Morse theory and higher torsion invariants I
Morse theory and higher torsion invariants I
Morse theory and higher torsion invariants I
complex.tex
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\input morse.def

\document

\Kapitel\ComplexKapitel=1 Generalised Thom-Smale complexes

Let~$M\to B$ be a smooth bundle with compact fibres,
and let~$h\colon M\to\R$ be a Morse function on each fibre.
We show that one can patch together locally defined Thom-Smale
complexes for~$h$ to obtain a finite-dimensional vector bundle~$V\to B$
with a flat superconnection~$A'$,
and a cochain map~$I\colon\(\Omega^*(M;F),d^M\)\to\(\Omega^*(B;V),A'\)$
Together with the action of the Morse function on~$V$,
we obtain an example of a family Thom-Smale complex
as defined in \MVBDef.

We start by reviewing in \ComplexLocSubsection\
the construction of a Thom-Smale complex~$\(\Omega^*(B;V),A'\)$
associated to a family of fibre-wise Morse-Smale gradient fields,
cf.~\BGzwei, Chapter~5.
In \SimplifyingSubsection,
we state some simplifying assumptions on the local geometry of~$M$
and~$F$ near the set of fibre-wise critical points.
Dropping the Smale transversality condition,
we construct in \CWSubsection\ a particular CW-complex that
is adapted to a simplicial complex~$S$ on the base space~$B$.
The coboundary operator of the associated cochain complex
naturally leads to the notion of a flat ``simplicial''
superconnection~$\delta+a$,
see \CWBaseProp.
This is nothing but a ``system of higher homotopies''
as defined in chapter~3 of~\IBuch.
We also construct a quasi-isomorphism relating~$\nabla^F$
to~$\delta+a$.

In the next step,
we lift the simplicial superconnection to a neighbourhood of
the diagonal in~$B\times B$ in \GoodCoverSubsection.
In~\ComplexPatchSubsection,
we extend it to a flat ``mixed'' simplicial
and differential superconnection.
The construction in \SimplPatchProp\
is reminiscent of the spectral sequence proof
of de Rham's theorem that simiplicial cohomology and de Rham
cohomology are isomorphic.
Nevertheless, the situation here is more complicated
since flat superconnections obey a quadratic equation,
while the equation for cohomology is linear.
Note also that we have to construct an explicit quasi-isomorphism
from~$\Omega^*(M;F)$ to~$\Omega^*(B;F)$ with certain geometric properties
that will be crucial in \WittenKapitel;
this is done in \SimplPatchPropZwei.
Finally in \SmoothingSubsection,
we pull the purely differential part of the mixed superconnection back to~$B$
by a partition of unity,
to obtain a flat smooth superconnection on the Thom-Smale bundle~$V\to B$
in \PatchSuperconnectionDefinition.
The main properties of our construction are summarized in \PatchThm.


\Abschnitt\ComplexLocSubsection Local Thom-Smale complexes

Let~$p\colon M\to B$ be a fibre-bundle with compact fibre~$X$,
and let~$F\to M$ be a flat vector bundle,
giving rise to a de Rham complex~$\Omega^*(M;F)$.
Let~$h\colon M\to\R$ be a family of Morse functions on the fibres of~$p$.
and let~$C\subset M$ be the fibre-wise critical set of~$h$.
Then~$C$ forms a finite covering~$\widehat p=p|_C\colon C\to B$.
If~$C^j$ denotes the critical set of index~$j$,
then~$\widehat p^j=p|_{C^j}\colon C^j\to B$ is also a covering,
and~$\widehat p$ is the disjoint union of the coverings~$\widehat p_0$, $\widehat p_1$,
\dots

Let~$TX\to M$ denote the vertical tangent bundle,
and let~$T^sX\oplus  T^uX\to C$ be any decomposition of~$TX|_C$
such that the vertical second derivative~$d^2h\colon(TX)^2\to\R$ of~$h$
becomes positive (negative) definite on~$T^sX$ (on~$T^uX$),
and such that~$d^2h$ vanishes on~$T^sX\times T^uX$.
Then the orientation bundle~$o(T^uX)=\Lambda^{\max}T^uX$
is independent of the choice of such splitting.

\Definition\MorseBundleDef
Define vector bundles
$$\xalignat2
	\widehat V^j&=F|_{C^j}\otimes o(T^uX)\to C^j\;,&
	\widehat V&=\bigcup_{j\ge 0}\widehat V^j=F|_C\otimes o(T^uX)\to C\;,\\
	V^j&=\widehat p^j_*\widehat V^j\to B\;,&
	V&=\bigoplus_{j\ge 0}V^j=\widehat p_*\widehat V\to B\;,\\
		\Land
	\End^kV&=\bigoplus_{j\ge 0}\Hom\(V^j,V^{j+k}\)
	\subset\End V\;.
\endxalignat$$
Let~$\nabla^{\widehat V}$, $\nabla^V$ and~$\nabla^{\End V}$
be the flat connections on~$\widehat V$ and~$V$ respectively
that are induced by~$\nabla^F$.
\enddefinition

\Definition\NVHVDef
Let~$N^{\widehat V}\in\Gamma(\End\widehat V)$
denote multiplication by~$j$ on~$\widehat V^j$,
and let~$N^V\in\Gamma(\End^0V)$ be its pushdown to~$B$.
Multiplication by~$h|_C$ on~$\widehat V$ will be denoted by~$h^{\widehat V}$,
and its pushdown to~$B$ by~$h^V\in\Gamma\(\End^0V\)$.
Then both~$N^V$ and~$h^V$ extend to $\Omega^*(B)$-linear endomorphisms
of~$\Omega^*(B,V)$.
\enddefinition

An endomorphism~$e\in\Omega^*(B,\End V)$ will be called
{\em strictly $h^V$-increasing\/}
if~$e$ maps the $\lambda$-eigenspace of~$h^V$
to the sum of eigenspaces of eigenvalues strictly greater than~$\lambda$.
In other words,
such an operator maps subspaces~$\widehat V_c\subset V_b$ for~$c\in\widehat p^{-1}b$
to the sum of subspaces~$\widehat V_{c'}$ with~$c'\in\widehat p^{-1}b$
and~$h(c')>h(c)$.

\Definition\HIncDef
For~$b\in B$,
let~$\End_+V_b=\bigoplus_{k\in\Z}\End_+^kV_b$
be the space of strictly $h^V$-increasing endomorphisms of~$V_b$.
Let
$$\align
	\Gamma(\End_+V)
	&=\bigl\{\,e\in\Gamma(\End V)\bigm|e_b\in\End_+V_b
		\text{ for all }b\in B\,\bigr\}\;,\\
			\Land
	\Omega^*(B;\End_+V)
	&=\Omega^*(B)\otimes_{C^\infty(B)}\Gamma(\End_+V)\;.
\endalign$$
\enddefinition

Note however that the spaces~$\End_+V_b$ may not form vector bundles over~$B$
because~$h$ is not locally constant on~$C$.

\Definition\HorDegFiltDef
Let~$F^0(M;F)\supset F^1(M;F)\supset\dots$
and~$F^0(B;V)\supset F^1(B;V)\supset\dots$
be the filtrations of~$\Omega^*(M;F)$ and~$\Omega^*(B,V)$ respectively
by horizontal degree,
given by
$$\xalignat2
	&&F^p(M;F)
	&=\bigoplus_kF^{p,k}(M;F)\;,\\
		\text{with}&&
	F^{p,k}(M;F)
	&=\bigl\{\,\alpha\in\Omega^k(M;F)
		\bigm|\iota_{X_0}\dots\iota_{X_{k-p}}\alpha=0
			\text{ for all~$X_0$, \dots, }X_{k-p}
				\in\Gamma(TX)\,\bigr\}\\
		\text{and}&&
	F^p(B;V)
	&=\bigoplus_{q\ge p}\Omega^q(B;V)\;.
\endxalignat$$
\enddefinition

We can choose an open cover~$\Cal U=\{\,U_i\mid i\in\Cal I\,\}$,
such that we have vertical metrics~$g\TX_i$ on~$TX|_{p^{-1}(U_i)}$
over~$U_i$,
for which~$h$ becomes a fibre-wise Morse-Smale function.
Let~$M^u_i\to\widehat p^{-1}(U_i)\subset C$
with~$M^u_i|_c\subset M_{\widehat p(c)}$ denote
the bundle of unstable cells of the fibre-wise negative gradient flow
with respect to the metric~$g\TX_i$.

