(\bga_{\sM}\times\id_X)^*p_1(\FF_{\ft})
=
p_1(\FF^{\vir}_{\ft,\fs})-4(\iota\times\id_X)^*(\pi_X\times\id_X)^*\PD[\sV].
\end{equation}
\end{lem}

Let $\ga_1,\dots,\ga_b\in H_1(X;\RR)$ be a basis
with $\bga_1,\dots,\bga_b\in H^1(X;\RR)$ the basis
satisfying $\langle \bga_i^*,\bga_j\rangle=\delta_{ij}$.
Let $\bga^{J,*}_1,\dots,\bga^{J,*}_b\in H^1(M_\fs;\RR)$
be the related basis as defined in
\cite[Definition 2.2]{FL2a} (there written as $\br^*\bga^{J,*}_i$).
Standard computations (compare \cite[Lemma 4.10]{FLLevelOne}
and \cite[Corollary 4.7]{FL2b}),
Lemma \ref{lem:PDual}, and the equality
$$
c_1(\LL_{\fs})=\mu_\fs\times 1 +\sum_{i=1}^{b_1} \bga^{J,*}_i\times \bga^*_i
$$
from \cite[Lemma 2.24]{FL2a}
then yield the following.

\begin{cor}
\label{cor:CohomologyClasses}
Assume the \spinu structure $\ft_\ell$ admits a splitting
$\ft_\ell=\fs\oplus \fs\otimes L$,
so $c_1(L)=c_1(\ft)-c_1(\fs)$.
Let $\bga_\sM$ be the gluing map defined in \eqref{eq:GluingMap}
and $\iota$ the inclusion defined in
\eqref{eq:TopStratumOfVirtualInclusion}.  Then for
$x\in H_0(X;\ZZ)$ a generator,
$\bga_i\in H_1(X;\ZZ)$ a generator as above,
$\beta\in H_{2}(X;\RR)$, and  $[Y]\in H_3(X;\RR)$,
\begin{equation}
\label{eq:PullbacksOfMuAndNu}
\begin{aligned}
\bga_{\sM}^*\mu_c(x) &= -\iota^*\nu,
\\
\bga_{\sM}^*\mu_p(x)
&=
\iota^*\left(-\tquarter (2\mu_\fs-\nu)^2+S^\ell(x)\right),
\\
\bga_{\sM}^*\mu_p(\bga_i)
&=
\iota^*\left(-\thalf(2\mu_\fs-\nu)\bga^{J,*}_i+ S^\ell(\bga_i)\right),
\\
\bga_{\sM}^*\mu_p(\beta)
&=
\iota^*\left(-\thalf2\mu_\fs-\nu)\langle c_1(\ft)-c_1(\fs),\beta\rangle
-2\sum_{i<j}\mu_{\fs}(\bga_i\bga_j)\langle \bga_i^*\smile\bga_j^*,\beta\rangle
+S^\ell(\beta)\right),
\\
\bga_{\sM}^*\mu_p(\beta_3)
&=
\iota^*\left(-\sum_i (c_1(\ft)-c_1(\fs)\smile \bga^*_i,[Y]\rangle \mu_{\fs}(\bga_i)
+S^\ell(\beta)\right),
\end{aligned}
\end{equation}
\end{cor}


\subsection{The Seiberg-Witten obstruction Euler class}
\label{subsec:SWObstrEuler}

The computation of the Euler class of the Seiberg-Witten, or background,
component of the obstruction is identical to that given in
\cite[Lemma 4.11]{FLLevelOne}.

\begin{lem}
\label{lem:SWObstruction}
Let $r_\Xi$ be the complex rank of the background obstruction bundle
$\Upsilon^s_{\ft,\fs}\to \bar\sM^{\vir,*}_{\ft,\fs}$.  Then,
\begin{equation}
e(\Upsilon^s_{\ft,\fs}/S^1)=\iota^*(-\nu)^{r_\Xi}.
\end{equation}
\end{lem}

\subsection{The local instanton obstruction Euler class}
\label{subsec:InstObstrEulerLocal}

We first give a description of the Euler class
of the instanton component of the obstruction bundle,
$\Upsilon^i_{\ft,\fs}/S^1\to\sM^{\vir}_{\ft,\fs}/S^1$
restricted to $\sO(\ft,\fs,\sP)$.

We first introduce an additional cohomology class.
Let $\pi_{\fs,i}:M_{\fs}\times X^\ell\to M_{\fs}\times X$ be
defined by the identity on $M_{\fs}$ and projection from $X^\ell$
onto the $i$-th factor of $X$.  Then,
the Chern classes of the bundle $\oplus_{i=1}^\ell \pi_{\fs,i}^*\LL_{\fs}$
are invariant under the action of the symmetric group and thus define
cohomology classes
\begin{equation}
\label{eq:ChernClassesOfSymSWUniv}
c_{\fs,\ell,i}\in H^{2i}(M_{\fs}\times \Sym^\ell(X);\RR).
\end{equation}
In addition, we will write
$$
\iota_{\sP}: \Si(X^\ell,\sP)\to\Sym^\ell(X),
\quad
\tilde\iota_{\sP}:\Delta^\circ(X^\ell,\sP)\to X^\ell,
$$
for the inclusions.
Finally,
we use the generic term
\begin{equation}
\label{eq:SpinuCharacter}
c(\ft)\in H^\bullet(X;\RR),
\end{equation}
for rational characteristic classes of the bundle
$\Fr_{\CCl(T^*X)}(V)\to X$.
These characteristic classes include $c_1(\ft)$, $p_1(\ft)$, $p_1(X)$,
and $e(X)$.

\begin{prop}
\label{prop:LocalInstantonEuler}
The Euler class of the restriction of the obstruction bundle
$\Upsilon^i_{\ft,\fs}/S^1$ to $\sO(\ft,\fs,\sP)$ is given
by a polynomial in the cohomology classes
\begin{equation}
\nu, \quad
\pi_X^*\iota_{\sP}^*S^\ell(c(\ft)), \quad\text{and}\quad
\pi_{X,\fs}^*\iota_{\sP}^*c_{\fs,\ell,i},
\end{equation}
which is otherwise independent of $X$.
\end{prop}

To prove Proposition
\ref{prop:LocalInstantonEuler},
we begin by observing that the space $\sO(\ft,\fs,\sP)$ retracts to
the intersection of $\sO(\ft,\fs,\sP)$ with the subspace
\begin{equation}
\label{eq:SWTopStratumSubspace}
\tM_{\fs}\times_{\sG_{\fs}\times S^1}\Gl(\ft,\fs,\sP).
\end{equation}
We compute the pullback of the restriction of the
Euler class of the instanton obstruction to
the subspace \eqref{eq:SWTopStratumSubspace}
by the map,
\begin{equation}
\label{eq:LocalCoveringMap}
\tM_{\fs}\times_{\sG_{\fs}\times S^1}\tilde{\Gl}(\ft,\fs,\sP)
\to
\tM_{\fs}\times_{\sG_{\fs}\times S^1}\Gl(\ft,\fs,\sP)
\end{equation}
defined by the projection
$$
\tilde{\Gl}(\ft,\fs,\sP)=
\Fr(\ft,\fs,\sP)\times_{\tG(\sP)}\prod_{P\in\sP} M^{s,\natural}_{spl,|P|}(\delta_P)
\to
\Gl(\ft,\fs,\sP),
$$
where the group $\tG(\sP)$ is defined in
\eqref{eq:DefineGluingDataBundleStructureGroup}
(essentially by omitting the symmetric group component of $G(\sP)$).
Observe that the preceding map appears in the diagram,
$$
\begin{CD}
\tilde{\Gl}(\ft,\fs,\sP) @>>> \Gl(\ft,\fs,\sP)
\\
@VVV @VVV
\\
\Delta^\circ(X^\ell,\sP) @>>> \Si(X^\ell,\sP)
\end{CD}
$$
For each $P\in\sP$,
the projection $\tilde{\Gl}(\ft,\fs,\sP)\to\Delta^\circ(X^\ell,\sP)$
fits into a diagram
\begin{equation}
\label{eq:DefineProjectionOfGluingParamBundleOntoFactor}
\begin{CD}
\tilde{\Gl}(\ft,\fs,\sP) @> \pi_{\tilde{\Gl},P} >>
\Fr(\fg_0)\times_X\Fr(TX)\times_{\SO(3)\times\SO(4)}
M^{s,\natural}_{spl,|P|}(\delta_P)
\\
@VVV @VVV
\\
\Delta^\circ(X^\ell,\sP) @> \pi_P >> X
\end{CD}
\end{equation}
Then, we have the following:

\begin{lem}
\label{lem:LocalInstantonPullbackToCover}
The pullback of the restriction of
$\Upsilon^i_{\ft,\fs}/S^1$ to $\sO(\ft,\fs,\sP)$
by the covering map \eqref{eq:LocalCoveringMap}
splits into a direct sum,
$\oplus_{P\in\sP}\pi_{\tilde{\Gl},P}^*\Upsilon^i_{\ft,\fs}(P)/S^1$,
where the bundle
\begin{equation}
\Upsilon^i_{\ft,\fs}(P)/S^1
\to
\tM_{\fs}\times_{\sG_{\fs}\times S^1}
\Fr(\fg_\ell)\times_X\Fr(TX)\times_{\SO(3)\times\SO(4)}
M^{s,\natural}_{spl,|P|}(\delta_P)
\end{equation}
is defined by
\begin{equation}
\label{eq:FactorOfInstantonEuler}
\begin{CD}
\tM_{\fs}\times_{\sG_{\fs}\times S^1}
\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}\Upsilon^i_{spl,|P|}
\\
@VVV
\\
\tM_{\fs}\times_{\sG_{\fs}\times S^1}
\Fr(\fg_\ell)\times_X\Fr(TX)\times_{\SO(3)\times\SO(4)}
\barM^{s,\natural}_{spl,|P|}(\delta_P)
\end{CD}
\end{equation}
and the map $\pi_{\tilde{\Gl},P}$ is defined in the diagram
\eqref{eq:DefineProjectionOfGluingParamBundleOntoFactor}.
\end{lem}

The $S^1$ action in \eqref{eq:FactorOfInstantonEuler}
reflects the action of $S^1$ on
the infinite-dimensional obstruction space.
Thus, following  \cite[Lemma 3.7]{FLLevelOne}, the $S^1$
action on
\begin{equation}
\label{eq:FactorOfInstantonEuler1}
\tM_{\fs}\times_{\sG_{\fs}}
\Fr_{\CCl(T^*X)}(V)\times_{\Spinu(4)}
\Upsilon^i_{spl,|P|}
\end{equation}
is given by the diagonal action of the action by
$\varrho_{\fs}(e^{-i\theta})$ on $\Fr_{\CCl(T^*X)}(V)$
and by scalar multiplication with weight one on the fibers of
$\Upsilon^i_{spl,|P|}$.

Lemma \ref{lem:LocalInstantonPullbackToCover}
implies that to compute the restriction
of the instanton obstruction bundle to the subspace
\eqref{eq:SWTopStratumSubspace},
it suffices to characterize the Euler class of
the bundle \eqref{eq:FactorOfInstantonEuler}.
We use the notation
$$
\Gl(\ft,\fs,P)
=
\Fr(\fg_\ell)\times_X\Fr(TX)\times_{\SO(3)\times\SO(4)}
M^{s,\natural}_{spl,|P|}(\delta_P)
$$
Then, the base of the bundle $\Upsilon^i_{\ft,\fs}(P)/S^1$
admits a product decomposition:
\begin{equation}
\label{eq:XFactorProductDecomp}
\tM_{\fs}\times_{\sG_{\fs}\times S^1}\Gl(\ft,\fs,P)
\cong
M_{\fs}\times S^1\backslash\Gl(\ft,\fs,P).
\end{equation}
Define
\begin{equation}
\label{eq:XFactorObstruction}
\Upsilon^i_{X,P}/S^1
=
M_{\fs}\times
S^1\backslash\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}\Upsilon^i_{spl,|P|}
\to
M_{\fs}\times S^1\backslash\Gl(\ft,\fs,P).
\end{equation}
As in the discussion of the $S^1$ action on
\eqref{eq:FactorOfInstantonEuler1},
the $S^1$ action in \eqref{eq:XFactorObstruction}
is given by the diagonal action of the action by
$\varrho_{\fs}(e^{-i\theta})$ on $\Fr_{\CCl(T^*X)}(V)$
and by scalar multiplication on the fibers of
$\Upsilon^i_{spl,|P|}$.

\begin{lem}
\label{lem:RelateInstantonObstructionFactors}
Let $\LL_{\fs}\to M_{\fs}\times X$ be the universal Seiberg-Witten
line bundle defined in ??.
Let $\pi_{X,\fs}:M_{\fs}\times S^1\backslash\Gl(\ft,\fs,P)\to M_{\fs}\times X$
be the projection.  Then,
\begin{equation}
\label{eq:RelateInstantonObstructionFactors}
\Upsilon^i_{\ft,\fs}(P)/S^1
\cong
\pi_{X,\fs}^*\LL_{\fs}^*\otimes\Upsilon^i_{X,P}/S^1.
\end{equation}
\end{lem}

\begin{proof}
By the technical result \cite[Lemma 3.27]{FL2a},
the tensor product in \eqref{eq:RelateInstantonObstructionFactors}
can be described as:
\begin{equation}
\label{eq:FiberProductToTensor}
\left(
\left(
\tM_{\fs}\times_{\sG_{\fs}} X\times S^1
\right)_{M_{\fs}\times X}\Upsilon^i_{X,P}/S^1
\right)/S^1,
\end{equation}
where $S^1$ acts diagonally on the factor of $S^1$ and
on the fiber of $\Upsilon^i_{X,P}/S^1$.
Because the $S^1$ quotient in the definition of $\Upsilon^i_{X,P}/S^1$
acts anti-diagonally (see the remarks following
\eqref{eq:XFactorObstruction}), we can then say that
the final $S^1$ quotient in \eqref{eq:FiberProductToTensor}
acts diagonally on the factor of $S^1$ and by the action
$\varrho_{\fs}(e^{i\theta})$ on the factor of $\Fr_{\CCl(T^*X)}(V)$
in the definition of $\Upsilon^i_{X,P}/S^1$.
One then defines a bundle map from the fibered product
\eqref{eq:FiberProductToTensor} to the bundle
$\Upsilon^i_{\ft,\fs}(P)/S^1$ by
$$
\left(
(A_0,\Phi_0,x,e^{i\theta},\tilde F_u,[A,F_s,\Psi]
\right)
\to
\left(
(A_0,\Phi_0,e^{-i\theta}\tilde F_u,[A,F_s,\Psi]
\right),
$$
where $(A_0,\Phi_0)\in\tM_{\fs}$, $x\in X$, $e^{i\theta}\in S^1$,
$\tilde F_u\in\Fr_{\CCl(T^*X)}(V)|_x$, and
$[A,F_s,\Psi]\in\Upsilon^i_{spl,|P|}$.
Observe that the map is well-defined because the diagonal
$S^1$ action described earlier in the proof vanishes in
the product $e^{-i\theta}\tilde F_u$.
The equivariance of this map with respect to the $\sG_{\fs}$ action is
correct because the action of $\sG_{\fs}$ defining $\LL_{\fs}$ is
given by:
$$
\left(u,(A_0,\Phi_0,x,z)\right)\to
\left( u_*(A_0,\Phi_0),x,u(x)^{-1}z\right),
$$
and thus using $e^{-i\theta}\tilde F_u$ takes the preceding
action of $\sG_{\fs}$ to the desired diagonal action on $\tilde M_{\fs}$
and $\Fr_{\CCl(T^*X)}(V)$.
\end{proof}

Next, we characterize the (rational)
Euler class of $\Upsilon^i_{X,P}/S^1$.  To this end, we
introduce the cohomology class,
\begin{equation}
\label{eq:XFactorS1Chern}
\nu_{\Gl}\in H^2(M_{\fs}\times S^1\backslash\Gl(\ft,\fs,P)),
\end{equation}
(denoted $\nu_\ft$ in \cite[Definition 5.22]{FLLevelOne})
to be the first Chern class of the $S^1$ bundle,
\begin{equation}
M_{\fs}\times \Gl(\ft,\fs,P)
\to
M_{\fs}\times S^1\backslash\Gl(\ft,\fs,P).
\end{equation}
In \cite[Lemma 5.23]{FLLevelOne}, the equation
\begin{equation}
\label{eq:RelateNus}
\nu=\nu_{\Gl}+2\pi_{X,\fs}^*c_1(\LL_{\fs})
\end{equation}
is proven.  The advantage of the bundle \eqref{eq:XFactorS1Chern}
over $\pi_{X,\fs}^*\LL_{\fs}$ is that the former pulls back
by the projection $M_{\fs}\times\S^1\backslash\Gl(\ft,\fs,P)
\to S^1\backslash\Gl(\ft,\fs,P)$.

\begin{lem}
\label{lem:XFactorOfLocalInstantonEuler}
The rational Euler class,
$$
e(\Upsilon^i_{X,P}/S^1)
\in
H^{2|P|}(M_{\fs}\times S^1\backslash\Gl(\ft,\fs,P);\RR),
$$
is given by a polynomial in $\pi_X^*c(\ft)$ and
$\nu_{\Gl}$ which is otherwise independent of $X$.
\end{lem}

\begin{proof}
Observe that the bundle $\Upsilon^i_{X,P}/S^1$ is the pullback by the projection
$$
M_{\fs}\times S^1\backslash\Gl(\ft,\fs,P)
\to
S^1\backslash\Gl(\ft,\fs,P)
$$
of the bundle
\begin{equation}
\label{eq:SimplifyBundle}
S^1\backslash\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}\Upsilon^i_{spl,|P|}
\to
S^1\backslash\Gl(\ft,\fs,\sP).
\end{equation}
The lemma will then follow if we can that the Euler class of the
bundle \eqref{eq:SimplifyBundle} satisfies the conclusion of the lemma.