\Definition\IntMapDef
Set
$$\xalignat2
	\hat I_i&\colon\Omega^*(M;F)
	\to\Omega^*\(\widehat p^{-1}(U_i);\widehat V|_{\widehat p^{-1}(U_i)}\)\;,&
	\alpha&\mapsto\int_{M^u_i/C}\alpha\;,\\
		\Land
	I_i&\colon\Omega^*(M;F)
	\to\Omega^*\(U_i;V|_{U_i}\)\;,&
	\alpha&\mapsto\int_{M^u_i/B}\alpha\;.
\endxalignat$$
\enddefinition

Note that~$\hat I_i$ and~$I_i$ are $\Omega^*(B)$-linear
in the sense that
	$$\hat I_i(p^*\alpha\wedge\beta)=\widehat p^*\alpha\wedge\hat I_i(\beta)
		\Mand
	I_i(p^*\alpha\wedge\beta)=\alpha\wedge I_i(\beta)\;.$$
By~\BGzwei, Theorem~5.5,
these maps are~$C^\infty$ under the simplifying assumptions
of \SimplifyingSubsection.

Summarizing the results of~\BGzwei, Sections~5.4 and~5.5,
in particular Theorem~5.8,
we obtain the following characterisation of~$I_i$.

\Theorem\LocTSCThm.
There exists a unique coboundary operator~$\del_i\in\Gamma\(\End_+^1V|_{U_i}\)$
such that the superconnection~$A'_i=\del_i+\nabla^V$ on~$V$ is flat
and
	$$I_i\Bigl(\Omega^*\(p^{-1}(U_i),F|_{p^{-1}(U_i)}\),\nabla^F\Bigr)
	\to\bigl(\Omega^*(U_i,V|_{U_i}),A'_i\bigr)$$
is a quasi-isomorphism of complexes
respecting the filtrations by horizontal degree.
If we denote the spectral sequences associated to the filtered complexes above
by~$\(E_k^*(M;F),d_k\)$ and~$\(E_k^*(U_i;V),d_k\)$ respectively,
then~$I_i$ induces isomorphisms
	$$I_{i*}\colon\(E_k^*(M;F),d_k\)
	\buildrel\sim\,\over\longrightarrow\(E_k^*(U_i;V),d_k\)
	\Rtext{for~$k\ge 1$.}$$

We will be mainly interested in the $E_0$- and $E_1$-terms of these
spectral sequences.
Let~$T^HM\subset TM$ be a horizontal subbundle complementary to~$TX$,
and let~$\Omega\in\Omega^2\(B,p_*TX\)$ be the associated fibre bundle
curvature.
Then the operators~$d^M$ on~$\Omega^*(M)$ and~$\nabla^F$ on~$\Omega^*(M;F)$
split naturally as
	$$d^M=d^X+\nabla^{\Omega^*(X)}-\iota_\Omega
		\Mand
	\nabla^F=\nabla^{F|_X}+\nabla^{\Omega^*(X;F)}-\iota_\Omega$$
with respect to the splitting~$\Omega^*(M)=\Omega^*\(B;\Omega^*(X)\)$
and~$\Omega^*(M;F)=\Omega^*(B;\Omega^*X;F)$ defined by~$T^HM$.
The operators~$d^X$ and~$\nabla^{F|_X}$ induce
operators on the quotients
	$$F^p(M)/F^{p+1}(M)\cong\Omega^p\(B;\Omega^*(X)\)
		\Mand
	F^p(M;F)/F^{p+1}(M;F)\cong\Omega^p\(B;\Omega^*(X;F)\)$$
respectively,
which are independent of~$T^HM$.
We have
	$$\(E_0^p(M;F),d_0\)
	\cong\Bigl(\Omega^p\(B;\Omega^*(X;F)\),\nabla^{F|_X}\Bigr)\;,
		\Mand
	\(E_0^p(U_i;V),d_0\)
	\cong\(\Omega^p(U_i;V),\del_i\)\;.
	\Formel\LocTSCEnullFormel$$

\Definition\HBundleDef
Let~$H=H\dR^*(M/B;F)$ denote the bundle of the fibre-wise cohomology
with its natural flat Gau\ss-Manin connection~$\nabla^H$.
Similarly,
let~$H_i=H^*(V;\del_i)$,
and let~$\nabla^{H_i}$ be the connection induced by~$\nabla^V$.
\enddefinition

By flatness of~$A'_i$,
the operator~$\del_i$ is parallel.
If follows that~$\nabla^{H_i}$ is well-defined and flat.
As in \BGzwei,
we have
	$$\(E_1^*(M;F),d_1\)
	\cong\(\Omega^*(B;H),\nabla^H\)\;,
		\Mand
	\(E_1^*(U_i;V),d_1\)
	\cong\(\Omega^*(U_i;H_i),\nabla^{H_i}\)\;.$$
By \LocTSCThm,
the integration map gives an isomorphism of these $E_1$-terms.
In particular,
there is an isomorphism of flat vector bundles
	$$I_{i*}
	\colon\(H|_{U_i},\nabla^H\)
	\to\(H_i,\nabla^{H_i}\)\;.$$



\Abschnitt\SimplifyingSubsection Some simplifying assumptions

In this section,
we construct vertical metrics~$g\TX$,
horizontal subbundles~$T^HM$, and metrics~$g^F$ on~$F$,
satisfying certain properties
that we will use throughout this section and \WittenKapitel.

Let~$p\colon M\to B$ be a smooth bundle with compact fibres,
and let~$h\colon M\to\R$ be a smooth function that is Morse
on each fibre of~$p$.
Let~$\widehat p=p|_C\colon C\to B$ be the finite covering of~$B$
by the fibre-wise critical points of~$h$.
Let~$(F,\nabla^F)\to M$ be a flat vector bundle.

For a Euclidean vector bundle~$\pi\colon(E,g^E)\to M$,
define
	$$D_rE=\bigl\{\,e\in E\bigm|\abs e<r\,\bigr\}\;.$$
A Euclidean connection~$\nabla^E$ on~$E$ with curvature~$F^E$
naturally defines a horizontal subbundle~$T^HE$ of the tangent bundle~$TE$
to its  total space,
such that the fibre bundle curvature~$\Omega^E\colon\pi^*TM\to E$
takes the form
	$$\Omega^E(v,w)_e=-[\bar v,\bar w]^\perp_e=F^E_{v,w}e\;,
	\Formel\VectorBundleCurvFormel$$
where~$\bar v$, $\bar w\in\Gamma(T^HE)$ are horizonttal lifts
of vector fields~$v$, $w$ on~$M$,
and~$e\in E$ is a point on the total space.

If~$g\TX$ is a vertical metric on~$TX\to M$,
let~$T^sX\subset TX|_C$ ($T^uX\subset TX|_C$)
denote the sum of all positive (negative) eigenspaces
of the second derivative~$\nabla\nabla h\in\Gamma(\End TX|_C)$,
so the bundle~$\pi\colon TX|_C\to C$ splits as~$TX|_C=T^sX\oplus T^uX$.
Let~$U_r(C)$ denote the fibre-wise neighbourhood of~$C$ of radius~$r$,
and let~$\exp^X\colon TX\to M$ be the fibre-wise Riemannian exponential map.

\Proposition\SimplifyingAssumptions.
There exists a horizontal subbbundle~$T^HM\subset TM$,
a vertical metric~$g\TX$ on~$TX$,
a metric~$g^F$ on~$F$,
and a Euclidean connection~$\nabla^{TX|_C}$ respecting
the splitting~$TX|_C=T^sX\oplus T^uX$,
with the following property.
For each compact subset~$K\subset B$ there exists an~$\eps>0$,
such that
\roster
\item
over~$K$,
the fibre-wise exponential map~$\exp^X|_{D_{2\eps}TX|_{C}}$
induces an isometry onto its image~$U_{2\eps}(C)$,
and its differential maps~$T^H(TX)$ to~$T^HM|_{U_{2\eps}(C)}$;
\item
the Morse function~$h|_{U_{2\eps}(C)}$ is for~$c\in C|_K$
and~$(z^s,z^u)\in D_{2\eps}TX_c$
given by
	$$h\(\exp^X_c(z^s,z^u)\)
	=h(c)+{\abs{z^s}^2-\abs{z^u}^2\over2}\;;$$
\item
over~$K$,
the flat bundle~$\(F,\nabla^F,g^F\)|_{U_{2\eps}(C)}$
is isometric to~$\pi^*\(F|_C,\nabla^F|_C,g^F|_C\)$.
\endroster

\Proof
This follows easily because the classical Morse lemma holds for families.
\qed\enddemo

If conditions~(1) -- (3) above hold for~$T^HM$, $g\TX$ and~$g^F$,
we will say that~$T^HM$, $g\TX$ and~$g^F$ satify the simplifying assumptions
of \SimplifyingSubsection.
We assume throughout this section that this is the case.
For simplicity,
we may assume (at least for the purpose of estimates)
that~$B$ is compact,
so that we may fix an~$\eps>0$ as in \SimplifyingAssumptions\
for all of~$B$.

In particular,
the closures of the $2\eps$-tubular neighbourhoods of different leaves
of the critical covering~$\widehat p\colon C\to B$ do not intersect,
and~$h$ differs by at most~$2\eps^2$ from~$h|_C$ along the fibres
of these tubular neighbourhoods.
For each flow line~$\gamma$ of the negative gradient flow,
this implies that~$h(c_-)-h(c_+)>4\eps^2$,
where~$c_\pm=\lim_{t\to\pm\infty}\gamma(t)\in C$.




\Abschnitt\CWSubsection A CW complex for a fibre-wise Morse function

In \ComplexLocSubsection,
we recalled from~\BGzwei\ how to associate a family of CW complexes
to a fibre-wise Morse function~$h$
in the special case that there exists a fibre-wise Morse-Smale
gradient field for~$h$.
In this subsection,
we similarly construct a CW complexes on~$M$ over a simplicial complex~$S$
on~$B$ for general fibre-wise Morse functions.
With respect to these complexes,
the map~$p\colon M\to B$ will be cellular,
and the preimage of any simplex~$\abs\sigma$ in~$M$
will consist of one cell~$M_\alpha(\sigma)$ of dimension~$k+\ind h$
for each leaf~$C_\alpha(\sigma)$
of the critical covering~$\widehat p\colon C\to B$.

Let the {\em standard $k$-simplex\/} be a fixed closed affine
$k$-simplex~$\Delta_k\subset\R^k$,
whose corners will be enumerated~$\Delta_k(0)$, \dots, $\Delta_k(k)$.
Then for each finite sequence of integers~$0\le i_0\le\dots\le i_l\le k$
there is a unique affine map~$f_{i_0\dots i_l}\colon\Delta_l\to\Delta_k$
mapping~$\Delta_l(j)$ to~$\Delta_k(i_j)$.
We call~$f_{i_0\dots i_l}$ a {\em face\/} of~$\Delta_k$
if~$i_0<\dots<i_l$,
and a {\em degeneracy\/} otherwise.

Let~$x_0$, \dots, $x_k$ are barycentric coordinates on~$\Delta_k$,
i.e., $x_j$ is an affine function on~$\Delta_k\subset\R^k$
that takes the value~$\delta_{ij}$ at~$\Delta_k(i)$,
and~$X_0+\dots+x_k=1$.
Let~$w_{ij}$ denote the constant vector field on~$\R^k$
with~$w_{ij}=\Delta_k(j)-\Delta_k(i)\in\R^k$,
so
	$$w_{ij}(x_l)
	=\cases
		-1 &\text{for~$l=i$,}\\
		1  &\text{for~$l=j$, and}\\
		0  &\text{otherwise.}
	\endcases\Formel\WkDef$$
We recall that~$\Delta_k$ is oriented
by fixing the oriented base~$w_{0,1}$, \dots, $w_{0,k}$,
and that~$\partial\Delta_k$ is oriented
such that prefixing an oriented base of~$\partial\Delta_k$
with an outward normal vector gives an oriented base of~$\Delta_k$.
Then we have
	$$\partial\Delta_k
	=\sum_{j=0}^k(-1)^jf_{0\dots\smash{\widehat j}\dots k}\Delta_{k-1}\;.$$

\Definition\SimplexDef
A {\em smoothly embedded $k$-simplex\/} in~$B$
is a map~$\sigma\colon\Delta_k\to B$
that extends to a smooth embedding of some small neighbourhood of~$\Delta_k$
in~$\R^k$ into~$B$.
Let~$\abs\sigma$ be the image of~$\sigma$,
and write~$\sigma_j=\sigma(\Delta_k(j))\in B$ for~$0\le j\le k$.

A {\em smooth ordered simplicial complex\/}~$S$ on~$B$
consists of sets~$S_k$ of smooth $k$-simplices in~$B$ for each~$k\ge 0$,
%together with a total ordering on~$S_0$,
and of {\em face maps\/}~$F_{i_0,\dots,i_l}\colon S_k\to S_l$
for each finite sequence of integers~$0\le i_0<\dots<i_l\le k$,
such that
	$$F_{i_0,\dots,i_l}(\sigma)
	=\sigma\circ f_{i_0,\dots,i_l}
	\colon\Delta_l\to B\;,$$
and~$B$ is the disjoint union of the interiors of the images
of all simplices in~$S$.
%and the vertices of each $k$-simplex~$\sigma$ are ordered,
%i.e., we have~$\sigma_0<\dots<\sigma_k$ in~$S_0$.
We say that~$\sigma'\in S_l$ is a {\em face\/} of~$\sigma\in S_k$,
if~$\sigma'=F_{i_0,\dots,i_l}(\sigma)$ for some face map~$F_{i_0,\dots,i_l}$.
A {\em simplicial subset\/} of~$B$ is a closed subset
that is a union of images of simplices of~$S$.
\enddefinition

We will now fix a smooth ordered simplicial complex~$S$ on~$B$.
Let us assume that~$S$ is so fine that each simplex in~$S$
is uniquely determined by the set of its vertices.
We will write shortly~$\sigma_{i_0,\dots,i_l}$
for the face~$F_{i_0,\dots,i_l}(\sigma)$,
thereby identifying vertices~$\sigma_j$ with their images in~$B$.
Note that we will not need degerated simplices or degeneracy maps
for most of the following constructions.