The kernel of the projection
\begin{equation}
\label{eq:GroupProjection}
(\Ad^u_{\SO(3)},\Ad^u_{\SO(4)}):
\Spinu(4)\to\SO(3)\times\SO(4)
\end{equation}
is the central $S^1$ in $\Spinu(4)$.  By the identity
(see \cite[Equation (3.14)]{FLLevelOne})
$$
\Fr_{\CCl(T^*X)}/S^1=\Fr(\fg_\ell)\times_X\Fr(TX).
$$
(where the $S^1$ is the kernel of the homomorphism \eqref{eq:GroupProjection}),
we can rewrite the space $\Gl(\ft,\fs,P)$ as
$$
\Gl(\ft,\fs,P)
=
\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}M^{s,\natural}_{spl,|P|}(\delta_P)
$$
where $\Spinu(4)$ acts on  $M^{s,\natural}_{spl,|P|}(\delta_P)$
via the projection \eqref{eq:GroupProjection}
and the action of $\SO(3)\times\SO(4)$ on
$M^{s,\natural}_{spl,|P|}(\delta_P)$.
Let $\tilde\iota_u:\Fr_{\CCl(T^*X)}(V_\ell)\to\ESpinu(4)$ be the classifying
map, appearing in the diagram:
$$
\begin{CD}
\Fr_{\CCl(T^*X)}(V_\ell) @> \tilde\iota_u >> \ESpinu(4)
\\
@VVV @VVV
\\
X @> \iota_u >> \BSpinu(4)
\end{CD}
$$


Recall from \cite[\S 4]{AtiyahJones},
that the map $[A]\to [D_A]$
where $D_A$ is the Dirac operator defined
by the connection $A$ defines a continuous map,
$$
f_\ka: \sB^s_\ka\to\sF_\ka,
$$
where $\sF_\ka$ is the space of Fredholm operators of index $\ka$.
The space $\sF_\ka$ is homotopic to $\BU=\lim_n\BU(n)$.
Although there is no index bundle defined for the Dirac operators
parameterized by $\sB^s_\ka$ because it is not compact, the
restriction of the pullback of the universal Chern
classes in $\BU$ by $f_\ka$
to the subspace $\sK^s_\ka\subset\sB^s_\ka$
where the cokernel of the Dirac operator vanishes
are equal to the Chern class of the
vector bundle defined by $\ind(\bD)$ on this subspace.
By property (iv) in Proposition \ref{prop:ExistenceOfSplicedEndsIndex},
the Chern classes of the bundle
$$
\Upsilon^i_{spl,|P|}
\to
M^{s,\natural}_{spl,|P|}
$$
are given by the pullback of the universal Chern classes
in $H^\bullet(\BU)$ by the map $f_{|P|}$.
By the $\Spinu(4)$-equivariance of the Dirac operator, the
map $f_\ka$ extends to a map
$$
F_\ka: \ESpinu(4)\times_{\Spinu(4)}\sB^s_\ka \to\sF_\ka,
$$
such that the restriction of $F_\ka$ to
$$
\ESpinu(4)\times_{\Spinu(4)}\sK^s_\ka
$$
is the composition of a classifying map for the index bundle
with the inclusion $\BU(\ka)\to\BU$.
We conclude that the classifying map for the bundle
$$
\Upsilon^i_{X,P}
=
\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}\Upsilon^i_{spl,|P|}
\to
\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}M^{s,\natural}_{spl,|P|}(\delta_P),
$$
factors through the composition
\begin{equation}
\label{eq:ClassifyingMapFactoring}
\begin{CD}
\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}M^{s,\natural}_{spl,|P|}(\delta_P)
@>\tilde\iota_u\times\iota_M >>
\ESpinu(4)\times_{\Spinu(4)}\sB^s_{|P|}
@> F_{|P|}>>
\sF_{|P|}
\end{CD}
\end{equation}
where $\iota_M:M^{s,\natural}_{spl,|P|}(\delta_P)\to\sB^s_{|P|}$
is the inclusion.  The bundle map
$\tilde\iota_u\times\iota_M$ descends to the $S^1$ quotients:
$$
\tilde\iota_u\times\iota_M:
S^1\backslash
\left(
\Fr_{\CCl(T^*X)}(V_\ell)\times_{\Spinu(4)}M^{s,\natural}_{spl,|P|}(\delta_P)
\right)
\to
S^1\backslash
\left(\ESpinu(4)\times_{\Spinu(4)}\sB^s_{|P|}\right)
$$
Then, because the rational cohomology of $\sB^s_{|P|}$
is trivial (see \cite[Lemma 5.1.14]{DK}), the rational
cohomology of the space
$$
S^1\backslash
\left(\ESpinu(4)\times_{\Spinu(4)}\sB^s_{|P|}\right)
$$
is generated by the first Chern class of the $S^1$ action
and cohomology classes pulled back by the projection
$$
S^1\backslash
\left(\ESpinu(4)\times_{\Spinu(4)}\sB^s_{|P|}\right)
\to
\BSpinu(4).
$$
Under the map $\tilde\iota_u\times\iota_M$,
the first Chern class of the $S^1$ action pulls back to
$\nu_{\Gl}$ while the cohomology classes pulled back
from $\BSpinu(4)$ pullback to characteristic classes of
$\Fr_{\CCl(T^*X)}(V)$, which we are denoting by $c(\ft)$.
The conclusion of the lemma then follows from the factoring
of the classifying map for the bundle
\eqref{eq:SimplifyBundle} given in
\eqref{eq:ClassifyingMapFactoring}.
\end{proof}


\begin{rmk}
Unlike the case $\ell=1$, the action of $\sG_{\fs}$ on
$S^1\backslash\Gl(\ft,\fs,\sP)$ is not trivial because there
can be more than one frame in the definition of $\Gl(\ft,\fs,\sP)$.
That is, if $F_1$ and $F_2$ are frames over separate points
$x_1$ and $x_2$
in $X$, the action of $S^1$ identifies $(F_1,F_2)$ with
$(e^{i\theta}F_1,e^{i\theta}F_2)$.  There can be $u\in\sG_\fs$
with $u(x_1)\neq u(x_2)$.  For such $u\in\sG_\fs$,
$(uF_1,uF_2)$ would not be identified with $(F_1,F_2)$ by the $S^1$ action.
\end{rmk}



\begin{proof}[Proof of Proposition \ref{lem:LocalInstantonPullbackToCover}]
By Lemmas \ref{lem:LocalInstantonPullbackToCover},
\ref{lem:RelateInstantonObstructionFactors}, and
\ref{lem:XFactorOfLocalInstantonEuler}, we see that the Euler class
of the pullback, by the map
\eqref{eq:LocalCoveringMap}, of the restriction of $\Upsilon^i_{\ft,\fs}/S^1$
to the space \eqref{eq:SWTopStratumSubspace} is given by
\begin{equation}
\label{eq:LocalEulerClass1}
\prod_{P\in\sP}
\pi_{\tilde{\Gl},P}^* e(\Upsilon^i_{\ft,\fs}(P)/S^1)
=
\prod_{P\in\sP}
\pi_{\tilde{\Gl},P}^*  e(\pi_{X,\fs}^*\LL_{\fs}^*\otimes\Upsilon^i_{X,P}/S^1).
\end{equation}
Now, by the splitting principal, we can assume that
$\Upsilon^i_{X,P}/S^1=\oplus_{i\in P}L_i$, where $L_i$ is a complex
line bundle and all the symmetric polynomials in $c_1(L_i)$ are given
by polynomials in $c(\ft)$ and,
by \eqref{eq:RelateNus}, in $\nu-2\pi_{X,\fs}^*c_1(\LL_\fs)$.
Observe that $\pi_{X,\fs}\circ\pi_{\tilde{\Gl},P}=\pi_{\fs,i}\circ\pi_X$,
so the symmetric polynomials in $\pi_{\tilde{\Gl},P}^*\pi_{X,\fs}^*\LL_{\fs}$
are given by the Chern classes $\tilde\iota_{\sP}^*\pi_X^*c_{\fs,\ell,i}$
defined in \eqref{eq:ChernClassesOfSymSWUniv}.
Hence, we can rewrite \eqref{eq:LocalEulerClass1} as
\begin{equation}
\label{eq:LocalEulerClass2}
=
\prod_{P\in\sP}
\prod_{i\in P}
\left(\tilde\iota_{\sP}^*\pi_{\fs,i}^*c_1(\LL_{\fs})\right)
\pi_{\tilde{\Gl},P}^*c_1(L_i).
\end{equation}
One then verifies the preceding product can be expressed in terms
of symmetric polynomials in $c_1(L_i)$ and
$\pi_{\fs,i}^*c_1(\LL_\fs)$.
\end{proof}



\subsection{The global instanton obstruction Euler class}
\label{subsec:InstObstrEulerGlobal}
We now piece together the local computations of the preceding
section to give a global characterization of the instanton
obstruction $\Upsilon^i_{\ft,\fs}/S^1\to\sM^{vir}_{\ft,\fs}/S^1$.

\begin{prop}
\label{prop:GlobalInstantonEulerClass}
The Euler class of the instanton obstruction bundle
$\Upsilon^i_{\ft,\fs}/S^1\to\sM^{vir}_{\ft,\fs}/S^1$
is given by a polynomial in
$\iota^*\nu$,
$\iota^*\pi_X^*S^\ell(c(\ft))$, and
$\iota^*\pi_{X,\fs}^*c_{\fs,\ell,i}$
which is otherwise independent of $X$.
\end{prop}

\begin{proof}
The construction of the bundle
$\Upsilon^i_{spl,\ka}\to M^{s,\natural}_{spl,\ka}(\delta)$
and the $\Spinu(4)$ equivariance of the diagram
\eqref{eq:OverlapObstructionCommuting} implies that
the particular classifying maps of the local instanton obstruction
bundles which admit the factorization
\eqref{eq:ClassifyingMapFactoring}
can be chosen to be equal on the overlap
given in Lemma \ref{lem:InstantonObstrCommDiagr}.
Hence, the local equality of cohomology classes given
in Proposition \ref{prop:LocalInstantonEuler} is an equality
of cocycles and thus an equality of global cohomology classes.
\end{proof}

\begin{defn}
We write $e(\Upsilon^i_{\ft,\fs}/S^1)$ for the real cohomology class
given by the Euler class of $\Upsilon^i_{\ft,\fs}/S^1$ as described
in Proposition \ref{prop:GlobalInstantonEulerClass}.
\end{defn}


\subsection{Extensions}
\label{subsec:Extensions}
We define extensions of the cohomology classes $\bga_{\sM}^*\mu_p(\beta)$,
$\bga_{\sM}^*\mu_c$, and $e(\Upsilon^i_{\ft,\fs}/S^1)$
from the top stratum $\sM^{\vir}_{\ft,\fs}/S^1$ to
$\bar\sM^{\vir,*}_{\ft,\fs}/S^1$ as follows.

\begin{defn}
\label{defn:ExtendedCohomologyClasses}
For $\beta\in H_\bullet(X;\RR)$, let
\begin{equation}
\label{eq:DefineExtendeMuP}
\barmu_p(\beta)\in H^{4-\bullet}(\bar\sM^{\vir,*}_{\ft,\fs}/S^1;\RR),
\end{equation}
be the cohomology class obtained by omitting the pullback
$\iota^*$ on the right-hand-side of \eqref{eq:PullbacksOfMuAndNu}.
Similarly, let
\begin{equation}
\barmu_c=-\nu\in H^2(\bar\sM^{\vir,*}_{\ft,\fs}/S^1;\RR),
\end{equation}
be the cohomology class obtained by omitting
the pullback
$\iota^*$ on the right-hand-side of \eqref{eq:PullbacksOfMuAndNu}.
Let
\begin{equation}
\label{eq:ExtendedInstEuler}
\bar e_i\in H^{2\ell}(\bar\sM^{\vir,*}_{\ft,\fs}/S^1;\RR)
\end{equation}
be the extensions of $e(\Upsilon^i_{\ft,\fs}/S^1)$
obtained by omitting the pullback $\iota^*$ from the expression in
Proposition \ref{prop:GlobalInstantonEulerClass}.
\end{defn}

\begin{rmk}
Note that there is no need to extend the Euler class of the Seiberg-Witten
obstruction bundle, $e(\Upsilon^s_{\ft,\fs}/S^1)$,
as it is already defined on $\bar\sM^{\vir,*}_{\ft,\fs}/S^1$
by Lemma \ref{lem:SWObstruction}.
\end{rmk}

%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Article: INTERSECTION
%File: link.tex
%Last update: March 2, 2002 (Tom)
%Spellcheck:  March 2, 2002 (Tom)
%Todo:

\section{The link}
\label{sec:Link}
We now use the Thom-Mather structure defined in \S \ref{subsec:TMStr}
to construct an ambient link, $\bar\bL^{\vir}_{\ft,\fs}$,
given by the boundary of a neighborhood of $M_{\fs}\times\Sym^\ell(X)$
in $\bar\sM^{\vir}_{\ft,\fs}/S^1$
and then the actual link $\bar\bL_{\ft,\fs}$, given by the boundary
of a neighborhood of $M_{\fs}\times\Sym^\ell(X)$
in $\bar\sM_{\ft}/S^1$.

We construct the link in \S \ref{subsec:AmbientLink} using
the tubular distance functions $t(\ft,\fs,\sP_i)$ constructed
in Lemma \ref{lem:GlobalSplicingTubularNgh}.
We use our understanding of the overlap maps to show that
the ambient link $\bar\bL^{vir}_{\ft,\fs}$ can be decomposed
into pieces
(enumerated by the strata of $\Sym^\ell(X)$)
each of which admits a fiber bundle structure.
Each piece is a smoothly-stratified space, with strata given by
smooth manifolds with corners.  The intersections of the
different pieces are given by these corners.
In \S \ref{subsec:Duality}, we show that the
intersection pairing appearing in
\eqref{eq:RawCobordismSum}
can be computed by a pairing of the appropriate cohomology classes
with the fundamental class of the ambient link.
We reduce the pairing with the fundamental class of the ambient
link to one with a simpler subspace,
$\bar{\bB\bL}^{\vir}_{\ft,\fs}$ in \S \ref{subsec:ReduceToBase}.
This subspace is also a union of pieces
each of which is a smoothly-stratified space with the strata given
by manifolds with corners.  In addition, each piece
of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$ is a fiber bundle over
a stratum of $\Sym^\ell(X)$.
In \S \ref{subsec:Boundaries}, we describe the intersections of
the pieces of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$ which are given by the corners as
described above.

For use in this section and the following,
pick a representative from each conjugacy class of partitions of $N_\ell$
and enumerate them in such a way that
$$
\Si(X^\ell,\sP_i)\subset\cl \Si(X^\ell,\sP_j)
\quad\text{only if}\quad i<j.
$$
Write these partitions as $\sP_1,\sP_2,\dots,\sP_r$ so that
$\Si(X^\ell,\sP_1)$ is the lowest stratum and $\Si(X^\ell,\sP_r)$
is the highest.

\subsection{Defining the link}
\label{subsec:AmbientLink}
We define the link by first constructing an ambient link
in \S \ref{subsubsec:AmbientLink}.  We discuss
the orientations of the link in \S \ref{subsubsec:Orient}
which leads to a formal statement of the cobordism formula
\eqref{eq:RawCobordismSum} in Theorem \ref{thm:CobordismThm}.
Then in \S \ref{subsubsec:FiberBundle}, we give a more
detailed description of the pieces of the link, showing
how they admit a fiber bundle structure.

\subsubsection{The ambient link}
\label{subsubsec:AmbientLink}
The ambient link is defined as the boundary of the neighborhood of
$M_{\fs}\times\Sym^\ell(X)$ in $\bar\sM^{\vir}_{\ft,\fs}/S^1$.
The subspace $M_{\fs}\times\Sym^\ell(X)$ of $\bar\sM^{\vir}_{\ft,\fs}/S^1$
is the intersection of the zero-locus of
the function defined by the length of the fiber vector of
the vector bundle $N(\delta)\to M_{\fs}$,
\begin{equation}
\label{eq:NTubularDist}
t_N:\bar\sM^{\vir}_{\ft,\fs}/S^1\to [0,\delta],
\end{equation}
with the union of the zero-loci of
the tubular distance function $t(\ft,\fs,\sP_i)$
defined on $\sU(\ft,\fs,\sP_i)$ in
Lemma \ref{lem:GlobalSplicingTubularNgh}.
We will define the ambient link as the union of two pieces:
\begin{equation}
\label{eq:AmbientLink}
\bar\bL^{\vir}_{\ft,\fs}
=
\bar\bL^{\vir,s}_{\ft,\fs}\cup \bar\bL^{\vir,i}_{\ft,\fs}.
\end{equation}
The first piece, $\bar\bL^{\vir,s}_{\ft,\fs}$, is defined to
be the codimension-one subspace
\begin{equation}
\label{eq:DefineSWComponentOfLink}
\bar\bL^{\vir,s}_{\ft,\fs}
=
t_N^{-1}(\delta) \subset
\bar\sM^{\vir}_{\ft,\fs}/S^1.
\end{equation}
The definition of the instanton component of
the link, $\bar\bL^{\vir,i}_{\ft,\fs}$, is
more involved as it will be the union of the hypersurfaces
of the local functions $\tilde t(\ft,\fs,\sP)$.
Pick a small constant $\eps_i$ for each stratum
$\Si(X^\ell,\sP_i)$ (these will be chosen
to be generic with respect to requirements obtained later).
Then, we define the local link by
\begin{equation}
\label{eq:DefineLinkStratum}
\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)
=
\sU(\ft,\fs,\sP_j)\cap t(\ft,\fs,\sP_j)^{-1}(\eps_j)
-
\cup_{i\neq j}t(\ft,\fs,\sP_i)^{-1}([0,\eps_i)).
\end{equation}
Taking the `hypersurface', $t(\ft,\fs,\sP_j)^{-1}(\eps_j)$
makes the subspace $\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$
a link for $M_{\fs}\times\Si(X^\ell,\sP_j)$.
Removing the subspaces $t(\ft,\fs,\sP_i)^{-1}([0,\eps_i))$
from $\sU(\ft,\fs,\sP_j)$ ensures that
$\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$ is disjoint from
the other strata of reducibles $M_{\fs}\times\Si(X^\ell,\sP_i)$
(for $i\neq j$).
We then define the instanton component of the link:
\begin{equation}
\label{eq:InstantonLinkComponent}
\bar\bL^{\vir,i}_{\ft,\fs}
=
\cup_i \ \bar\bL^{\vir,i}_{\ft,\fs}(\sP_i).
\end{equation}

The next lemma follows immediately from the construction.

\begin{lem}
\label{lem:BoundaryOfNeigh}
The ambient link $\bar\bL^{\vir}_{\ft,\fs}$ defined in
\eqref{eq:AmbientLink}, \eqref{eq:DefineSWComponentOfLink}, and
\eqref{eq:InstantonLinkComponent}
is the boundary of a neighborhood of $M_{\fs}\times \Sym^\ell(X)$
in $\bar\sM^{\vir}_{\ft,\fs}/S^1$.
For generic choices of the constants $\eps_i$,
the intersection of
$\bar\bL^{\vir}_{\ft,\fs}$ with each stratum of
$\bar\sM^{\vir}_{\ft,\fs}/S^1$ is a smooth manifold with corners.
\end{lem}

We can now define the link:

\begin{defn}
\label{defn:DefineLink}
The link, $\bar\bL_{\ft,\fs}$, of $M_{\fs}\times\Sym^\ell(X)$ in
$\bar\sM_{\ft}/S^1$ is
defined to be the intersection of the ambient link
$\bar\bL^{\vir}_{\ft,\fs}$ defined in
\eqref{eq:AmbientLink} with the zero-locus of the obstruction section.
\end{defn}

We refer to the intersection of any two or more of the pieces
$\bar\bL^{\vir,s}_{\ft,\fs}$ or
$\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$ as an edge.
For generic choices of the parameters $\delta,\eps_i$,
the intersection of each of these edges with the strata
of $\bar\sM^{vir}_{\ft,\fs}/S^1$
will also be a smooth manifold with corners.


\begin{lem}
\label{lem:DefiningLink}
For generic values of the parameters $\delta,\eps_i$ used to define
the ambient link, the link $\bar\bL_{\ft,\fs}$ is the
boundary of a neighborhood of $M_{\fs}\times\Sym^\ell(X)$ in
$\bar\sM_{\ft}/S^1$.  The link $\bar\bL_{\ft,\fs}$ is
smoothly-stratified
subspace of $\bar\sM_{\ft}/S^1$;
the intersection of $\bar\bL_{\ft,\fs}$ with each stratum of
$\bar\sM_{\ft}/S^1$ and each edge
of $\bar\bL^{\vir}_{\ft,\fs}$ is a smooth, oriented manifold
with corners.  The intersections of $\bar\bL_{\ft,\fs}$ with
the lower strata of $\bar\sM_{\ft}/S^1$ have codimension
at least two in $\bar\bL_{\ft,\fs}$.
\end{lem}

The following result then translates immediately from
\cite{FLLevelOne}.

\begin{lem}
\label{lem:GRIntersect0}
Assume $w\in H^2(X;\ZZ)$ is such that $w\pmod{2}$ is good
in the sense of Definition \ref{defn:Good}.
Given a \spinu structure $\ft$ on $X$ and  a \spinc
structure $\fs$ on $X$ satisfying $\ell(\ft,\fs)\ge 0$
and $w_2(\ft)\equiv w\pmod{2}$,
there are positive constants $\eps_0$ and
$\delta_0$ such that the following hold for all generic choices of
$\eps_i\leq \eps_0$ and $\delta\leq\delta_0$ defining
$\bar\bL^{\vir}_{\ft,\fs}$.
\begin{itemize}
\item
The link $\bar\bL_{\ft,\fs}$ is disjoint from $\bar M^w_\ka$ and
$\bar\sM_{\ft}^{\red}$ in the stratification
\eqref{eq:StratificationCptPU(2)Space} of $\bar\sM_{\ft}/S^1$.
\item
For all $z\in\AAA(X)$ and positive integers $\eta$
satisfying \eqref{eq:DimensionCondition}, and for
$\bar\sV(z)$ and $\bar\sW$  the geometric representatives
discussed in \S \ref{subsec:Cohomology},
the intersection
\begin{equation}
\label{eq:Int}
\bar\sV(z)\cap\bar\sW^{\eta}\cap\bar\bL_{\ft,\fs}
\end{equation}
is a finite collection of points
contained in the top stratum $\bL_{\ft,\fs}$ of
$\bar\bL_{\ft,\fs}\subset\bar\sM_{\ft}/S^1$,
and is disjoint from  the edges  of
$\bar\bL^{\vir}_{\ft,\fs}$.
\end{itemize}
\end{lem}



\subsubsection{Orientations of the link}
\label{subsubsec:Orient}
To define an intersection number from the intersection
\eqref{eq:Int}, it is necessary to discuss orientations of the link.
An orientation for $\sM_{\ft}$ determines one for $\bL_{\ft,\fs}$ through
the convention introduced in \cite[Equations (2.16), (2.16) \& (2.25)]{FL2b} by
considering $\bL_{\ft,\fs}$ as a boundary of
$(\sM_{\ft}-\bga(\sM_{\ft,\fs}^{\stab}))/S^1$. Specifically, at a point
$[A,\Phi]\in \bL_{\ft,\fs}$, if
\begin{itemize}
\item
$\vec r\in
T\sM_{\ft}^{*,0}$ is an outward-pointing radial vector with respect to the
open neighborhood $\sM_{\ft}\cap\bga(\sM_{\ft,\fs}^{\stab})$
 and complementary to the tangent space of
$\bL_{\ft,\fs}$,
\item
$v_{S^1}\in T\sM_{\ft}^{*,0}$ is
tangent to the orbit of $[A,\Phi]$ under the (free) circle action (where
$S^1\subset\CC$ has its usual orientation), and
\item
$\la_{\sM}\in\det(T\sM_{\ft}^{*,0})$ is an orientation for $T\sM_{\ft}$
at $[A,\Phi]$,
\end{itemize}
then we define an orientation $\la_{L}$ for
$T\bL_{\ft,\fs}$ at $[A,\Phi]$ by
\begin{equation}
\label{eq:QuotientBoundaryOrientation}
\la_{\sM}
=
-v_{S^1}\wedge\vec r \wedge \tilde{\la}_{L},
\end{equation}
where the lift $\tilde\la_{L}\in \Lambda^{\max-2}(T\sM_{\ft}^{*,0})$ at
$[A,\Phi]$ of $\la_{L}\in\det(T\bL_{\ft,\fs})\subset
\Lambda^{\max-1}(T(\sM_{\ft}^{*,0}/S^1))$,
obeys $\pi_*\tilde\la_{L}=\la_{L}$, if $\pi:\sM_{\ft}\to\sM_{\ft}/S^1$ is
the quotient map.

\begin{defn}
\label{defn:BdryOrnLink}
If $O$ is an orientation for
$\sM_{\ft}$, we call the orientation for $\bL_{\ft,\fs}$ related to
$O$ by equation \eqref{eq:QuotientBoundaryOrientation} the {\em boundary
orientation defined by $O$\/}.
\end{defn}

The {\em standard orientation\/}
 for $\bar\bL_{\ft,\fs}$ is defined
in \cite[Definition 3.12]{FLLevelOne}.  The standard orientation
arises from the local topology of $\bar\bL^{\vir}_{\ft,\fs}$
and thus is more natural for computations on this space.
The standard and boundary orientations are related in
the following.

\begin{lem}
\label{lem:OrientationFactor}
\cite[Lemmas 3.13 \& 3.14]{FLLevelOne}
If $\ft$ and $\ft_\ell$ are \spinu structures on $X$
satisfying
$$
p_1(\ft)=p_1(\ft_\ell)-4\ell,\quad
c_1(\ft)=c_1(\ft_\ell), \quad\text{and}\quad
w_2(\ft)=w_2(\ft_\ell).
$$
Let $\Om$ be a homology orientation and
let $w$ be an integral lift of $w_2(\ft)$.  If $\ft_\ell$ admits a splitting
$\ft_\ell=\fs\oplus \fs\otimes L$, then the standard orientation for
$\bar\bL_{\ft,\fs}$ and the boundary orientation for $\bar\bL_{\ft,\fs}$ defined
through the orientation $O^{\asd}(\Om,w)$ for $\sM_{\ft}$ differ by a
factor of
\begin{equation}
\label{eq:OrientChangeFactor}
(-1)^{o_{\ft}(w,\fs)},
\quad\text{where}\quad o_{\ft}(w,\fs)
=
\textstyle{\quarter}(w-c_1(L))^2.
\end{equation}
\end{lem}


For all $z\in\AAA(X)$ and positive integers $\eta$
satisfying \eqref{eq:DimensionCondition}, and for
$\bar\sV(z)$ and $\bar\sW$  the geometric representatives
discussed in \S \ref{subsec:Cohomology}, we can
then define the intersection number
\begin{equation}
\label{eq:IntNumber}
\#\left(\bar\sV(z)\cap\bar\sW^{\eta}\cap\bar\bL_{\ft,\fs}\right)
\end{equation}
to be the oriented count of points in the intersection
\eqref{eq:Int}, using the standard orientation of $\bar\bL_{\ft,\fs}$.
This yields the following formal expression of the cobordism
formula.

\begin{thm}
\label{thm:CobordismThm}
Let $\ft$ be a \spinu structure on a smooth, oriented four-manifold
$X$.  Let $z\in\AAA(X)$ and $\eta$ a non-negative integer
satisfy
\begin{equation}
\label{eq:DimensionCondition}
\deg(z)+2\eta=\dim\sM^{*,0}_{\ft}-2.
\end{equation}
Assume there is $w\in H^2(X;\ZZ)$
satisfying $w_2(\ft)\equiv w\pmod 2$ and
which is good in the sense of
Definition  \ref{defn:Good}.
Then the intersection numbers \eqref{eq:IntNumber} satisfy
\begin{equation}
\label{eq:RawCobordismSum1}
\#\left(\bar\sV(z)\cap \bar\sW^{n_a-1}\cap \bar\bL^w_{\ft,\ka}\right)
=
-\sum_{\fs\in\Spinc(X)} (-1)^{o_{\ft}(w,\fs)}\
\# \left( \bar\sV(z)\cap \bar\sW^{n_a-1}\cap \bar\bL_{\ft,\fs}\right),
\end{equation}
where $\bar\bL^w_{\ft,\ka}$
is the link of the anti-self-dual connections in $\bar\sM_{\ft}/S^1$
from \cite[Definition 3.7]{FL2a}.
\end{thm}

\subsubsection{The fiber bundle structure}
\label{subsubsec:FiberBundle}
We now describe how the fiber bundle structure of the pieces
$\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$ of the link.
Observe that the functions $t(\ft,\fs,\sP_j)$, while nominally
defined on the subspaces $\sU(\ft,\fs,\sP)$ of $\bar\sM^{\vir}_{\ft,\fs}/S^1$
can also be understood as functions on subsets of $\Sym^\ell(X)$ by
virtue of the inclusion,
$$
N(\delta)\times\Sym^\ell(X) \subset \bar\sM^{\vir}_{\ft,\fs}/S^1,
$$
and by virtue of the equality in item (2) of Lemma
\ref{lem:GlobalSplicingTubularNgh}, the restriction of
$t(\ft,\fs,\sP_j)$ to
$$
\left(N(\delta)\times\Sym^\ell(X)\right) \cap \sU(\ft,\fs,\sP)
$$
is equal to the pullback by the projection
$$
N(\delta)\times\Sym^\ell(X) \to \Sym^\ell(X).
$$
of the function $t(X^\ell,\sP)$
defined in Lemma \ref{eq:XTubularDistFunc}.