\Remark\SimplTransRem
By the definition of a smoothly embedded $k$-simplex~$\sigma$ in~$B$,
we can pull the bundle~$M\to B$ back to a neighbourhood~$U(\Delta_k)$
of~$\Delta_k$ in~$\R^k$.
In particular,
we can say a vector field on~$p^{-1}(\abs\sigma)\subset M$ is smooth
if its pullback extends smoothly to the pullback of~$M$ to~$U(\Delta_k)$.
If two smooth submanifolds of~$p^{-1}(\abs\sigma)$ intersect,
we can say that they are transversal if their pullbacks
extend to smooth submanifolds over~$U(\Delta_k)$ that intersect transversally.
This makes sense even if the original intersection
took place in the boundary of~$p^{-1}(\abs\sigma)$.
\endremark

For each $k$-simplex~$\sigma\in S_k$,
let~$P(\sigma)=C|_{\abs\sigma}/\abs\sigma$
be the set of leaves of the covering~$\widehat p\colon C\to B$
restricted to~$\abs\sigma$.
For~$\alpha\in P(\sigma)$,
let~$C_\alpha(\sigma)$ be the corresponding leaf of~$C|_{\abs\sigma}$.
Let~$h_\alpha$ be restriction of~$h$ to~$C_\alpha(\sigma)$,
regarded as a function on~$\abs\sigma$,
and let~$\ind_\alpha$ denote the fibre-wise Morse index of~$h$
at~$C_\alpha(\sigma)$.
We choose the simplical structure on~$B$ fine enough such that
	$$\abs{h_\alpha(q')-h_\alpha(q)}
	<{\eps^2\over2}
	\Rtext{for all~$\alpha\in P(\sigma)$ and all~$q$, $q'\in\abs\sigma$.}
	\Formel\CWFineAss$$
Let us define a relation~$\prec_\sigma$ on~$P(\sigma)$ by setting
	$$\alpha\prec_\sigma\beta
		\Mtext{iff}
	\(h_\beta-h_\alpha\)(q)>2\eps^2
		\Rtext{for some~$q\in\abs\sigma$.}
	\Formel\CWPrecDef$$

\Remark\CWPrecRemark
\roster
\item
Assumption~\CWFineAss\ guarantees
that~$\prec_\sigma$ is a partial ordering on~$P(\sigma)$.
\item
If a $\sigma'$ is a face of~$\sigma$,
then~$\prec_\sigma$ refines on~$\prec_{\sigma'}$.
\item
Because of the simplifying assumptions of \SimplifyingSubsection,
the unstable cell~$M^u(C_\beta(\sigma))$
can only intersect the closed $2\eps$-tubular neighbourhood
of~$C_\alpha(\sigma)$
if~$\alpha\prec_\sigma\beta$.
In particular,
there is no negative gradient flow line with respect to~$g\TX_0$
from~$C_\beta(\sigma)$ to~$C_\alpha(\sigma)$ over some~$q\in\abs\sigma$
unless~$\alpha\prec_\sigma\beta$.
\endroster
\endremark

Using barycentric coordinates and the vector fields~$w_{ij}$ of~\WkDef,
we construct compatible vector fields on all simplices of~$S$.

\Proposition\WkDynProp.
The vector fields
	$$W_k=\sum_{0\le i<j\le k}x_i\,x_j\,w_{ij}$$
on the standard simplices~$\Delta_k$ have the following properties:
\roster
\item
whenever~$0\le i_0\le\dots\le i_l\le k$,
we have
	$$W_k\circ f_{i_0,\dots,i_k}
	=df_{i_0,\dots,i_k}\circ W_l\;,$$
in particular,
$W_k$ is tangential to all faces of~$\Delta_k$;
\item
$W_k$ vanishes only at the vertices~$\Delta_k(j)$ of~$\Delta_k$,
where it has non-degenerate zeros,
and the stable and unstable tangential subspaces
are spanned by~$w_{0,j}$, \dots, $w_{j-1,j}$
and by~$w_{j,j+1}$, \dots, $w_{j,k}$ respectively;
and
\item
the stable and unstable cells of~$W_k$ at~$\Delta_k(j)$
are given by
	$$M^s\(\Delta_k(j)\)
	=\abs{f_{0,\dots,j}}\setminus\abs{f_{0,\dots,j-1}}
		\Mand
	M^u\(\Delta_k(j)\)
	=\abs{f_{j,\dots,k}}\setminus\abs{f_{j+1,\dots,k}}\;.$$
\endroster

By~\Smale,
the vector field~$W_k$ can be obtained as a gradient field
of a Morse function on~$\Delta_k$ with respect to some Riemannian metric.

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\centerline{\NeueFigur\WkFig
\WkFig. The vector fields~$W_k$ on small standard simplices.}
\endinsert

\Proof
Because~$w_{ij}$ is tangent to all faces of~$\Delta_k$ containing~$i$ and~$j$,
and because~$x_ix_j$ vanishes on all other faces of~$\Delta_k$,
the vector field~$W_k$ is tangential to all faces of~$\Delta$.
Now~(1) follows by an elementary calculation.
Also, it is clear that~$W_k$ has non-degenerate zeros
at the vertices of~$\Delta_k$ of the type described in~(2).

To prove that~$W_k$ has no other zeros
and to determine the stable and unstable cells,
regard the affine function~$h_k=\sum_{j=0}^kj\,x_j$.
We have
	$$W_k(h_k)
	=\sum_{0\le i<j\le k}(j-i)\,x_i\,x_j\ge 0\;,
	\Formel\WkDiffFormel$$
with~``$=$'' only at the vertices of~$\Delta_k$,
which completes the proof of~(2).

By~\WkDiffFormel,
each trajectory~$\gamma(t)$ of~$W_k$ converges for~$t\to\pm\infty$
to a zero of~$W_k$.
By~(1), each face of~$\Delta_k$ is invariant under the flow of~$W_k$.
Since the dynamics of~$W_k$ near the vertices are given by~(2),
(3) now follows easily.
\qed\enddemo

Property~(1) guarantees the existence
of a piece-wise smooth vector field~$W_B$ on~$B$ such that
	$$W_B\circ\sigma=D\sigma\circ W_k$$
for all~$k$ and all~$\sigma\in S_k$.
Note that~$W_B$ looks like the negative gradient vector field
of a Morse function when restricted to the image of a single simplex.

Let~$\overline W_B$ be the horizontal lift of~$W_B$ to~$M$,
so~$\overline W_B$ is now a piece-wise smooth horizontal vector field on~$M$.
We fix a smooth cut-off function~$\phy_M\colon M\to[0,1]$
with support in~$U_{\eps}(C)$ and~$\phy_M|_{U_{\eps/2}(C)}=1$,
and we define vector field
	$$W=-\nabla\TX h+\phy_M\,\overline W_B\;.
	\Formel\WConstruction$$
Then~$W$ is smooth on~$p^{-1}(\abs\sigma)$ in the sense of \SimplTransRem\
for each simplex~$\sigma\in S$.
We denote by~$M^s_W(x,\sigma)$, $M^u_W(x,\sigma)$
the stable and unstable cells of the flow of~$W$
restricted to~$p^{-1}(\abs\sigma)$ at a critical point~$x$ of~$W$,
while~$M^s$, $M^u$ still denotes the fibre-wise stable and unstable cells
of~$-\nabla\TX h$.
For~$c\in\R$,
we define the sets
$$\align
	h^{-1}(h_\alpha+c)
	&=\bigl\{\,x\in p^{-1}(\abs\sigma)
		\bigm|h(x)=h_\alpha\(p(x)\)+c\,\bigr\}\;,\\
	h^{-1}(h_\alpha+c,\infty)
	&=\bigl\{\,x\in p^{-1}(\abs\sigma)
		\bigm|h(x)>h_\alpha\(p(x)\)+c\,\bigr\}\;,\\
	\Land
	h^{-1}(-\infty,h_\alpha+c)
	&=\bigl\{\,x\in p^{-1}(\abs\sigma)
		\bigm|h(x)<h_\alpha\(p(x)\)+c\,\bigr\}
	\subset p^{-1}(\abs\sigma)\;.
\endalign$$