With this understanding, define
\begin{equation}
\label{eq:DefineK}
K_j=\Si(X^\ell,\sP_i) - \cup_{i<j}\ t(X^\ell,\sP_i)^{-1}([0,\eps_i)).
\end{equation}

Recall that the open subspace $\sU(\ft,\fs,\sP_j)$ is a neighborhood
of $N_{\ft,\fs}(\delta)/S^1\times \Si(X^\ell,\sP_j)$ in the fiber
bundle
\begin{equation}
\label{eq:LocalFiberBundle}
\begin{CD}
\prod_{P\in\sP}\barM^{s,\natural}_{spl,|P|}(\delta_P)
@>>>
\tilde N(\delta_P)\times_{\sG_{\fs}\times S^1}\Gl(\ft,\fs,\sP)
\\
@. @V \pi(\ft,\fs,\sP_j) VV
\\
@. N(\delta)/S^1\times \Si(X^\ell,\sP_j)
\end{CD}
\end{equation}
We will write
\begin{equation}
\label{eq:LocalFiber}
\barM(\sP)=
\prod_{P\in\sP}\barM^{s,\natural}_{spl,|P|}(\delta_P)
\end{equation}
for the fiber of \eqref{eq:LocalFiberBundle}.

The first step in describing the fiber bundle structure
of $\bL^{\vir,i}_{\ft,\fs}(\sP_j)$ in $\sU(\ft,\fs,\sP_j)$ is
the following description of the base of the bundle.

\begin{lem}
\label{lem:InstLinkLowerStrCompl}
The space $K_j$ defined in \eqref{eq:DefineK} is a compact,
codimension-zero submanifold with corners
of $\Si(X^\ell,\sP_j)$.
Then, the subspace
\begin{equation}
\label{eq:RemoveLowerTrivials}
\sU(\ft,\fs,\sP_j)
-
\cup_{i<j}\ t(\ft,\fs,\sP_i)^{-1}([0,\eps_i))
\end{equation}
is given by the intersection of $\sU(\ft,\fs,\sP_j)$
with the restriction of the fiber bundle
\eqref{eq:LocalFiberBundle} to $N(\delta)/S^1\times K_j$.
In addition, the subspace \eqref{eq:RemoveLowerTrivials}
is disjoint from the subspace $N(\delta)/S^1\times K_i$
for all $i<j$.
\end{lem}

\begin{proof}
By \eqref{eq:TMForTubularReds}, for $i<j$ the function
$t(\ft,\fs,\sP_i)$ is constant on the fibers of the projection
map $\pi(\ft,\fs,\sP_j)$.  Hence, restricting $\pi(\ft,\fs,\sP_j)$
to $\pi(\ft,\fs,\sP_j)^{-1}(K_j)$ does not change the fibers of
this projection.  The equality between $t(\ft,\fs,\sP_i)$
and $t(X^\ell,\sP_j)$ in item (2) of Lemma
\ref{lem:GlobalSplicingTubularNgh} then completes the proof.
\end{proof}

The function $t(\ft,\fs,\sP_j)$ on $\sU(\ft,\fs,\sP_j)$
is defined by a function,
\begin{equation}
\label{eq:TubularDistanceFunctionOnFibers}
t_f(\sP_j)=\sum_{P\in\sP_j}|P|\tilde\la_{|P|}^2:
\barM(\sP)=
\prod_{P\in\sP_j}\barM^{s,\natural}_{spl,|P|}(\delta_P)
\to
[0,\infty)
\end{equation}
(see \eqref{eq:DefineTubularDistanceFunction})
 on the fibers of
\eqref{eq:LocalFiberBundle}.
Because $t_f(\sP_j)$ is $G(\sP_j)$-invariant
it extends to a function on \eqref{eq:LocalFiberBundle};
this function is $t(\ft,\fs,\sP_j)$.
Because the open subspace $\sU(\ft,\fs,\sP_j)$ is defined
by a subspace of the fiber bundle $\Gl(\ft,\fs,\sP_j)$ in
\eqref{eq:LocalFiberBundle}, the following lemma is
necessary to ensure that working with the subspace
$\sU(\ft,\fs,\sP_j)$ does not restrict the fibers
of the local link $\bar\bL^{vir,i}_{\ft,\fs}(\sP_j)$.


\begin{lem}
\label{lem:SmallConst}
For any compact $K\Subset \Si(X^\ell,\sP_j)$, there is $\eps_K$
such that the intersection
$$
\sU(\ft,\fs,\sP_i) \cap
\pi(\ft,\fs,\sP_j)^{-1}\left( N(\delta)/S^1\times K\right),
$$
contains $t(\ft,\fs,\sP_j)^{-1}([0,\eps_K))
\cap \pi(\ft,\fs,\sP_j)^{-1}\left( N(\delta)/S^1\times K\right)$.
\end{lem}

\begin{proof}
The lemma follows immediately from the compactness of $K$.
\end{proof}

Lemma \ref{lem:SmallConst} allows us to define a link
of $N(\delta)/S^1\times K_j$ using a subspace
of \eqref{eq:LocalFiberBundle} defined by
replacing the fiber $\barM(\sP_j)$ with the subspace
\begin{equation}
\label{eq:DefineFiberLink}
\barM(\sP_j,\eps_j)=t_f(\sP_j)^{-1}(\eps_j)\subset\barM(\sP_j)
\end{equation}
where $t_f(\sP_j)$ is defined in \eqref{eq:TubularDistanceFunctionOnFibers}.
The resulting subspace will be $t(\ft,\fs,\sP_j)^{-1}(\eps_j)$:

\begin{lem}
Let $\eps_j>0$ be a small constant given by Lemma \ref{lem:SmallConst}
for the space $K_j$ defined in \eqref{eq:DefineK}.
The subspace
\begin{equation}
\label{eq:RemoveLowerTrivialsPlus}
t(\ft,\fs,\sP_j)^{-1}(\eps_j)
-
\cup_{i<j}\ t(\ft,\fs,\sP_i)^{-1}([0,\eps_i))
\end{equation}
is the codimension-one subspace of $\sU(\ft,\fs,\sP_j)$,
disjoint from $N(\delta)/S^1\times \Si(X^\ell,\sP_i)$ for all
$i\le j$ and is
given by
\begin{equation}
\label{LocalInstantonLink0}
\begin{CD}
\barM(\sP_j,\eps_j)
@>>>
\tilde N(\delta)\times_{\sG_{\fs}\times S^1}
\Fr(\ft,\fs,\sP_j,g_{\sP_j})|_{K_j}\times_{G(\sP_j)}\barM(\sP,\eps_j)
\\
@. @V \pi(\ft,\fs,\sP_j) VV
\\
@. N(\delta)/S^1\times K_j
\end{CD}
\end{equation}
where $\barM(\sP_j,\eps_j)$  is defined in
\eqref{eq:DefineFiberLink}.
\end{lem}

\begin{proof}
The subspace \eqref{eq:RemoveLowerTrivialsPlus}
is disjoint from the subspaces
$N(\delta)/S^1\times \Si(X^\ell,\sP_i)$ for $i<j$ by
Lemma \ref{lem:InstLinkLowerStrCompl} and from
$N(\delta)/S^1\times \Si(X^\ell,\sP_j)$ because
$$
N(\delta)/S^1\times \Si(X^\ell,\sP_j)
=
t(\ft,\fs,\sP_j)^{-1}(0).
$$
The subspace \eqref{eq:RemoveLowerTrivialsPlus} is a
sub-fiber bundle of \eqref{eq:LocalFiberBundle} because,
as noted above,
the function $t(\ft,\fs,\sP_j)$ is defined
by extending a function on the fibers over the bundle.
\end{proof}



The subspace \eqref{eq:RemoveLowerTrivialsPlus} still intersects
$M_{\fs}\times\Sym^\ell(X)$, but only along higher strata
$M_{\fs}\times\Si(X^\ell,\sP_i)$ where $i>j$.
We remove this intersection by removing neighborhoods
of  the subspace $t(\ft,\fs,\sP_i)^{-1}(0)\cap\sU(\ft,\fs,\sP_j)$,
giving the definition of $\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$
found in \eqref{eq:DefineLinkStratum}.
To understand the subspace $\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$
of \eqref{eq:RemoveLowerTrivialsPlus},
we must examine the function $t(\ft,\fs,\sP_i)$
on $\sU(\ft,\fs,\sP_j)\cap\sU(\ft,\fs,\sP_i)$.

Recall from Proposition \ref{prop:XOverlapControl}
that the intersection $\sU(\ft,\fs,\sP)\cap \sU(\ft,\fs,\sP')$
(where $\sP<\sP')$
is given by an open subspace of the fiber bundle
\begin{equation}
\label{eq:OverlapFiberBundle}
\begin{CD}
\tilde N(\delta)\times_{\sG_\fs\times S^1}
\Fr(\ft,\fs,\sP,g_{\sP})
\times_{G(\sP)} \coprod_{\sP''\in [\sP<\sP']}\barM(\sP,\sP'')
\\
@VVV
\\
N(\delta)/S^1\times \Si(X^\ell,\sP)
\end{CD}
\end{equation}
where the fiber has components (up to a quotient by the symmetric
subgroup $\fS(\sP)$)
\begin{equation}
\barM(\sP,\sP'')
=
\prod_{P\in\sP}
\left(
\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P)
\times\prod_{Q\in\sP''_P} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
\right).
\end{equation}
There is an $\fS(\sP)$-equivariant projection map,
\begin{equation}
\label{eq:StratumFiberProjection}
\coprod_{\sP''\in [\sP<\sP']}
\barM(\sP,\sP'')
\to
\coprod_{\sP''\in [\sP<\sP']}
\prod_{Q\in\sP''} \barM^{s,\natural}_{spl,|Q|}(\delta_Q).
\end{equation}
If $\tilde \la_Q$ is the modified scale parameter
on $\barM^{s,\natural}_{spl,|Q|}(\delta_Q)$
defined in \eqref{eq:TMScaleR4}), then
the function,
\begin{equation}
\label{eq:UpperStratumFiberFunction}
\sum_{Q\in\sP'} |Q|\tilde\la_Q^2
:\prod_{Q\in\sP''} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
\to [0,\infty)
\end{equation}
extends to a unique
$\fS(\sP)$-invariant
function on the image of the projection \eqref{eq:StratumFiberProjection}.
The pullback of this $\fS(\sP)$-invariant function by the
projection map \eqref{eq:StratumFiberProjection} defines
a function,
\begin{equation}
\label{eq:OverlapStratumFiberFunction}
t_{o,f}(\sP,[\sP']):
\coprod_{\sP''\in [\sP<\sP']}
\barM(\sP,\sP'')
\to [0,\infty).
\end{equation}
The function $t_{o,f}(\sP,[\sP'])$
is invariant under the action of
the group $G(\sP)$ and thus defines a function on the bundle
\eqref{eq:OverlapFiberBundle}:
\begin{equation}
\label{eq:PullbackFiberFunction}
\tilde t_o(\sP,[\sP']):
\tilde N(\delta)\times_{\sG_\fs\times S^1}
\Fr(\ft,\fs,\sP,g_{\sP})
\times_{G(\sP)} \coprod_{\sP''\in [\sP<\sP']}\barM(\sP,\sP'')
\to
[0,\infty).
\end{equation}

\begin{lem}
\label{lem:IdentifyUpperTubFunctionOnOverlap}
Let $\sP<\sP'$ be partitions of $N_\ell$.
Let $\rho^{N,\ft,\fs,u}_{\sP,[\sP']}$ be the upwards overlap
map appearing in \eqref{eq:GlobalSplicingCD}.
The pullback
$t(\ft,\fs,\sP')\circ \rho^{N,\ft,\fs,u}_{\sP,[\sP']}$
is defined on the subspace
$\sU(\ft,\fs,\sP)\cap \sU(\ft,\fs,\sP')$ of the fiber bundle
\eqref{eq:OverlapFiberBundle}
and is equal to the function $\tilde t_o(\sP,[\sP'])$ from
\eqref{eq:PullbackFiberFunction}.
\end{lem}

\begin{proof}
The lemma follows immediately from the definition of the
function $t(\ft,\fs,\sP')$ and the observation
that the overlap
map $\rho^{N,\ft,\fs,u}_{\sP,[\sP']}$ is the identity on
the spaces $\barM^{s,\natural}_{spl,|Q|}(\delta_Q)$.
\end{proof}

To understand the space \eqref{eq:DefineLinkStratum},
we now discuss the behaviour of the function
$\tilde t_o(\sP,[\sP'])$ under the downwards transition
map $\rho^{\ft,\fs,d}_{\sP,[\sP']}$ from \eqref{eq:GlobalSplicingCD}.
On the component of the intersection
$\sU(\ft,\fs,\sP)\cap \sU(\ft,\fs,\sP')$ corresponding to
the partition $\sP''\in [\sP<\sP']$, the map
$\rho^{\ft,\fs,d}_{\sP,[\sP']}$ is defined by the inclusion
of fibers (see the discussion around \eqref{eq:SplicingDataFactor})
defined the product (over $P\in\sP$) of splicing maps (of connections
to the trivial connection on $\RR^4$):
\begin{equation}
\label{eq:DownwardOverlapFiberMap}
\coprod_{\sP''\in [\sP<\sP']}
\barM(\sP,\sP'')
\to
\barM(\sP)=
\prod_{P\in\sP}
\bar\sM^{s,\natural}_{spl,|P|}(\delta_P).
\end{equation}
Let
\begin{equation}
\label{eq:OverlapFiber}
U_f(\sP,[\sP']) \subset \barM(\sP)
\end{equation}
be the image of the inclusion \eqref{eq:DownwardOverlapFiberMap}.
(Note that the map \eqref{eq:DownwardOverlapFiberMap} is actually
injective only after taking the quotient of the domain and image
by $\fS(\sP)$; we omit that quotient from the notation since it
is not relevant to this discussion.)
Let
\begin{equation}
\label{eq:OverlapTubDist}
t_f(\sP,[\sP']): U_f(\sP,[\sP'])\to [0,\infty)
\end{equation}
be the function  whose pullback
by the inclusion \eqref{eq:DownwardOverlapFiberMap}
is equal to the
function $t_{o,f}(\sP,[\sP'])$ from
\eqref{eq:OverlapStratumFiberFunction}.
We can characterize the function $t_f(\sP,[\sP'])$
as follows.
Let
$$
t(\Theta,\sP'_P): \sU(\Theta,\sP'_P)\subset \barM^{s,\natural}_{spl,|P|}
\to
[0,\infty),
$$
be the tubular distance function defined in \eqref{eq:DefineTubular1}.
These function (pulled back by the appropriate projections) define
a function
\begin{equation}
\label{eq:OverlapTubDist1}
\sum_{P\in\sP} |P| t(\Theta,\sP'_P)^2:
\barM(\sP)=\prod_{P\in\sP}\barM^{s,\natural}_{spl,|P|}
\to [0,\infty),
\end{equation}
which is independent of $\sP'\in [\sP<\sP']$
and we have the following:

\begin{lem}
\label{lem:DescribeOverlapTubDist}
The function $t_f(\sP,[\sP'])$
defined in \eqref{eq:OverlapTubDist}
is equal to the function defined in
\eqref{eq:OverlapTubDist1}.
\end{lem}

\begin{proof}
The lemma follows form the definition of $t_f(\sP,[\sP'])$
and the definition \eqref{eq:DefineTubular1}.
\end{proof}


We then have:

\begin{lem}
\label{lem:UpperStratTubOnLower}
For $i>j$,
the function $t(\ft,\fs,\sP_i)$ on
the subspace $\sU(\ft,\fs,\sP_i)\cap\sU(\ft,\fs,\sP_j)$
is defined by the extension of
the $G(\sP)$-invariant function
$t_f(\sP,[\sP'])$ from the fiber $\sU_f(\sP,[\sP'])$
to the subspace $\sU(\ft,\fs,\sP_i)\cap\sU(\ft,\fs,\sP_j)$ of the fiber
bundle \eqref{eq:LocalFiberBundle}.
\end{lem}

For all $k>j$ we define
\begin{equation}
\label{eq:DefineLinkFiber0}
\barM(\sP_j,[\sP_k],\eps)=
\barM(\sP_j) - t_f(\sP_j,[\sP_k])^{-1}([0,\eps)).
\end{equation}
Let $\beps_j=(\eps_j,\eps_{j+1},\dots,\eps_r)$.
Then define
\begin{equation}
\label{eq:DefineLinkFiber}
\barM(\sP_j,\beps_j)=
\barM(\sP_j,\eps_j)
\cap
\left(
\cap_{k>j} \barM(\sP_j,[\sP_k],\eps_k)
\right)
\subset
\barM(\sP).
\end{equation}


We can then conclude with the following description of the
fiber bundle structure of
$\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$.

\begin{lem}
\label{lem:LocalLinkFiberBundle}
The local instanton link $\bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)$ defined
in \eqref{eq:DefineLinkStratum}
is the subbundle of the restriction of
the fiber bundle \eqref{eq:LocalFiberBundle} to
$N(\delta)/S^1\times K_j$ defined by the subspace
\eqref{eq:DefineLinkFiber} of
the fiber $\barM(\sP_j)$:
\begin{equation}
\label{LocalInstantonLink}
\begin{CD}
\barM(\sP_j,\beps_j)
@>>>
\tilde N(\delta_P)\times_{\sG_{\fs}\times S^1}
\Fr(\ft,\fs,\sP_j,g_{\sP_j})\times_{G(\sP_j)}\barM(\sP,\beps)
\\
@. @V \pi(\ft,\fs,\sP_j) VV
\\
@. N(\delta)/S^1\times K_j
\end{CD}
\end{equation}
This local instanton link
is disjoint from $N(\delta)/S^1\times \Sym^\ell(X)$.
\end{lem}



\subsection{Duality and the link}
\label{subsec:Duality}
We now discuss how the intersection number,
\begin{equation}
\#\left(\bar\sV(z)\cap\bar\sW^{\delta_c} \cap \bar\bL_{\ft,\fs}\right)
\end{equation}
can be expressed in terms of pairing a cohomology class with a homology class.

The proof of the following is analogous to that
\cite[Proposition 5.2]{FLLevelOne}

\begin{thm}
\label{thm:Duality}
For $\beta\in H_\bullet(X;\RR)$,
let
$$
\barmu_p(\beta), \barmu_c,
\quad\text{and}\quad\bar e_i
\in H^\bullet(\bar\sM^{\vir,*}_{\ft,\fs}/S^1;\RR)
$$
be the cohomology classes from Definition \ref{defn:ExtendedCohomologyClasses}.
Let $\bar e_s=e(\Upsilon^s_{\ft,\fs}/S^1)$ be the Euler class of the
Seiberg-Witten obstruction bundle.
If $d(\ft)=\dim\sM^{*,0}_{\ft,\fs}$, then
let $[\bL^{vir}_{\ft,\fs}]\in H_{d(\ft)-2}(\bar\sM^{\vir,*}_{\ft,\fs}/S^1;\RR)$
be the homology class defined in \eqref{eq:DefineAmbientLinkFund}.
Then
\begin{equation}
\label{eq:Duality}
\#\left(\bar\sV(z)\cap\bar\sW^{\delta_c} \cap \bar\bL_{\ft,\fs}\right)
=
\langle
\barmu_p(z)\smile \barmu_c^{\delta_c}\smile \bar e_i \smile\bar e_s,
[\bL^{vir}_{\ft,\fs}]
\rangle.
\end{equation}
\end{thm}

\subsubsection{The homology class of the ambient link}
We now define the homology class of the ambient link,
$$
[\bL^{vir}_{\ft,\fs}]\in H_{d(\ft)-2}(\bar\sM^{\vir,*}_{\ft,\fs}/S^1;\RR).
$$
The union of the lower strata in $\bL^{vir}_{\ft,\fs}$ will be
denoted $\bL^{sing}_{\ft,\fs}\subset\bL^{vir}_{\ft,\fs}$.
By \cite[Propositions 3.2]{Quinn}, $\bL^{sing}_{\ft,\fs}\subset\bL^{vir}_{\ft,\fs}$ is
a neighborhood deformation retract, so there is an open neighborhood
$U\subset \bL^{vir}_{\ft,\fs}$ such that
\begin{enumerate}
\item
$\bL^{vir}_{\ft,\fs}-U$ is a topological manifold with boundary,
\item
The neighborhood $U$ retracts onto
$\bL^{sing}_{\ft,\fs}$.
\end{enumerate}
The neighborhood $U$ and the retraction of $U$ onto
$\bL^{sing}_{\ft,\fs}$ can also be constructed by noting
that (i) the property of being a neighborhood deformation retraction
is a local property and that (ii) the pre-image of the lower strata of
$\barM^{s,\natural}_{\ka}(\delta)$ in the bubbletree compactification of
$M^{s,\natural}_{\ka}(\delta)$ as defined in \cite{FeehanGeometry}
form the boundary of a topological manifold and thus are a
neighborhood deformation retraction.
Let
$$
\jmath: \left(\bL^{vir}_{\ft,\fs},\emptyset\right)
\to
\left(\bL^{vir}_{\ft,\fs},U\right)
$$
be the map of pairs.  Because $U$ retracts to the codimension-four
subspace $\bL^{sing}_{\ft,\fs}$,
$$
\jmath_*:
H_{d(\ft)-2}\left(\bL^{vir}_{\ft,\fs},\emptyset\right)
\to
H_{d(\ft)-2}\left(\bL^{vir}_{\ft,\fs},U\right)
$$
is an isomorphism.  If
$$
[\bL^{vir}_{\ft,\fs}-U,\rd(\bL^{vir}_{\ft,\fs}-U)]
\in
H_{d(\ft)-2}\left(\bL^{vir}_{\ft,\fs},U\right)
$$
is the relative fundamental class of the(topological) manifold with
boundary $\bL^{vir}_{\ft,\fs}-U$, then we define
\begin{equation}
\label{eq:DefineAmbientLinkFund}
[\bL^{vir}_{\ft,\fs}]
=\jmath_*^{-1}[\bL^{vir}_{\ft,\fs}-U,\rd(\bL^{vir}_{\ft,\fs}-U)],
\end{equation}
to be the fundamental class of the ambient link.

\begin{proof}[Proof of Theorem \ref{thm:Duality}]
The proof is analogous to that of
\cite[Proposition 5.2]{FLLevelOne} using relative Euler classes
of the obstruction sections and representatives of the cohomology
with compact support along the geometric representatives.
\end{proof}

\subsection{Reducing to a submanifold}
\label{subsec:ReduceToBase}
We now show how to reduce pairings with the homology class
$[\bL^{vir}_{\ft,\fs}]$ defined in \eqref{eq:DefineAmbientLinkFund}
to pairings with the homology class of a subspace which eliminates
the topology of the normal bundle $N(\delta)\to M_{\fs}$.


The deformation retraction $N(\delta)\to M_{\fs}$.
defines a deformation retraction of $\bar\bL^{\vir,i}_{\ft,\fs}$
to the subspace
\begin{equation}
\label{eq:InstantonLinkComponentBase}
\bar\bB\bL^{\vir}_{\ft,\fs}
=
\cup_i \ \bar\bB\bL^{\vir}_{\ft,\fs}(\sP_i)
=
\cup_i \left( t_N^{-1}(0)\cap \bL^{\vir,i}_{\ft,\fs}(\sP_i)\right)
\end{equation}
where the function $t_N$ is defined in
\eqref{eq:NTubularDist}.
Essentially, $\bar\bB\bL^{\vir}_{\ft,\fs}(\sP_i)$ is defined
by replacing $\tilde N(\delta)$ with $\tilde M_{\fs}$ in
the definition \eqref{eq:DefineLinkStratum}.  Hence,
we have the following fiber bundle description of
$\bar\bB\bL^{\vir}_{\ft,\fs}(\sP_i)$.

\begin{lem}
\label{lem:LocalBaseLinkFiberBundle}
The space
\begin{equation}
\label{eq:DefineLocalBaseLink}
\bar\bB\bL^{\vir}_{\ft,\fs}(\sP_j)
=
t_N^{-1}(0)\cap \bar\bL^{\vir,i}_{\ft,\fs}(\sP_j)
\end{equation}
is the restriction of the fiber bundle \eqref{LocalInstantonLink}
to the subspace
$$
M_{\fs}\times K_j \subset N(\delta)/S^1\times K_j
$$
of the base and is thus given by:
\begin{equation}
\label{eq:LocalBaseLinkFibration}
\begin{CD}
\barM(\sP_j,\beps_j)
@>>>
\tilde M_{\fs}\times_{\sG_{\fs}\times S^1}
\Fr(\ft,\fs,\sP_j,g_{\sP_j})\times_{G(\sP_j)}\barM(\sP,\beps)
\\
@. @V \pi(\ft,\fs,\sP_j) VV
\\
@. M_{\fs}\times K_j
\end{CD}
\end{equation}
\end{lem}


Let $\bar{\bB\bL}^{\vir}_{\ft,\fs}\subset\bar\bL^{\vir}_{\ft,\fs}$ be
the subspace defined in \eqref{eq:InstantonLinkComponentBase}.
We observe that $\bar\bL^{\vir,i}_{\ft,\fs}$ is a
disk bundle over $\bar{\bB\bL}^{\vir}_{\ft,\fs}$.
A fundamental class,
\begin{equation}
[\bar{\bB\bL}^{\vir}_{\ft,\fs}]
\in H_{d(\ft)-2r_N-2}(\bar\bL^{\vir}_{\ft,\fs};\RR),
\end{equation}
(where $r_N$ is the complex rank of the bundle $N_{\ft_0,\fs}\to M_{\fs}$)
is defined exactly as is done in \eqref{eq:DefineAmbientLinkFund}.
We then have:

\begin{prop}
\label{prop:ReductionToBase}
Let $d_s=\dim M_{\fs}$.
Let $s_j(N)\in H^{2j}(M_{\fs})$ and $r_N$ be the Segre classes
and complex rank (respectively) of the
bundle $N_{\ft_0,\fs}\to M_{\fs}$.
Let $k$ and $m$ be non-negative integers satisfying
$k+2m=d(\ft)-2$.
For any $\alpha\in H^k(M_{\fs}\times \Sym^\ell(X);\RR)$, we
have the equality
\begin{equation}
\label{eq:ReduceToBase}
\begin{aligned}
{}&
\langle
\nu^m\smile \pi_{X,\fs}^*\alpha,[\bar\bL^{\vir}_{\ft,\fs}]
\rangle
\\
\quad
&
=
\sum_{j=0}^{d_2/2}
(-1)^{r_N+j}
\langle
\nu^{m-r_N-j}
\smile
\pi_{\fs}^*s_j(N)
\smile
\pi_{X,\fs}^*\alpha,
[\bar{\bB\bL}^{\vir}_{\ft,\fs}]
\rangle
\end{aligned}
\end{equation}
\end{prop}


\begin{proof}
See \cite[\S 5.2]{FLLevelOne}.
\end{proof}

\begin{rmk}
\label{rmk:GeneralSegreClasses}
The Segre classes $s_i(N)$ have been computed under some
assumptions on $H^1(X;\RR)$ in
\cite[Lemma 4.11]{FL2a}.
From \cite[Theorem 3.29]{FL2a} one can see
that in general, these Segre classes
will be given by a universal polynomial in $\mu_{\fs}(x)$
and $\mu_{\fs}(\bga_i)$ with coefficients depending only on
the indices $n_s'$ and $n_s''$ which depend only on $p_1(\ft_0)$ and
$c_1(\ft_0)$.
\end{rmk}

\subsection{Boundary of link components}
\label{subsec:Boundaries}
Each of the  pieces $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ as
$\sP_i$ is a stratified space where each stratum
is a smooth manifold with corners.
We describe this corner structure by denoting
\begin{equation}
\label{eq:DefineBoundaryOne}
\rd_{k_1}\dots\rd_{k_r}\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)
=
\left(
\cap_{u=1}^r
t(\ft,\fs,\sP_{k_u})^{-1}(\eps_{k_u})
\right)
\cap \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j).
\end{equation}
Observe that the intersection of the
pieces of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$ is described by
the equality:
\begin{equation}
\label{eq:MatchingBoundaries}
\rd_k\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)
=
\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)
\cap
\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_k)
=
\rd_j\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_k)
\end{equation}
which follows immediately from the definition.
If
$$
\cl\Si(X^\ell,\sP_j)\cap \Si(X^\ell,\sP_k)
=\emptyset
\quad\text{and}\quad
\Si(X^\ell,\sP_j)\cap\cl\Si(X^\ell,\sP_k)
=\emptyset,
$$
then the boundaries in \eqref{eq:MatchingBoundaries} are empty.
By the symmetry of the definition
\eqref{eq:DefineBoundaryOne} with respect to $k_u$ and
the equality \eqref{eq:MatchingBoundaries}, it
suffices to describe the boundary
in \eqref{eq:DefineBoundaryOne} under the assumption
that $k_1< k_2<\dots< k_r<j$.

\begin{lem}
\label{lem:CornerDescription}
Let $K_j$ be the subspace defined in \eqref{eq:DefineK}.
For $k_1< k_2<\dots< k_r<j$, define
$\rd_{k_1}\dots\rd_{k_r}K_j$ by
\begin{equation}
\label{eq:DefineBoundaryOfCompactum}
\rd_{k_1}\dots\rd_{k_r}K_j
=
\left(
\cap_{u=1}^r
t(X^\ell,\sP_{k_u})^{-1}(\eps_{k_u})
\right)
\cap K_j
\subset \Si(X^\ell,\sP_j).
\end{equation}
Then, the corner $\rd_{k_1}\dots\rd_{k_r}\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)$
is the restriction of the fiber bundle
$$
\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)\to K_j
$$
to the subspace $\rd_{k_1}\dots\rd_{k_r}K_j\subset K_j$.
\end{lem}

\begin{proof}
The lemma follows immediately from the definitions
\eqref{eq:DefineBoundaryOne} and
\eqref{eq:DefineBoundaryOfCompactum}
and the relation between the functions $t(\ft,\fs,\sP_i)$ and
$t(X^\ell,\sP_i)$ given in the proof of
Lemma \ref{lem:InstLinkLowerStrCompl}.
\end{proof}