\Proposition\WDynProp.
The zeros of~$W$ are precisely the points~$\widehat p^{-1}(\abs\sigma)\in C$
for~$\sigma\in S_0$.
For each~$k$ and each $k$-simplex~$\sigma\in S_k$,
the flow of~$W$ preserves both~$p^{-1}(\abs\sigma)\subset M$
and~$C(\abs\sigma)$.
If~$W$ vanishes at~$x\in p^{-1}(\abs\sigma)$,
then there is a coordinate chart~$\psi\colon U\to V$ of~$M$ around~$x$
that maps~$p^{-1}(\abs\sigma)\cap U$ diffeomorphically
onto~$\([0,\infty)^k\times\R^n\)\cap V$,
such that~$W$ is in this coordinates given as
	$$(D\psi\circ W)|_{p^{-1}(\abs\sigma)\cap U}
	=\biggl(\pm \psi^1\,{\del\over\del\psi^1}
			\pm\psi^2\,{\del\over\del\psi^2}\biggr)
		\circ W|_{p^{-1}(\abs\sigma)\cap U}\;.\tag1$$
\parr
For~$\alpha\in P(\sigma)$,
let~$x\in C_\alpha(\abs\sigma)$ project to the $j$-th vertex~$\sigma_{j}$
of~$\abs\sigma$.
Then we have
$$\split
	M^s_W(x,\sigma)
		\cap h^{-1}\biggl(-\infty,h_\alpha+{3\over2}\,\eps^2\biggr)
	&=D_{\sqrt3\,\eps}T^sX|_{C_\alpha(\abs{\sigma_{0,\dots,j}}
			\setminus\abs{\sigma_{0,\dots,j-1}})}\;,\\
	\Land
	M^u_W(x,\sigma)
		\cap h^{-1}\biggl(h_\alpha-{3\over2}\,\eps^2,\infty\biggr)
	&=D_{\sqrt3\,\eps}T^uX|_{C_\alpha(\abs{\sigma_{j,\dots,k}}
			\setminus\abs{\sigma_{j+1,\dots,k}})}\;.
\endsplit\tag2$$
Moreover,
if~$y$ is another critical point of~$W$
and~$M^u_W(x,\sigma)\cap M^s_W(y,\sigma)\ne\emptyset$,
then either~$\beta=\alpha$ or~$\beta\prec_\sigma\alpha$.

\Proof
The first two claims of the proposition follow directly from our construction.
Let~$x$ be a zero of~$W$ with~$p(x)=\sigma_j$.
We can choose a coodinate chart of~$B$
that restricts to the barycentric coordinates of~$\abs\sigma$,
except the $j$th.
Composing with~$\hat p$,
we obtain a chart~$\psi_0$ of~$C$ around~$x$.
Finally,
we obtain the desired chart by trivialising~$T^sX$ and~$T^uX\to C$
by parallel translation with respect to the connection~$\nabla^{TX|_C}$
of \SimplifyingAssumptions,
along paths starting from~$x$ that are radial in the chart~$\psi_0$.

Let~$\sigma\in S_k$ be a $k$-simplex,
and let~$h_k$ be the function constructed in the proof of \WkDynProp,
which we now regard as a function on~$\abs\sigma\subset B$.
Then for some sufficiently large~$C>0$,
we clearly have
	$$W(h-C\,h_k)\le 0\;,$$
with~``$=$'' only at the zeros of~$W$.
This shows that each non-constant flow line~$\gamma(t)$
converges to zeros of~$W$ for~$t\to\pm\infty$.

We will restrict our attention to the unstable cells of~$W$ and~$-\nabla\TX h$,
the stable case being completely analogous.
By our simplifying assumptions
and because~$\phy_M\,\overline W_B$ is horizontal,
$W$ is tangential to the subset~$D_{2\eps}T^uX$
of the $2\eps$-tubular neighbourhood~$U_{2\eps}\(C_\alpha(\abs\sigma)\)$,
and restricted to~$U_{2\eps}\(C_\alpha(\abs\sigma)\)$,
the unstable cells of~$x$ is
	$$D_{2\eps}T^uX|_{C_\alpha(\abs{\sigma_{0,\dots,j}}
			\setminus\abs{\sigma_{j+1,\dots,k}})}\;.$$

\midinsert
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\centerline{\NeueFigur\WDynFig
\WDynFig. The vector field~$W$ near the fibre-wise critical set of~$h$.}
\endinsert


Now regard a typical flow-line~$\gamma(t)$ of~$W$
with~$\lim_{t\to-\infty}\gamma(t)=x$.
Unless~$\gamma(t)\in C_\alpha(\abs\sigma)$ for one~$t\in\R$
(and thus for all~$t$),
the flow-line~$\gamma$ will leave~$U_{2\eps}\(C_\alpha(\abs\sigma)\)$
at a certain times~$t_0$
with
	$$h\(\gamma(t_0)\)
	<h_\alpha\(p(\gamma(t_0))\)-2\eps^2\;.$$
Because~$\supp(\phy_M)\subset U_\eps(C)$,
the trajectory~$\gamma$ will now follow the fibre-wise flow of~$-\nabla\TX h$
until it enters~$U_\eps\(C_\beta(\abs\sigma)\)$ for some~$\beta$
Let~$t_1>t_0$ be the time where~$\gamma(t)$
enters the larger neighbourhood~$U_{2\eps}\(C_\beta(\abs\sigma)\)$,
and let~$\gamma(t_1)=(z^s,z^u)\in\del D_{2\eps}TX|_{C_\beta(\abs\sigma)}$.
Since~$\gamma(t)$ hits~$U_\eps\(C_\beta(\abs\sigma)\)$,
clearly
	$$\abs{z^u}<\eps
		\Mand
	\abs{z^s}
	=\(4\eps^2-\abs{z^u}^2\)^{\textstyle{1\over2}}
	>\sqrt 3\,\eps\;,$$
so by \SimplifyingAssumptions~(2),
	$$h_\alpha\(p(\gamma(t_0))\)-\eps^2>h\(\gamma(t_0)\)
	>h\(\gamma(t_1)\)
	>h_\beta\(p(\gamma(t_1))\)+\eps^2\;,$$
where of course~$p(\gamma(t_0))=p(\gamma(t_1))\in\abs\sigma$.
In particular, we have~$\beta\prec_\sigma\alpha$ by \CWPrecDef.

Repeating this argument starting from the time where~$\gamma(t)$
leaves~$U_{2\eps}\(C_\beta(\abs\sigma)\)$,
we check that~$\gamma(t)$ passes the $\eps$-tubular neighbourhoods
of a finite number of critical levels
that form a decreasing sequence
with respect to the partial ordering~$\prec_\sigma$.
The vertical movement inside the larger $2\eps$-tubular
neighbourhood of any of these fixpoints
will decrement~$h\(\gamma(t)\)$ by at least~$2\eps^2$,
while the horizontal movement introduced by~$\phy_M\,W_B$
can increment~$h\(\gamma(t)\)$ by at most~${\eps^2/2}$ by~\CWFineAss.
This shows in particular
that~$h\(\gamma(t)\)<h_\alpha\(p(\gamma(t))\)$ for~$t>t_0$,
and as claimed,
	$$%\multline
	M^u_W(x,\sigma)
		\cap h^{-1}\biggl(h_\alpha-{3\over2}\,\eps^2,\infty\biggr)
	=D_{\sqrt3\,\eps}T^uX|_{C_\alpha(\abs{\sigma_{0,\dots,j}}
			\setminus\abs{\sigma_{j+1,\dots,k}})}\;.
		\quad\qed%\\
%	=M^u\(C_\alpha(\abs{\sigma_{j,\dots,k}}
%			\setminus\abs{\sigma_{j+1,\dots,k}})\)
%		\cap h^{-1}(h_\alpha-\eps^2,\infty)\;.
%	\quad\qed
%\endmultline
$$
\enddemo

In order to describe the closures of the stable and unstable cells
geometrically,
we need the notion of submanifolds with conical singularities.
We recall the recursive definition in~\Laudenbach.