We further extend the description of the corner
\eqref{eq:DefineBoundaryOne} in Lemma \ref{lem:CornerDescription}
by describing the corner $\rd_{k_1}\dots\rd_{k_r}K_j\subset K_j$.
We begin by describing the corner $\rd_kK_j$ where $k<j$.
First, observe that for $\eps_k$ sufficiently small
$\rd_k K_j$ is in the image of the exponential map with
domain an open subspace of
\begin{equation}
\label{eq:NormalOfLowest}
\Fr(TX^\ell,\sP_k,g_{\sP_k})|_{K_k}
\times_{G(T,\sP)} Z(\sP_k)/\fS(\sP_k),
\end{equation}
where the fiber $Z(\sP_k)$ is given by
$$
Z(\sP_k)=\prod_{P\in\sP_k} Z_P(\delta_P).
$$
The function $t(X^\ell,\sP_k)$ is defined by
the $G(T,\sP)$-invariant functions on the fiber,
\begin{equation}
\label{eq:FiberFunction1}
m_{\sP,[\sP]}= \sum_{P\in\sP}|P| t(Z_P)^2: Z(\sP_k)\to [0,\infty)
\end{equation}
where $t(Z_P)$ is defined in Lemma \ref{lem:R4TMScale}.
(The motivation for the appearance of two copies of $\sP$ in $m_{\sP,[\sP]}$ will
be apparent shortly.)
%Then,
%define
%$$
%Z(\sP_k,\eps_k)=m_{\sP_k,[\sP_k]}^{-1}(\eps_k)\subset Z(\sP_k).
%$$
For other partitions $\sP'$ with $\sP_k<\sP'$,
a neighborhood of $K_k$ in $\Si(X^\ell,\sP')$ is given by
restricting the exponential map $e(X^\ell,g_{\sP_k})$ to
the subspace of \eqref{eq:NormalOfLowest} given by
the subspace of the fiber,
\begin{equation}
\label{eq:OverlapFiber1}
\coprod_{\sP''\in [\sP_k<\sP']}\prod_{P\in\sP_k}\Delta^\circ(Z_P,\sP''_P)
\subset Z(\sP_k).
\end{equation}
A neighborhood of this subspace of the fiber $Z(\sP_k)$ is given
by the union, over $\sP''\in [\sP_k<\sP']$, of the images of the exponential map
\begin{equation}
\label{eq:OverlapFiberExp1}
\prod_{P\in\sP_k} e(Z_P,\sP''_P):
\prod_{P\in\sP_k}
\left(
\Delta^\circ(Z_P,\sP''_P)
\times\prod_{P''\in\sP''}Z_{P''}
\right)
\to
Z(\sP_k)
\end{equation}
where $e(Z_P,\sP''_P)$ is defined in \eqref{eq:NormalOfR4Diagonal}.
On the domain of the map \eqref{eq:OverlapFiberExp1}, we
can define a function
\begin{equation}
\label{eq:DefineFiberFunction2}
m_{\sP,[\sP']}=
\sum_{P''\in\sP''}|P''| (t(Z_{P''})\circ\pi_{P''})^2
\end{equation}
where $\pi_{P''}$ is the projection from the domain of the map
\eqref{eq:OverlapFiberExp1} to $Z_{P''}$.
We also write $m_{\sP,[\sP']}$ for the pushforward of this function
to the image of the map \eqref{eq:OverlapFiberExp1}.
One can see that the function $m_{\sP,[\sP']}$ is invariant under the choice
of $\sP''\in [\sP<\sP']$.


\begin{lem}
\label{lem:CompareTubFunc}
For $\sP\le\sP'$, the pullback of the function $t(X^\ell,\sP')$
by the exponential map $e(X^\ell,\sP)$ to the domain
\eqref{eq:NormalOfLowest} is equal to the function defined
by the $G(\sP)$-invariant function $m_{\sP,[\sP']}$ on the
fiber $Z(\sP)$.
\end{lem}

\begin{proof}
The lemma follows from an examination of the diagram
\eqref{eq:OverlapNormalBundle} and the definition of $t(X^\ell,\sP')$.
Compare the discussion prior to Lemma
\ref{lem:IdentifyUpperTubFunctionOnOverlap}.
\end{proof}

Hence, for $\beps=(\eps_1,\dots,\eps_r)$ the choice of parameters
used to define the link, we define a subspace of the fiber $Z(\sP_k)$
by
\begin{equation}
\label{eq:CompactSubspOfFiber}
Z(\sP_k,\beps)=
Z(\sP_k) - \cup_{i\ge k} m_{\sP_k,[\sP_i]}^{-1}([0,\eps_i)).
\end{equation}
For $k\le k_1<\dots<k_r$, we define
\begin{equation}
\label{eq:FiberBoundary}
\rd_{k_1}\dots \rd_{k_r} Z(\sP_k,\beps)
=
Z(\sP_k)\cap
\left( \cap_{i=1}^r m_{\sP_k,[\sP_{k_i}]}^{-1}(\eps_{k_i}) \right).
\end{equation}
Finally, recall from Lemma \ref{lem:RestricedExponentialMapDomain}
that a neighborhood of $K_k$ in
$\Si(X^\ell,\sP_j)$ is parameterized by the subspace of
\eqref{eq:NormalOfLowest} given (up to symmetric group quotients)
by the subspace of the fiber:
\begin{equation}
\label{eq:DefineDiagonalsFiber2}
Z(\sP_k,[\sP_j])=
\coprod_{\sP''\in [\sP_k,\sP_j]}
\prod_{P\in\sP_k}
\Delta^\circ(Z_P(\delta_P),\sP''_P).
\end{equation}
Hence, for $k\le k_1<\dots<k_r<j$ we define
\begin{equation}
\label{eq:CornerFiberOfSymProd}
\begin{aligned}
Z(\sP_k,[\sP_j],\beps)&=
Z(\sP_k,\beps)
\cap
Z(\sP_k,[\sP_j]),
\\
\rd_{k_1}\dots \rd_{k_r} Z(\sP_k,[\sP_j],\beps)
&=
\rd_{k_1}\dots \rd_{k_r} Z(\sP_k,\beps)
\cap
Z(\sP_k,[\sP_j]).
\end{aligned}
\end{equation}
The above discussion yields the following result.

\begin{lem}
\label{lem:ParamBoundaryOfXCompactum}
For $k_1<k_1<\dots <k_r<j$, the subspace
$\rd_{k_1}\dots\rd_{k_r}K_j$ is given by the image under
the exponential map $e(X^\ell,\sP_{k_1})$ of the subspace
\begin{equation}
\label{eq:FiberBundleBoundaryOfCompactum}
\Fr(TX^\ell,\sP_{k_1},g_{\sP_{k_1}})
\times_{G(T,\sP_{k_1})}
\rd_{k_1}\dots \rd_{k_r} Z(\sP_{k_1},[\sP_j],\beps)/\fS(\sP_{k_1})
\end{equation}
of the domain \eqref{eq:NormalOfLowest}.
\end{lem}

The following relation between the boundaries
$\rd_jK_i$ and $\rd_j\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
when $i<j$ will also be useful.

\begin{lem}
\label{lem:SecondBoundaryDescript}
For $i<j$, let $\barM(\sP_i,\beps_i)$ be the space
defined in \eqref{eq:DefineLinkFiber}.
Let $t_f(\sP_i,[\sP_j])$ be the function defined in
\eqref{eq:OverlapTubDist}.
Define
\begin{equation}
\label{eq:UpwardsFiberBoundary}
\rd_j\barM(\sP_i,\beps_i)
=
t_f(\sP_i,[\sP_j])^{-1}(\eps_j)\cap \barM(\sP_i,\beps_i)
 \subset \barM(\sP_i,\beps_i).
\end{equation}
Then, the boundary $\rd_j\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$,
defined in \eqref{eq:DefineBoundaryOne},
is the fiber subbundle,
\begin{equation}
\label{eq:UpwardsFiberBoundaryFibration}
\begin{CD}
\rd_j\barM(\sP_i,\beps_i)
@>>>
\tilde M_{\fs}\times_{\sG_{\fs}\times S^1}
\Fr(\ft,\fs,\sP_i,g_{\sP_i})\times_{G(\sP_i)}\rd_j\barM(\sP_i,\beps_i)
\\
@. @V \pi(\ft,\fs,\sP_i) VV
\\
@. M_{\fs}\times K_i
\end{CD}
\end{equation}
of the fiber bundle $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
given in \eqref{eq:LocalBaseLinkFibration}.
\end{lem}

\begin{proof}
The lemma follows from the description in Lemma
\ref{lem:UpperStratTubOnLower} of the relation
between the functions $t(\ft,\fs,\sP_j)$ and
$t_f(\sP_i,[\sP_j])$.
\end{proof}

We now give another description of the
fiber $\rd_j\barM(\sP_i,\beps_i)$
appearing in \eqref{eq:UpwardsFiberBoundary}.
First, because the function $t_f(\sP_i,[\sP_j])$
is defined on $U_f(\sP_i,[\sP_j])$ in
\eqref{eq:OverlapTubDist},
we have the inclusion
$$
\rd_j\barM(\sP_i,\beps_i)
\subset
U_f(\sP_i,[\sP_j]),
$$
where the subspace $U_f(\sP_i,[\sP_j])$ is defined in
\eqref{eq:OverlapFiber}.  Because $U_f(\sP_i,[\sP_j])$
is defined
as the image of the injective (up to symmetric group
action) splicing map
in \eqref{eq:DownwardOverlapFiberMap},
there is a homeomorphism
(up to symmetric group action)
between $U_f(\sP_i,[\sP_j])$
and
$$
\coprod_{\sP''\in [\sP_i<\sP_j]}
\prod_{P\in\sP_i}
\left(
\Delta^\circ(Z_{|P|}(\delta_P),\sP''_P)
\times\prod_{Q\in\sP''_P} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
\right).
$$
This homeomorphism defines a projection map:
\begin{equation}
\label{eq:FiberBoundaryProjectionMap}
U_f(\sP_i,[\sP_j])
\to
Z(\sP_i,[\sP_j])/\fS(\sP_i)
\end{equation}
where $Z(\sP_i,[\sP_j])$ is defined in
\eqref{eq:DefineDiagonalsFiber2}.
In addition, because
\begin{align*}
\coprod_{\sP''\in [\sP_i<\sP_j]}
\prod_{P\in\sP}
\prod_{Q\in\sP''_P} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
&=
\coprod_{\sP''\in [\sP_i<\sP_j]}
\prod_{Q\in\sP''} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
\\
&=
\coprod_{\sP''\in [\sP_i<\sP_j]}\barM(\sP''),
\end{align*}
where $\barM(\sP'')$ is defined in \eqref{eq:LocalFiber},
there is a projection map
\begin{equation}
\label{eq:FiberBoundaryProjectionMapToConn}
U_f(\sP_i,[\sP_j])
\to
\coprod_{\sP''\in [\sP_i<\sP_j]}
\barM(\sP'')/\fS(\sP_i).
\end{equation}
We then have:

\begin{lem}
\label{lem:FiberBoundaryDescription1}
For $i<j$, if
$$
U_f(\sP_i,[\sP_j],\beps_i)=U_f(\sP_i,[\sP_j])\cap\barM(\sP_i,\beps_i).
$$
then the restriction of the projection map
\eqref{eq:FiberBoundaryProjectionMap} to
$U_f(\sP_i,[\sP_j],\beps_i)$ defines a projection map
\begin{equation}
\label{eq:FiberBoundaryProjectionMap1}
U_f(\sP_i,[\sP_j],\beps_i)
\to
Z(\sP_i,[\sP_j],\beps)/\fS(\sP_i),
\end{equation}
where the subspace $Z(\sP_i,[\sP_j],\beps)$ is defined
in \eqref{eq:CornerFiberOfSymProd}.
\end{lem}

\begin{proof}
The subspace $\barM(\sP_i,\beps_i)$ is
defined by the functions $t_f(\sP_i,[\sP_k])$
in \eqref{eq:DefineLinkFiber0} and
\eqref{eq:DefineLinkFiber}.
The subspace $Z(\sP_i,[\sP_j],\beps_i)$
is defined in \eqref{eq:CompactSubspOfFiber}
by the functions $m_{\sP_i,[\sP_k]}$ in
\eqref{eq:DefineFiberFunction2}.
The projection map \eqref{eq:FiberBoundaryProjectionMap},
is given by a product of the projection maps
$\pi(\Theta,\sP''_P)$ defined in
Lemma \ref{eq:SplicedEndR4Projection}.
The tubular distance functions $t(\Theta,\sP''_P)$ are
constant on the fibers of these projections by
\eqref{eq:R4SplicedEndProjectionTM}.
The lemma then follows from the relation between
the functions $t_f(\sP_i,[\sP_k])$
and $t(\Theta,\sP''_P)$ in Lemma
\ref{lem:DescribeOverlapTubDist}
and the relation between $t(\Theta,\sP'_P)$
and the function $m_{\sP,[\sP'']}$
given by the definitions of these function.
\end{proof}

A similar argument yields the following
characterization of the fiber
$\rd_j\barM(\sP_i,\beps_i)$.


\begin{lem}
\label{lem:BoundaryFiberDescription2}
For $i<j$, the restriction of the projection map
\eqref{eq:FiberBoundaryProjectionMap} to
the fiber $\rd_j\barM(\sP_i,\beps_i)$ appearing in
\eqref{eq:UpwardsFiberBoundary} defines a projection map
$$
\rd_j\barM(\sP_i,\beps)
\subset
\rd_jZ(\sP_i,[\sP_j],\beps),
$$
to the subspace $\rd_jZ(\sP_i,[\sP_j],\beps_i)$
defined in \eqref{eq:CornerFiberOfSymProd}.
\end{lem}

We now characterize the fiber of the projection
map \eqref{eq:FiberBoundaryProjectionMap}
appearing in Lemmas \ref{lem:FiberBoundaryDescription1}
and \ref{lem:BoundaryFiberDescription2}.

\begin{lem}
\label{lem:FiberBoundaryDescription3}
For $i<j$, the fiber $\rd_j\barM(\sP_i,\beps_i)$ appearing in
\eqref{eq:UpwardsFiberBoundary} is homeomorphic to
the subspace of
\begin{equation}
\coprod_{\sP''\in [\sP_i<\sP_j]}
\prod_{P\in\sP_i}
\left(
\rd_jZ(\sP_i,[\sP_j],\beps)
\times\prod_{Q\in\sP''_P} \barM^{s,\natural}_{spl,|Q|}(\delta_Q)
\right)
\end{equation}
given by the pre-image of the subspace
$$
\coprod_{\sP''\in [\sP_i<\sP_j]}
\barM(\sP'',\beps_j)/\fS(\sP_i)
$$
under the projection map \eqref{eq:FiberBoundaryProjectionMapToConn}.
\end{lem}

\begin{proof}
The fiber $\rd_j\barM(\sP_i,\beps_i)$
is defined by the functions $t_f(\sP_i,[\sP_k])$ where $k>i$.
For $j>k>i$, the constraint $t_f(\sP_i,[\sP_k])^{-1}((\eps_k,\infty))$,
is given, as described in Lemma \ref{lem:FiberBoundaryDescription1},
by the restriction of the projection map
\eqref{eq:FiberBoundaryProjectionMap} to the subspace
$\rd_jZ(\sP_i,[\sP_j],\beps)$.  For $k\ge j>i$,
the function $t_f(\sP_i,[\sP_k])$ on
$U_f(\sP_i,[\sP_j])$
and the function $t_f(\sP'',[\sP_k])$
(for $\sP''\in [\sP_i<\sP_j]$) on
$U_f(\sP'',[\sP_k])$ are both defined by
pullback of the function $t(\ft,\fs,\sP_k)$
(see Lemma \ref{lem:IdentifyUpperTubFunctionOnOverlap}).
Hence, the constraints $t_f(\sP_i,[\sP_k])^{-1}((\eps_k,\infty))$
for $k\ge j>i$ are as described in the statement of the Lemma.
\end{proof}

%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Article: INTERSECTION
%File: comp.tex
%Last update: March 4th, 2002 (Tom)
%Spellcheck:  March 4th, 2002 (Tom)
%Todo:

\section{The computation}
\label{sec:Comp}


In this section, we perform the computation leading to a proof of
the following theorems which form the technical heart of this paper.