\Definition\SMCSDef
A stratified subset~$\Sigma=(\Sigma_0,\dots,\Sigma_k)$
is a {\em submanifold with conical singularities\/} of dimension~$k$,
if
\roster
\item
for any~$0\le i\le k$, the set~$\Sigma_i\setminus\Sigma_{i+1}$
is a smooth submanifold of~$M$,
\item
around each point~$x\in\Sigma_i\setminus\Sigma_{i+1}$,
there is a $C^\infty$-coordinate chart~$U_x\to\R^m$ of~$M$
that maps~$U_x\cap(\Sigma_0,\dots,\Sigma_i)$ to
	$$\R^{k-i}\times\(T_0,\dots,T_i\)\;,$$
where~$T=\(T_0,\dots,T_i,\)\subset\R^{m-k+i}$
is a submanifold with conical singularities, and
\item
around each~$x\in\Sigma_k$,
there is a $C^1$-coordinate chart~$U'_x\to\R^m$ of~$M$
that maps~$\Sigma\cap U'_x$ to the cone of a $(k-1)$-dimensional submanifold
of~$S^{m-1}$ with conical singularities.
\endroster
Two submanifolds~$\Sigma$ and~$T\subset M$
with conical singularities are said to intersect {\em transversally\/}
iff~$\Sigma_i\setminus\Sigma_{i+1}$ intersects~$T_j\setminus T_{j+1}$
transversally for all~$i$, $j$.
\enddefinition

\Remark\SMCSTransRem
Let~$\Sigma=(\Sigma_0,\dots,\Sigma_k)$
be a submanifold with conical singularities of~$M$,
and let~$T\subset M$ be a submanifold.
Suppose that~$T$ intersects a stratum~$\Sigma_j\setminus\Sigma_{j+1}$
transversally.
Then there is a neighbourhood~$U$ of~$T\cap(\Sigma_j\setminus\Sigma_{j+1})$
in~$M$
such that~$T\cap U$ intersects~$\Sigma_i\cap U$ transversally in~$U$
for all~$i\le j$.
\endremark

Note that~$\abs\sigma\subset B$ and~$p^{-1}(\abs\sigma)\subset M$
are submanifolds with conical singularities.
We now want to take the unstable cells of~$W$ to form a CW-complex on~$M$
whose cells are submanifolds with conical singularities.
By the following generalisation of a result of Laudenbach,
it suffices to deform~$W$ in such a way that a suitable version
of the Smale transversality condition along the lines of \SimplTransRem\ holds.
For a fixed simplex~$\sigma\in S$,
let~$h'\colon\abs\sigma\to\R$ be a smooth function
such that~$h'(q)$ is a regular value of~$h|_{p^{-1}(q)}$
for all~$q\in\abs\sigma$.


\Proposition\SimplLaudenbachProp.
Let~$\sigma\in S$ be a fixed simplex.
If the stable cells~$M^u_W(x,\sigma)$
intersect the unstable cells~$M^s_W(y,\sigma)$ of~$W$
transversally in~$p^{-1}(\abs\sigma)\subset M$
in the sense of \SimplTransRem,
then~$\overline{M^u_W(x,\sigma)}$ and~$\overline{M^s_W(y,\sigma)}$
are submanifolds of~$M$ with conical singularities.
\parr
If for a fixed zero~$x$ of~$W$ over~$\abs\sigma$,
all~$M^u_W(x',\sigma)$
with~$x'\in\overline{M^u_W(x,\sigma)}\cap h^{-1}(h',\infty)$
meet all~$M^s_W(y,\sigma)$ with~$y\in h^{-1}(h',\infty)$
transversally in the sense of \SimplTransRem,
then~$\overline M^u_W(x,\sigma)\cap h^{-1}(h')$
is a submanifold of~$h^{-1}(h')$ with conical singularities.
The strata of~$\overline M^u_W(x,\sigma)\cap h^{-1}(h')$
are of the form~$M^u_W(x',\sigma')\cap h^{-1}(h')$,
where~$\sigma'$ is a face of~$\sigma$
and~$x'\in\overline{M^u_W(x,\sigma)}\cap p^{-1}(\abs{\sigma'})
\cap h^{-1}(h',\infty)$.



\Proof
First, assume that for one of the unstable cells~$M^u_W(x,\sigma)$,
all~$M^u_W(x',\sigma)$ with~$x'\in\overline{M^u_W(x,\sigma)}$
meet all~$M^s_W(y,\sigma)$ transversally in the sense of \SimplTransRem,
for a fixed~$\sigma\in S_k$ and a zero~$x$ of~$W$ over~$p^{-1}(\abs\sigma)$.
In particular,
we can pull~$M$ and~$W$ back to~$\Delta_k$ by~$\sigma$,
and we can extend the pullbacks~$\bar p\colon\overline M\to\Delta_k$
and~$\overline W$
smoothly over a neighbourhood~$U$ of~$\Delta_k$ in~$\R^k$,
preserving all transversal intersections.
Because~$\overline W$ takes the form of \WDynProp~(1)
near zeros of~$\overline W$,
Laudenbach's arguments in~\Laudenbach\ imply
that the closure~$\overline{\overline M^u_{\overline W}(\bar x,U)}$
of the unstable cell at the pullback~$\bar x$ of~$x$
is a submanifold with conical singularities
whose strata are unions of other unstable cells. 

Let~$H$ be an affine hyperplane
that contains a codimension~$1$ face of~$\Delta_k$.
Me may assume that~$\bar p^{-1}(H)$ is preserved by the flow.
Assume first that~$\bar p(\bar x)\notin H$,
then~$\overline{\overline M^u_{\overline W}(\bar x,U)}$
may hit~$\bar p^{-1}(H)$,
but it cannot extend across~$\bar p^{-1}(H)$.
On the other hand,
assume that~$\bar p(\bar x)\in H$.
Then it is easy to see
that either~$\overline M^u_{\overline W}(\bar x,U)$
is contained in~$\bar p^{-1}(H)$,
or~$\overline M^u_{\overline W}(\bar x,U)$
intersects~$\bar p^{-1}(H)$ transversally.
Similar statement hold for all
zeros~$\bar x'\in\overline{\overline M^u_{\overline W}(\bar x,U)}$.
This implies that~$\overline{\overline M^u_{\overline W}(\bar x,U)}
\cap\bar p^{-1}(\Delta_k)$
is again a submanifold with conical singularities,
which has some extra strata given as intersections of unstable cells
with $\bar p$-preimages of faces of~$\Delta_k$.

Going back to~$M$,
we know now that the strata of~$\overline{M^u_W(x,\sigma)}$
are unions of sets of the form
	$$M^u_W(x',\sigma)\cap p^{-1}(\abs{\sigma'})
	=M^u_W(x',\sigma')\;,
	\Formel\MuWStrataFormel$$
where~$\sigma'$ is a face of~$\sigma$,
and~$x'\in\overline{M^u_W(x,\sigma)}\cap p^{-1}(\abs{\sigma'})$
is a zero of~$W$.
A similar statement holds if we reverse the roles of the stable
and unstable cells.

%For an intermediate step in the proof of \SimplSmaleProp\ below,
Finally assume that~$h'\colon\abs\sigma\to\R$ is a smooth function
such that~$h'(q)$ is a regular value of~$h|_{p^{-1}(q)}$
for all~$q\in\abs\sigma$.
We also assume that for one of the unstable cells~$M^u_W(x,\sigma)$,
all~$M^u_W(x',\sigma)$
with~$x'\in\overline{M^u_W(x,\sigma)}\cap h^{-1}(h',\infty)$
meet all~$M^s_W(y,\sigma)$ with~$y\in h^{-1}(h',\infty)$
transversally in the sense of \SimplTransRem.
Then the same argument as above implies
that~$\overline M^u_W(x,\sigma)\cap h^{-1}(h')$
is a submanifold of~$h^{-1}(h')$ with conical singularities,
whose strata are of the form~$M^u_W(x',\sigma')\cap h^{-1}(h')$,
for~$\sigma'$ a face of~$\sigma$
and~$x'\in\overline{M^u_W(x,\sigma)}\cap p^{-1}(\abs{\sigma'})
\cap h^{-1}(h',\infty)$.
\qed\enddemo


We use \SimplLaudenbachProp\ inductively
to vertically deform~$W$
such that Smale's transversality condition holds in~$p^{-1}(\abs\sigma)$
in the sense of \SimplTransRem.