\begin{thm}
\label{thm:LinkPairing}
Let $\ft$ be a \spinu structure on a smooth four-manifold $X$
with $b^1(X)=0$.
Assume that $\fs$ is a \spinc structure with
$M_{\fs}\times\Sym^\ell(X)$ a subset of the space of ideal
monopoles $I\sM_{\ft}$ defined in \eqref{eq:idealmonopoles}.
Let $\bar\bL_{\ft,\fs}$ be the link of the reducible
$\SO(3)$ monopoles given by $\fs\in\Spinc(X)$
as defined in Definition \ref{defn:DefineLink}.
For $z=h^{\delta-2m}x^m\in\AAA(X)$ and $\eta$ a non-negative integer
satisfying
$$
\deg(z)+2\eta=\dim\sM_{\ft}-2,
$$
let $\bar\sV(z)$ and $\bar\sW^{\eta}$ be the geometric
representatives defined in \S \ref{subsec:Cohomology}.
Then
\begin{equation}
\label{eq:LinkIntersectionFormula}
\begin{aligned}
{}&
\#\left( \bar\sV(z)\cap\bar\sW^{\eta}\cap\bar\bL_{\ft,\fs}\right)
\\
&\quad
=
\SW_X(\fs)
\sum_{i=0}^{ \min(\ell,\left[\frac{\delta-2m}{2}\right])}
\left(
q_{\delta,\ell,m,i}(c_1(\fs)-c_1(\ft),c_1(\ft))Q_X^i
\right) (h).
\end{aligned}
\end{equation}
where $q_{\delta,\ell,m,i}$ are degree $\delta-2m-2i$
homogeneous polynomials which are universal functions
of the constants given in Theorem \ref{thm:MainThm}.
\end{thm}

We note that a similar result can also be achieved
by the methods of this paper without
the assumption that $b^1(X)=0$ and $z=h^{\delta-2m}x^m$,
but the resulting expression becomes considerably more
complicated.
While the precise expression for the intersection number in
\eqref{eq:LinkIntersectionFormula} is still unknown, we note
the following important result,
referred to as the `multiplicity conjecture' in
\cite{FL2b,FKLM, FLLevelOne}, which holds even without the
simplifying assumptions.

\begin{thm}
\label{thm:Multiplicity}
Let $X$ be a smooth, oriented four-manifold.
Assume that $\ft$ is a \spinu structure on $X$ and $\fs$ is
a \spinc structure on $X$ such that $M_{\fs}\times\Sym^\ell(X)$
is contained in the space of ideal monopoles $I\sM_{\ft}$
defined in \eqref{eq:idealmonopoles}.
Let $\bar\bL_{\ft,\fs}$ be the link of $M_{\fs}\times\Sym^\ell(X)$
given in Definition \ref{defn:DefineLink}.
Then,
if $\SW_{X,\fs}(\om)$ vanishes for all $\om\in\AAA_2(X)$, then
$$
\#\left( \bar\sV(z)\cap\bar\sW^{\delta}\cap\bar\bL_{\ft,\fs}\right)=0.
$$
\end{thm}

To prove Theorems \ref{thm:LinkPairing} and \ref{thm:Multiplicity},
we begin by observing that
Theorem \ref{thm:Duality} gives the equality
\begin{equation}
\label{eq:IntersectionToCohom1}
\#\left(\bar\sV(z)\cap\bar\sW^{\delta_c} \cap \bar\bL_{\ft,\fs}\right)
=
\langle
\barmu_p(z)\smile \barmu_c^{\delta_c}\smile \bar e_i \smile\bar e_s,
[\bar\bL^{vir}_{\ft,\fs}]
\rangle.
\end{equation}
The expression for the $\mu$-classes in Corollary \ref{cor:CohomologyClasses},
the expression for $\bar e_s$ in Lemma \ref{lem:SWObstruction},
the expression for $\bar e_i$ in Proposition
\ref{prop:GlobalInstantonEulerClass}, and the equality between
the homology classes $[\bar\bL^{vir}_{\ft,\fs}]$ and
$[\bar{\bB\bL}^{vir}_{\ft,\fs}]$ in
Proposition \ref{prop:ReductionToBase} implies that the
intersection number in \eqref{eq:IntersectionToCohom1} can be
reduced to pairings of the form
\begin{equation}
\label{eq:ReducedPairing1}
\langle
\nu^a\smile\pi_X^*S^\ell(\beta)\smile \pi_{\fs}^*\mu_s(z_2),
[\bar{\bB\bL}^{vir}_{\ft,\fs}]
\rangle
\end{equation}
where $\beta\in H^\bullet(X;\RR)$ and $z_2\in \AAA_2(X)$.

We wish to compute the pairings \eqref{eq:ReducedPairing1} by
writing it as a sum over pairings with the pieces
$\bar{\bB\bL}^{vir}_{\ft,\fs}(\sP_i)$ and then use the fiber bundle
structure in Lemma \ref{lem:LocalBaseLinkFiberBundle} of each of these pieces.
To do this, we must replace the cohomology classes in
\eqref{eq:ReducedPairing1} by cohomology classes with compact support
away from the boundaries $\rd_j\bar{\bB\bL}^{vir}_{\ft,\fs}(\sP_i)$.
Such a replacement requires a choice and this choice must be
done consistently on each piece.  We encode these choices in the
following geometric data.  We define a quotient,
$\check{\bB\bL}^{vir}_{\ft,\fs}$, of
$\bar{\bB\bL}^{vir}_{\ft,\fs}$ by replacing the boundaries
$\rd_j\bar{\bB\bL}^{vir}_{\ft,\fs}(\sP_i)$  with spaces
of codimension greater than or equal to two.
The cohomology classes in \eqref{eq:ReducedPairing1} pull back
from cohomology classes on this quotient.
Because the image of the boundaries in the quotient has
codimension greater than or equal to two, we can
define cohomology classes with compact support by pulling back
cocycles from the quotient.  Using these cocycles is
equivalent to computing \eqref{eq:ReducedPairing1} using the
fundamental class of $\check{\bB\bL}^{vir}_{\ft,\fs}$
which, because the image of the boundaries has codimension two,
can be written as a sum of fundamental classes of pieces.

In \S \ref{subsec:Quotient}, we construct the quotient space
$\check{\bB\bL}^{vir}_{\ft,\fs}$.  This construction shows
that the images of each piece $\bar{\bB\bL}^{vir}_{\ft,\fs}(\sP_i)$
in the quotient has a fiber bundle structure identical to that
of $\bar{\bB\bL}^{vir}_{\ft,\fs}(\sP_i)$ but with a compact
fiber and base.  In \S \ref{subsec:Pullbacks}, we show how
to use the quotient to perform the desired computations.
Finally in \S \ref{subsec:Proofs} we give the proofs of the
main theorems of the paper.



\subsection{The quotient space}
\label{subsec:Quotient}
We wish to take advantage of the fiber bundle structure
\begin{equation}
\label{eq:FiberBundle}
\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j) \to K_j
\end{equation}
and our control over the structure group of this fibration
to compute cohomological pairings with $\bar{\bB\bL}^{\vir}_{\ft,\fs}$.
To do this, we must write the pairing
in \eqref{eq:ReducedPairing1}
with the homology class $[\bar{\bB\bL}^{\vir}_{\ft,\fs}]$ as a sum over
pairings with some kind of homology class
representing the space $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)$.
However,  the pieces $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)$ have boundaries
as described in \S \ref{subsec:Boundaries} and so would only define
relative homology classes.
To overcome this, we describe a quotient of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$
obtained by leaving the interiors of the pieces $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)$
and replacing the boundaries
$\rd_k\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)$ with quotients obtained by
deleting part of the gluing data.

To be more precise, for $A\subset X$ a topological
space and $f:A\to B$ a
continuous map, we define the {\em compression of $X$ along $f$\/}
to be the quotient space obtained from $X$ by identifying points
in $A$ which have the same image under $f$.  The following is an elementary
exercise.

\begin{lem}
\label{lem:MappingConeReplacement}
If $A\subset X$ has a neighborhood $N_A$
homeomorphic to a mapping cone of some map $p:Y\to A$, then the compression
of $X$ along $f$ is homeomorphic to the space obtained by replacing
$N_A$ with the mapping cone of the map $f\circ p$.
\end{lem}

We will define the quotient of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$ to be the
compressions of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$ along a projection
map from $\rd_k\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)$ given by deleting the
data in the fiber of $\rd_k K_j\to K_k$
(given by \eqref{eq:FiberBundleBoundaryOfCompactum}).
Three observations
then complete the computation.
First, the quotient is the union of the quotients of the pieces,
$\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$, and the intersection of
any two such pieces has codimension-four.
Second, the cohomology classes in \eqref{eq:IntersectionToCohom1} pull back
from this quotient and hence the computation can be done on the quotient.
Third, the quotient pieces, $\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$, all
admit the desired fiber bundle structure, \eqref{eq:FiberBundle},
albeit over a larger space, with the same structure group.


\subsubsection{The quotients of $K_i$}
We begin by defining a quotient $\check K_i$ of the compact subspaces $K_j$
of $\Sym^\ell(X)$ defined in \eqref{eq:DefineK}.
We define the quotients inductively.
As before, we have enumerated the partitions respecting the
ordering of $\Si(X^\ell,\sP_i)$.
For $j<i$, write
\begin{equation}
\label{eq:SymProjectionShortNotation}
\pi_{j,i}:\rd_jK_i\to K_j
\end{equation}
for the projection
defined by the restriction of $\pi(X^\ell,\sP_j)$ to the subspace
in Lemma \ref{lem:ParamBoundaryOfXCompactum}.  For $k<j<i$, the
equality
\begin{equation}
\label{eq:MultipleSymProjectionEquality}
\pi_{k,i}=\pi_{k,j}\circ \pi_{j,i}
\end{equation}
follows from the Thom-Mather equality \eqref{eq:DiagonalsTM1} which is
shown to hold in Lemmas
\ref{lem:DiagonalsTM1}-\ref{lem:FlatteningMetrics2}.


If $\sP_1$ is the lowest partition, $\check{K_1}=K_1$.
For all $j$ with $1<j$, let $\check K_j(1)$ be the
compression of $K_j$ along the map $\pi_{1,j}:\rd_1K_j\to K_1$.
Let $\rd_i\check K_j(1)$ be the image of $\rd_i K_j$ under
the projection $K_j\to \check K_j(1)$.
The equalities
\begin{equation}
\label{eq:EqualProjections}
\pi_{1,j}\circ\pi_{j,i}=\pi_{1,i}
\quad\text{for $1<j<i$}
\end{equation}
imply that the projections $\pi_{j,i}$ descend to
maps on the quotients,
$\check\pi_{j,i}(1):\rd_i\check K_i(1)\to \check K_j(1)$.
Define $\check K_2=\check K_2(1)$.

Continue inductively.  Assume that
\begin{enumerate}
\item
For all $i=1,\dots,r$,
a quotient $\check K_i(1,2,\dots,p)$ of $K_i$ has been defined,
\item
If $i>j$ and $\rd_j \check K_i(1,2,\dots,p)$ is the image of
$\rd_jK_i$ under the quotient map, then the map
$\pi_{j,i}:\rd_jK_i\to K_j$ induces a map
$$
\check\pi_{j,i}(1,2,\dots,p):\rd_j\check K_i(1,2,\dots,p)\to \check K_j(1,2,\dots,p),
$$
\item
The quotient $\check K_i(1,2,\dots,p)$ is the compression of
$\check K_i(1,2,\dots,p-1)$ along the map
$\check \pi_{k-1,i}(1,\dots,p-1):\rd_{k-1}K_i(1,2,\dots,p-1)$
\end{enumerate}
Because the maps $\check\pi_{j,i}(1,2,\dots,p)$ are defined by
quotients of the maps $\pi_{j,i}$, the analogue of the
equality \eqref{eq:EqualProjections} holds for $i>j>k$:
\begin{equation}
\label{eq:EqualProjections1}
\check \pi_{k,j}(1,\dots,p)\circ
\check \pi_{j,i}(1,\dots,p)=\check \pi_{k,i}(1,\dots,p).
\end{equation}
For $i>p+1$, we then define
$\check K_i(1,\dots,p+1)$ to be the compression
of $\check K_i(1,\dots,p)$ along the map
$$
\check \pi_{p+1,i}(1,\dots,p):\rd_{p+1}\check K_i(1,\dots,p)
\to
K_{p+1}(1,\dots,p).
$$
For $i\le p+1$, we define $\check K_i(1,\dots,p+1)=\check K_i(1,\dots,p)$.
The equality \eqref{eq:EqualProjections1} implies that the maps
$\check \pi_{j,i}(1,\dots,p)$ descend to define maps
$\check \pi_{j,i}(1,\dots,p+1)$ on the quotients,
completing the induction.

Define $\check K_i=\check K_i(1,\dots,i-1)$.
We have the following characterization of $\check K_i$.



\begin{lem}
The quotient $\check K_i$ is homeomorphic to $\cl\Si(X^\ell,\sP_i)$.
\end{lem}

\begin{proof}
The lemma follows from the construction of $\check K_i$
and from Lemma \ref{lem:MappingConeReplacement}.
\end{proof}




\subsubsection{The quotients of $\bB\bL^{vir}_{\ft,\fs}(\sP_i)$}

The construction of the quotient
$\check {\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ is done in an identical
manner to that of $\check K_i$.
To control the resulting topology, we begin with the following lemma.


\begin{lem}
\label{lem:FibrationDescends}
The fiber bundle $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)\to K_i$
is the pullback of an orbifold fiber bundle
$\check {\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)\to\check K_i$
by the projection map $K_i\to\check K_i$.
\end{lem}

\begin{proof}
The fiber bundle
$$
\Fr(\ft,\fs,\sP_i,g_{\sP_i})\to \Delta^\circ(X^\ell,\sP_i)
$$
defined in \eqref{eq:DefineGluingDataBundle} extends from
the open diagonal $\Delta^\circ(X^\ell,\sP_i)$ to
its closure $\Delta(X^\ell,\sP_i)$.  The action of
the symmetric group, $\fS(\sP_i)/\Ga(\sP_i)$, is no longer free,
but only pseudo-free
on this extension and hence an orbifold fiber bundle.
The quotient space
$\check {\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ is defined by using
this extension of the above fiber bundle, the identification
of $\check K_i$ with $\Si(X^\ell,\sP_i)$, and leaving the
fiber unchanged.
\end{proof}

By the characterization of $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
as a pullback, there is a map
\begin{equation}
\label{eq:QuotientMap1}
\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)
\to
\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)
\end{equation}
which we refer to as the quotient map.
The following observation about the pseudo-fiber
bundle described above is an immediate consequence
of its construction.

\begin{lem}
\label{lem:S1OnQuotient}
There is a free $S^1$ action on the bundle
$\check {\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)\to\check K_i$
defined in Lemma \ref{lem:FibrationDescends} such
that the quotient map \eqref{eq:QuotientMap1}
is equivariant with respect to the $S^1$ action defining
the cohomology class $\nu$ on
$\bar {\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
and this $S^1$ action on the quotient.
\end{lem}


The quotient $\check {\bB\bL}^{\vir}_{\ft,\fs}$ of
$\bar{\bB\bL}^{\vir}_{\ft,\fs}$ is then defined, roughly speaking,
by taking the compressions of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$ along the
top map in the diagram for $i<j$
\begin{equation}
\label{eq:CompressionMap}
\begin{CD}
\rd_i \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)
@> \pi^{\bB}_{i,j} >>
\check {\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)|_{\check K_i}
\\
@V p VV @VVV
\\
\rd_i K_j @> \pi_{i,j}>> \rd_i\check K_j
\end{CD}
\end{equation}
where we understand $\rd_i \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)$ to
be a subspace of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$.



To carry out the construction of $\check {\bB\bL}^{\vir}_{\ft,\fs}$,
we use a downwards induction, first
taking the compression along the maps
$\pi^{\bB}_{i,r}$ (where $\sP_r$ is the top stratum).
That this may be done for more than one boundary
$\rd_j \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_r)$
(i.e. the map $\pi^{\bB}_{j,r}$ descends to the quotient
obtained from a compression along $\pi^{\bB}_{i,r}$)
follows from the equality \eqref{eq:MultipleSymProjectionEquality})
on the maps $\pi_{k,i}$ on $\rd_kK_i$ and the compatibility
of the trivializations of the bundle $\Fr(\ft,\fs,\sP_r)$
used to define the maps $\pi^{\bB}_{i,r}$.

To continue the induction, taking the quotient of
${\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$, for $r>i$ along
maps $\pi^{\bB}_{j,i}$, we must ensure that the maps
$\pi^{\bB}_{j,i}$ descend to the quotient of
${\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ obtained by
compressing along the map $\pi^{\bB}_{i,j}$ where
$i<j$.
We use the following result describing the compression
along the map $\pi^{\bB}_{i,j}$, for $i<j$,
as a compression of a map on a subspace of
$\rd_j \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$.

\begin{lem}
\label{lem:DescribeUpwardsBoundaryQuotient}
If $i<j$ and $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
is compressed along the map
$$
\pi^{\bB}_{i,j}:
\rd_i \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)
\to
\check {\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)|_{\check K_i}
$$
then the resulting quotient of
$\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ can be described
as a fiber bundle,
\begin{equation}
\label{eq:UpwardsFiberQuotient}
\begin{CD}
\check{M}_j(\sP_i,\beps_i)
@>>>
\tilde M_{\fs}\times_{\sG_{\fs}\times S^1}
\Fr(\ft,\fs,\sP_i,g_{\sP_i})\times_{G(\sP_i)}\check{M}(\sP_i,\beps_i)
\\
@. @V \pi(\ft,\fs,\sP_i) VV
\\
@. M_{\fs}\times \check K_i
\end{CD}
\end{equation}
where the fiber $\check{M}_j(\sP_i,\beps_i)$ is a compression
of $\barM(\sP_i,\beps_i)$ along the map
\begin{equation}
\label{eq:FiberCompressionMap}
\pi^{\bB,f}_{i,j}:\rd_j\barM(\sP_i,\beps_i) \to
\coprod_{\sP''\in [\sP_i<\sP_j]}
\barM(\sP'',\beps_j)/\fS(\sP_i)
\end{equation}
defined by the map
\eqref{eq:FiberBoundaryProjectionMapToConn} as described in
Lemma \ref{lem:FiberBoundaryDescription3}.
\end{lem}

\begin{proof}
The compression along the map $\pi^{\bB}_{i,j}$ is defined
by deleting the data in the fiber of the map
$\pi_{i,j}$ in the diagram \eqref{eq:CompressionMap}.
Assume the fiber lies over a point $\bx\in\rd_j\check K_i$
with pre-image in
$\rd_{k_1}\rd_{k_2}\dots \rd_{k_r} \rd_j K_i$
and whose pre-image is disjoint from all other corners
$\rd_{k_0}\rd_{k_1}\rd_{k_2}\dots \rd_{k_r} \rd_j K_i$.
Then,  the fiber of the map $\pi_{j,i}$ in the
diagram \eqref{eq:CompressionMap}
given by the fiber of the projection
$$
\rd_{k_1}\rd_{k_2}\dots \rd_{k_r} \rd_j K_i \to K_{k_1}
$$
given by the space
$\rd_{k_2}\dots \rd_{k_r} \rd_j Z(\sP_{k_1},[\sP_i],\beps)$
defined in \eqref{eq:CornerFiberOfSymProd}
(see Lemma \ref{eq:FiberBundleBoundaryOfCompactum}).
Using the equality of boundaries,
$$
\rd_i \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_j)
=
\rd_j \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)
$$
and the description of the fiber of
$\rd_j \bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ over $\bx$
in Lemma \ref{lem:FiberBoundaryDescription3}, we see
the compression is as described in the lemma.
\end{proof}

Proceeding inductively, one observes that
the quotient described in Lemma
\ref{lem:DescribeUpwardsBoundaryQuotient}
is still the pullback from a bundle (with the same fiber)
over $\check K_i$ because only the
fiber has been changed and not the underlying
bundle.
Thus, one may assume (inductively) that a
quotient,
$\tilde{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$,
of $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
has been obtained by all the compressions along
$\rd_j\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ for $i<j$ as described
in Lemma \ref{lem:DescribeUpwardsBoundaryQuotient}.
Because these compressions only alter the fiber of the bundle
$\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)\to K_i$,
the quotient $\tilde{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
still admits a fiber bundle structure
$\tilde{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)\to K_i$
which is the pullback of a bundle
$\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
as in Lemma \ref{lem:FibrationDescends}.
The bundle $\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
is a quotient of $\tilde{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
compressing along the maps in induced by $\pi^{\bB}_{k,i}$
induced on the quotient, allowing us to proceed inductively.

We summarize:

\begin{lem}
\label{lem:QuotientBundle}
Let
$$
\check{\Fr}(\ft,\fs,\sP_i,g_{\sP_i})\to\Delta(X^\ell,\sP_i)
$$
be the extension of the bundle
$$
\Fr(\ft,\fs,\sP_i,g_{\sP_i})\to\Delta^\circ(X^\ell,\sP).
$$
Then,
the quotient $\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
of $\bar{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ admits a fiber
orbifold bundle structure,
\begin{equation}
\label{eq:QuotientFiber}
\begin{CD}
\check{M}(\sP_i)
@>>>
\tilde M_{\fs}\times_{\sG_{\fs}\times S^1}
\check{\Fr}(\ft,\fs,\sP_i,g_{\sP_i})\times_{G(\sP_i)}\check{M}(\sP_i,\beps_i)
\\
@. @V \pi(\ft,\fs,\sP_i) VV
\\
@. M_{\fs}\times \cl\Si(X^\ell,\sP_i)
\end{CD}
\end{equation}
where the fiber $\check{M}(\sP_i)$ is obtained by compressions of
the fiber $\barM(\sP_i,\beps_i)$ of \eqref{eq:LocalBaseLinkFibration}
along the maps $\pi^{\bB,f}_{i,j}$ defined in
\eqref{eq:FiberCompressionMap}.
\end{lem}

The orbifold bundle structure is due to the non-trivial stabilizer
of $\fS(\sP)/\Ga(\sP)$ acting on $\bar{\Fr}(\ft,\fs,\sP)$ on the points in
$\Delta(X^\ell,\sP)-\Delta^\circ(X^\ell,\sP)$.
We then define
\begin{equation}
\label{eq:DefineBaseQuotient}
\check{\bB\bL}^{\vir}_{\ft,\fs}
=
\cup_i \check{\bB\bL}^{\vir}_{\ft,\fs},
\end{equation}
to be the quotient of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$
defined by the taking union of the quotients of the pieces
as described above.
Note that the same quotient is taken on each piece.

\begin{prop}
\label{prop:PropertiesOfQuotient}
The quotient $\check{\bB\bL}^{\vir}_{\ft,\fs}$
of $\bar{\bB\bL}^{\vir}_{\ft,\fs}$ defined above
has the following properties:
\begin{enumerate}
\item
The maps $\pi_X:\bar{\bB\bL}^{\vir}_{\ft,\fs}\to \Sym^\ell(X)$
and $\pi_{\fs}:\bar{\bB\bL}^{\vir}_{\ft,\fs}\to M_{\fs}$
descend to the quotient,
\item
There is a cohomology class
\begin{equation}
\label{eq:S1ClassOnQuotient}
\check \nu\in H^2(\check{\bB\bL}^{\vir}_{\ft,\fs})
\end{equation}
which pulls back to
the cohomology class $\nu\in H^2(\bar{\bB\bL}^{\vir}_{\ft,\fs})$
under the quotient map.
\end{enumerate}
\end{prop}

\begin{proof}
The first property is immediate from the construction.
The second property follows from Lemma
\ref{lem:S1OnQuotient}.
\end{proof}

\subsubsection{The fundamental classes}
We can define a fundamental class
$$
[\check{\bB\bL}^{\vir}_{\ft,\fs}]
\in H_{\max}(\check{\bB\bL}^{\vir}_{\ft,\fs})
$$
as was done in \eqref{eq:DefineAmbientLinkFund}.
However, because the intersections of the different
pieces $\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$
have codimension greater than or equal to two,
we can also define fundamental classes,
$$
[\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)]
\in H_{\max}(\check{\bB\bL}^{\vir}_{\ft,\fs}).
$$
If
$$
\pi_{\bB}:
\bar{\bB\bL}^{\vir}_{\ft,\fs}
\to
\check{\bB\bL}^{\vir}_{\ft,\fs}
$$
is the quotient map, then the fundamental classes defined
above satisfy:
\begin{equation}
(\pi_{\bB})_*
[\bar{\bB\bL}^{\vir}_{\ft,\fs}]
=
[\check{\bB\bL}^{\vir}_{\ft,\fs}],
\end{equation}
and
\begin{equation}
[\check{\bB\bL}^{\vir}_{\ft,\fs}]
=
\sum_i \ [\check{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)].
\end{equation}

The preceding discussion and the equality
$\nu=\pi_{\bB}^*\check\nu$ in
Proposition \ref{prop:PropertiesOfQuotient}
then imply:

\begin{lem}
\label{lem:ReduceToLocalPairings}
For any  $\mu_{\fs}\in H^\bullet(M_{\fs})$,
$S^\ell(\beta)\in H^\bullet(\Sym^\ell(X))$,
and $z_2\in\AAA_2(X)$,
the equality
\begin{equation}
\label{eq:ReduceToLocalPairings}
\begin{aligned}
{}&
\langle \nu^\delta\smile \pi_{\fs}^*\mu_{\fs}(z_2)\smile\pi_X^*S^\ell(\beta),
[{\bB\bL}^{\vir}_{\ft,\fs}]\rangle
\\
&\quad=
\sum_i
\langle \check\nu^\delta\smile \pi_{\fs}^*\mu_{\fs}(z_2)\smile\pi_X^*S^\ell(\beta),
[\check{\bB\bL}^{\vir}_{\ft,\fs}]\rangle.
\end{aligned}
\end{equation}
\end{lem}

\subsection{Pullback arguments}
\label{subsec:Pullbacks}
The pairings on the right-hand-side of
\eqref{eq:ReduceToLocalPairings} can be computed using a
pushforward-pullback argument as follows.
The pseudo-fiber bundle in \eqref{eq:UpwardsFiberQuotient}
is branch-covered by the fiber bundle,
\begin{equation}
\label{eq:BranchCoverFibration}
\begin{CD}
\check{M}(\sP_i,\beps_i)
@>>>
\tilde M_{\fs}\times_{\sG_{\fs}\times S^1}
\Fr(\ft,\fs,\sP_i,g_{\sP_i})\times_{\tG(\sP_i)}\check{M}(\sP_i,\beps_i)
\\
@. @V \pi(\ft,\fs,\sP_i) VV
\\
@. M_{\fs}\times \Delta(X^\ell,\sP_i)
\end{CD}
\end{equation}
Let $\hat{\bB\bL}^{\vir}_{\ft,\fs}(\sP_i)$ denote the fiber bundle
in \eqref{eq:BranchCoverFibration}.  (The actual degree of the
covering, $|\fS(\sP_i)/\Ga(\sP_i)|$, is not relevant to this
discussion.)  Hence, we have
\begin{equation}
\langle
\check\nu^\delta\smile \pi_{\fs}^*\mu_{\fs}\smile\pi_X^*S^\ell(\beta),
[\check{\bB\bL}^{\vir}_{\ft,\fs}]
\rangle
=
q_i
\langle
\check\nu^\delta\smile \pi_{\fs}^*\mu_{\fs}\smile\pi_X^*S^\ell(\beta),
[\hat{\bB\bL}^{\vir}_{\ft,\fs}]
\rangle
\end{equation}
for some rational number $q$.
Let $G_i=\tG(\sP_i)$.
Let $f_{\ft,\fs,i}:\Delta(X^\ell,\sP_i)\to \BG_i$ be the classifying
map.  We then have a diagram,
\begin{equation}
\label{eq:LocalPushPull}
\begin{CD}
\check{\bB\bL}^{\vir}_{\ft,\fs}
@> \tilde f_{\ft,\fs,i} >>
S^1\backslash\EG_i\times_{\tG(\sP_i)}\check{M}(\sP_i,\beps_i)
\\
@V\pi(\ft,\fs,\sP_i) VV
@VVV
\\
\Delta(X^\ell,\sP_i) @> f_{\ft,\fs,i} >> \BG_i
\end{CD}
\end{equation}
Observe that $\check\nu$ is the pullback of first
Chern class of the $S^1$ action on
$\EG_i\times_{\tG(\sP_i)}\check{M}(\sP_i,\beps_i)$.
Then, a pushforward pullback argument applied
to the diagram \eqref{eq:LocalPushPull} yields

\begin{lem}
\label{lem:LocalPushPull}
There are universal polynomials,
$m_{\ell,\delta,\ft,\fs,i}\in H^\bullet(M_{\fs}\times\Delta(X^{\ell},\sP_i))$,
in the characteristic classes of the \spinu structure $V_\ell$
and the
cohomology classes $\mu_{\fs}(\cdot)$ such that
\begin{equation}
\label{eq:FinalPairing}
\begin{aligned}
{}&\langle
\check\nu^\delta\smile \pi_{\fs}^*\mu_{\fs}(z_2)\smile\pi_X^*S^\ell(\beta),
[\hat{\bB\bL}^{\vir}_{\ft,\fs}]
\rangle
\\
&\quad=
\langle m_{\ell,\delta,\ft,\fs,i}\smile
\mu_{\fs}(z_2)\smile\pi_X^*S^\ell(\beta),[M_{\fs}\times \Delta(X^{\ell},\sP_i)]\rangle.
\end{aligned}
\end{equation}
\end{lem}


\subsection{Proof of the main theorem}
\label{subsec:Proofs}
We now prove Theorems
\ref{thm:LinkPairing},
\ref{thm:MainThm},
and \ref{thm:Multiplicity}.



\begin{proof}[Proof of Theorem \ref{thm:LinkPairing}]
The discussion prior to \eqref{eq:ReducedPairing1},
Lemma \ref{lem:ReduceToLocalPairings}, and
Lemma \ref{lem:LocalPushPull} imply that
the pairing on the left-hand-side of
\eqref{eq:LinkIntersectionFormula} can be reduced to
an algebraic expression in terms of the form appearing on
the right-hand-side of \eqref{eq:FinalPairing}.
The result then follows from a discussion on the type
of terms which will appear.
First, observe that factors of $Q_X(h)$ will arise only
from products of $S^\ell(h)\smile S^\ell(h)$.  There are
at most $(\delta-2m)/2$ of these factors.
In addition, the dimension of the highest
stratum in $\Sym^\ell(X)$ is $4\ell$, so we can have no
power of $S^\ell(h)\smile S^\ell(h)$ greater than $\ell$.
Thus, the power
of $Q_X(h)$ appearing in the formula must be less than
$\min[(\delta-2m)/2,\ell]$.

The factors of $\langle c_1(\fs)-c_1(\ft),h\rangle$
and $\langle c_1(\ft),h\rangle$ appearing in
\eqref{eq:LinkIntersectionFormula} can be understood
by the algebraic work in \cite[\S 6.1]{FLLevelOne}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:Multiplicity}]
The condition that $\langle \mu_{\fs}(z_2),[M_{\fs}]\rangle=0$
for all $z_2\in\AAA_2(X)$ and the expression on
the right-hand-side of
\eqref{eq:FinalPairing}, (to which the intersection number can
be reduced) implies that all pairings will be multiples
of $\SW_{X,\fs}(\cdot)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:MainThm}]
Theorem \ref{thm:MainThm} follows from
Theorem \ref{thm:LinkPairing} and the cobordism
formula \eqref{eq:RawCobordismSum1}.  The 
polynomials $p_{\delta,\ell,m,i}$
appearing in Theorem \ref{thm:MainThm}
differ from the
polynomials $q_{\delta,\ell,m,i}$ appearing
in Theorem \ref{thm:LinkPairing} 
by the factors of two appearing in 
\eqref{eq:ASDPairing} and factors of negative one
from the change of orientation expression 
in \eqref{eq:OrientChangeFactor} as
derived in the analogous formula in \cite{FLLevelOne}
(see \cite[Theorem 1.4]{FLLevelOne}).
\end{proof}

%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifx\undefined\bysame
\newcommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
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%end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

math0203047.extracted_bib
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