\Proposition\SimplSmaleProp.
After a small vertical perturbation of~$W$
outside~$U_{2\eps}(C)$
that is smooth on each~$p^{-1}(\abs\sigma)$,
the closures of the stable and unstable
cells~$\overline{M^u_W(x,\sigma)}$, $\overline{M^s_W(y,\sigma)}$
are submanifolds of~$M$ with conical singularities.
The strata of~$\overline{M^u_W(x,\sigma)}$
are unions of sets~$M^u_W(x',\sigma')$,
where~$\sigma'$ is a face of~$\sigma$,
and~$x'$ is a zero of~$W$.
If~$W$ already satisfies the transversality condition of \SimplTransRem\
over each closed simplex of a simplicial subset~$B_0\subset B$,
then~$W$ can be deformed in such a way
that~$W|_{\pi^{-1}(B_0)}$ is not affected.

\Proof
By \SimplLaudenbachProp,
it suffices to establish transversality in the sense above.
We proceed by induction of the degree~$k$ of the simplices of~$S$,
modifying~$W$ only over those $k$-simplices
where the the stable and unstable cells do not yet meet transversally.
Then the relativity statement above is automatically satisfied.
For~$k=0$,
we can achieve transversality by the classical argument of Smale (\Smale).

Thus assume that~$\sigma\in S_k$ with~$k\ge 1$,
and that transversality in the sense of \SimplTransRem\
already holds over each face of~$\partial\abs\sigma$.
Let~$x$, $y$
be two zeros of~$W$ on~$p^{-1}\(\abs\sigma\)$
with~$\{x\}=C_\alpha\(\sigma_{i}\)$ and~$\{y\}=C_\beta\(\sigma_{j}\)$.
If~$M^u_W(x,\sigma)\cap M^s_W(y,\sigma)\ne\emptyset$,
then~$0\le i\le j\le k$,
and
	$$p\(M^u_W(x,\sigma)\cap M^s_W(y,\sigma)\)
	\;\subset\;\abs{\sigma_{i,\dots,j}}
		\setminus\(\abs{\sigma_{i+1,\dots,j}}
			\cup\abs{\sigma_{i,\dots,j-1}}\)\;.$$

Unless~$i=0$ and~$j=k$,
the simplex~$\sigma_{i,\dots,j}$ is a proper face of~$\sigma$,
and we know by induction
that the intersection of~$M^u_W(x,\sigma)$ and~$M^s_W(y,\sigma)$ is transversal
over~$\abs{\sigma_{i,\dots,j}}$.
Let~$\gamma(t)\in M^u_W(x,\sigma)\cap M^s_W(y,\sigma)$
for some flow-line~$\gamma(t)$ of~$W$,
then~$\lim_{t\to-\infty}=x$ and~$\lim_{t\to\infty}=y$.
By definition of~$W_k$ and~$W$ and using \WkDynProp,
it is easy to check that
$$\split
	T_{\gamma(t)}M^u_W(x,\sigma)
	&\equiv T_{\gamma(t)}\(p^{-1}\abs{\sigma_{i,\dots,k}}\)
		\Rtext{mod }T_{\gamma(t)}\(p^{-1}\abs{\sigma_{i,\dots,j}}\)\\
		\Mand
	T_{\gamma(t)}M^s_W(y,\sigma)
	&\equiv T_{\gamma(t)}\(p^{-1}\abs{\sigma_{0,\dots,j}}\)
		\Rtext{mod }T_{\gamma(t)}\(p^{-1}\abs{\sigma_{i,\dots,j}}\)
\endsplit\Formel\FirstWTransFormel$$
for all~$t\in\R$.
Together with transversality over~$\abs{\sigma_{i,\dots,j}}$,
i.e.,
	$$T_{\gamma(t)}\(p^{-1}\abs{\sigma_{i,\dots,j}}\)
	\subset T_{\gamma(t)}M^u_W(x,\sigma)+T_{\gamma(t)}M^s_W(y,\sigma)\;,$$
we obtain transversality in~$p^{-1}(\abs\sigma)$.

Now assume that~$x\in C_\alpha(\abs\sigma)$ and~$y\in C_\beta(\abs\sigma)$
with~$p(x)=\sigma_0$ and~$p(y)=\sigma_k$.
By \WDynProp,
we must have either~$\beta=\alpha$ or~$\beta\prec_\sigma\alpha$.
In the first case,
the intersection takes place along
	$$C_\alpha\(\abs\sigma\setminus(\abs{\sigma_{1,\dots,k}}
		\cup\abs{\sigma_{0,\dots,k-1}})\)
	\Formel\SecondWTransFormel$$
and is clearly transversal by \WDynProp.

We are left with the case~$\{x\}=C_\alpha(\sigma_0)$,
$\{y\}=C_\beta(\sigma_k)$ with~$\beta\prec_\sigma\alpha$.
We start another induction over~$\alpha$,
where we complete~$\prec_\sigma$ to a total ordering~$<_\sigma$.
Fix~$\alpha$ and assume that~$M^u_W(x',\sigma)$
and~$M^s_W(y,\sigma)$ intersect transversally
for all zeros~$x$, $y$ of~$W$ with~$x\in C_{\alpha'}(\abs\sigma)$
for some~$\alpha'<_\sigma\alpha$,
and with~$y\in C(\abs\sigma)$.
This is in particular true for the smallest~$\alpha$.

Let~$S_{2\eps}T^uX=\del D_{2\eps}T^uX$
be the sphere bundle in~$T^uX$ of radius~$2\eps$.
We may fix a small open neighbourhood~$U_\alpha$
of~$S_{2\eps}T^uX|_{C_\alpha(\abs\sigma)}$ in the hypersurface
	$$h^{-1}\(h_\alpha-2\eps^2\)\cap p^{-1}(\abs\sigma)
	=\bigl\{\,z\in p^{-1}(\abs\sigma)
		\bigm|h(z)>h_\alpha\(p(z)\)-2\eps^2\,\bigr\}\;,$$
such that~$U_\alpha\cap U_{2\eps}\((C_\alpha'(\abs\sigma)\)=\emptyset$
for all~$\alpha'<_\sigma\alpha$.
Then we have in particular~$U_\alpha\cap M^u_W\(x',\sigma\)
=\emptyset$ for all~$x'\in C_{\alpha'}(\abs\sigma)$
with~$\alpha'<_\sigma\alpha$,
so a small modification of~$W$ along~$U_\alpha$
over the interior~$\inn\sigma$ of~$\abs\sigma$
will not destroy any transversal intersection established in a previous step.
Note that for~$U_\alpha$ small enough,
we get
	$$M^u_W(x,\sigma)\cap U_\alpha
	=S_{2\eps}T^uX|_{C_\alpha(\abs\sigma)}\;.$$

Our goal is to find a diffeomorphism~$F_\alpha$ of~$U_\alpha$
that is an arbitrarily small perturbation of~$\id_{U_\alpha}$,
that equals~$\id_{U_\alpha}$ outside
some compact subset of~$U_\alpha\cap p^{-1}(\inn\sigma)$,
and that preserves fibres of~$p$,
such that the intersections of
	$$F_\alpha\(S_{2\eps}T^uX|_{C_\alpha(\abs\sigma)}\)$$
with all~$M^s_W(y,\sigma)\cap U_\alpha$ become transversal in~$U_\alpha$.
We will then cut~$M$ along~$U_\alpha\subset h^{-1}(h_\alpha-2\eps^2)$
and glue the two sides back using~$F_\alpha$.
Clearly,
we can get the same effect by a suitable perturbation of~$W$
``just below''~$h^{-1}(h_\alpha-2\eps^2)$,
so that~$W|_{U_{2\eps}(C)}$ is unaffected.
Moreover,
\WDynProp\ still holds for the modified vector field~$W$.

By induction over~$\alpha$ and by \SimplLaudenbachProp,
we know that~$M^s_W(y,\sigma)\cap U_\alpha$
is a submanifold of~$h^{-1}(h_\alpha-2\eps^2)$ with conical singularities
for  all zeros~$y$ of~$W$,
whose strata are unions of sets of the form~$M^s_W(y',\sigma')\cap U_\alpha$.
%Note that a flow-line~$\gamma(t)$ of~$W$ may actually
%pass the hypersurface~$h^{-1}(h_\alpha-2\eps)$ several times
%due to the construction of~$W$,
%but if we choose~$U_\alpha$ small enough,
%we can make sure that it passes~$U_\alpha$ at most once.
Let~$\sigma'$ be a proper face of~$\sigma$,
then we know by induction
that all~$M^s_W(y,\sigma')\cap U_\alpha$
intersect~$S_{2\eps}T^uX|_{C(\abs\sigma')}$ transversally
in~$p^{-1}(\abs{\sigma'})$.
As above,
this implies that~$M^s_W(y,\sigma')\cap U_\alpha$
intersects~$S_{2\eps}T^uX|_{C(\abs\sigma)}$ transversally
in~$p^{-1}(\abs\sigma)$.

By \SMCSTransRem,
the intersection of~$M^s_W(y,\sigma)\cap U_\alpha$
with~$S_{2\eps}T^uX|_{C(\abs\sigma)}$ is transversal
in a neighbourhood of~$p^{-1}(\del\abs\sigma)$.
%This means that it suffices to modify~$W$ only over~$\inn\sigma$.
%thus preserving all modifications to~$W$ that we have performed in previous
%steps.
There clearly is a diffeomorphism~$F_\alpha$ as above
such that~$F_\alpha\(S_{2\eps}T^uX|_{C_\alpha(\abs\sigma)}\)$
intersects all~$M^s_W(y',\sigma)$ transversally,
if we allow~$F_\alpha$ to mix the fibres of~$p$.
However,
because~$p$ restricts
to a submersion~$S_{2\eps}T^uX|_{C_\alpha(\abs\sigma)}\to\abs\sigma$,
we can always take~$F_\alpha$ such that it preserves the fibres of~$p$.
After the gluing construction described above,
$M^u_W(x,\sigma)$ now intersects all~$M^s(y,\sigma)$ transversally
in a neighbourhood of~$U_\alpha$.
Moreover,
the glueing construction is equivalent to a small vertical modification of~$W$
over~$\inn\sigma$,
and ``just below''~$h^{-1}\(h_\alpha-2\eps^2\)$.
Since transversality is preserved by the flow of~$W$,
these intersections are everywhere transversal.
This finishes the last step of both inductions.
\qed\enddemo

For each~$k$, each $k$-simplex~$\sigma\in S_k$ and each~$\alpha\in P(\sigma)$,
let
	$$M_\alpha(\sigma)
	=\overline{M^u_W(x,\sigma)}
	\Formel\CWConstruction$$
denote the closure of the $W$-unstable manifold of the critical
point~$x\in C_\alpha(\sigma_{0})$ over~$\abs\sigma$.
Then~$M_\alpha(\sigma)$ is a submanifold of~$M$ with conical singularities
whose interior is homeomorphic to a $k+\ind_\alpha$-dimensional disc.
The following proposition shows that the~$M_\alpha(\sigma)$
form a CW-structure on~$M$ that behaves nicely with respect
to the projection~$p\colon M\to B$ and the simplicial complex~$S$ on~$B$.
We write~$M_\alpha(\sigma)$ for the generators of the corresponding
chain complex,
and use~$\partial$ to denote its boundary operator.
Let us for the moment assume that the bundle~$T^uX\to C$ is oriented,
so that all cells~$M_\alpha(\sigma)$ carry a natural orientation
that agrees with the orientation of~$\abs\sigma\times T^uX_{\sigma_{0}}$
at~$\sigma_{0}$.

\Proposition\CWBaseProp.
For~$\alpha$, $\beta\in P(\sigma)$,
there exist an integer~$a_{\alpha\beta}(\sigma)$,
such that
$$\align
	a_{\alpha\beta}(\sigma)
	&=0\Rtext{unless~$\beta\prec_\sigma\alpha$
			and~$\ind_\alpha+k=\ind_\beta+1$,}
	\tag 1\\
	\partial M_\alpha(\sigma)
	&=\sum_j(-1)^j M_\alpha(\sigma_{0\dots\smash{\widehat j}\dots k})%\tag2\\
%	&\qquad
		+\sum_j\sum_\beta(-1)^{k(j-1)}
			a_{\alpha\beta}(\sigma_{0\dots j})
			\,M_\beta(\sigma_{j\dots k})\;,\tag2\\
		\Land
	0&=
	\sum_j(-1)^ja_{\alpha\beta}(\sigma_{0\dots\smash{\widehat j}\dots k})%\tag3\\
%	&\qquad
	+\sum_j\sum_\gamma(-1)^{k(j-1)}a_{\alpha\gamma}(\sigma_{0\dots j})
		\,a_{\gamma\beta}(\sigma_{j\dots k})\;.\tag3
\endalign$$

Note that equation~(1) makes sure
that only cells of dimension~$k+\ind_\alpha-1$ enter the second sum in~(2).
The right hand side of~(2) reads like the dual of a ``simplicial''
superconnection,
consisting of the simplical boundary operator on~$B$
and a sum of products with simplicial cochains
valued in the endomorphisms of a bundle~$\hat p_*\(C\times\Z\)$
of free abelian groups.
Then~(3) simply says that this superconnection is flat.

\Remark\IgusaSignRem
In \IBuch, Definition~3.1.1,
Igusa defines {\em system of higher homotopies\/}
to be systems of endomorphism~$e(v_0,\dots,v_k)\in\End(V)$
for~$n\ge v_0\ge\dots\ge v_k\ge 0$,
satisfying a non-degeneracy axiom and
axioms similar to \CWBaseProp~(1) and~(3),
where~(3) above is replaced by
	$$\sum_{j=0}^k(-1)^j\(e(v_0,\dots,v_j)\,e(v_j,\dots,v_k)
		-e(v_0,\dots,\widehat{v_j},\dots,v_k)\)\;.$$
In particular,
if we fix a simplex~$\sigma\in S_n$ and set
	$$e_\sigma(v_0,\dots,v_k)
	=\cases
		(-1)^{\textstyle{k(k-1)\over2}}\,a(\sigma_{v_0,\dots,v_k})
			&\text{if~$v_0<\dots<v_k$, and}\\
		0	&\text{otherwise,}
	\endcases$$
we obtain a system of higher homotopies in the sense of Igusa.

If~$T^uX\to C$ is orientable,
let~$G=\{1\}$ be the trivial group,
otherwise take~$G=\{\pm1\}$.
Then the structure group of~$o(T^uX)$ is~$G$,
and the structure group of~$\hat p_*o(T^uX)\to B$ is a subgroup
of the group of $G$-monomial matrices,
i.e., the group of invertible matrices having precisely one non-zero
element in each row and each column,
which is an element of~$G$.
Putting the~$e_\sigma$ together for all simplices~$\sigma\in S$,
we get a functor~$\xi$ from the category of simplices of~$S$
to Igusa's Whitehead space~$\Wh(\Z,G)$
(which is constructed as a simplicial category),
and~$\xi$ is {\em $G$-monomial\/}
in the sense of Chapter~3.3 in~\IBuch.
\endremark


\demo{Proof\/ \rm of \CWBaseProp}
Let~$\sigma\in S_k$,
and let~$x$ be the critical point in~$C_\alpha$ over the vertex~$\sigma_0$.
Recall that by \SimplSmaleProp,
the strata of~$\overline{M^u_W(x,\sigma)}$
are unions of sets~$M^u_W(x',\sigma')$,
with~$\sigma'$ a face of~$\sigma$,
and~$x'\in\overline{M^u_W(x,\sigma)}\cap p^{-1}(\abs{\sigma'})$.
We now have to identify the codimension-$1$-stratum
of~$\overline{M^u_W(x,\sigma)}$.
It is a union of two types of unstable cells~$M^u_W(x',\sigma')$.
The first type has~$\sigma'=\sigma$,
so assume that~$x'\in C_\beta$ lies over the vertex~$\sigma_j$.
Then the dimension of~$M^u_W(x',\sigma)=M^u_W(x',\sigma_{j,\dots,k})$
is~$\ind_\beta+k-j$.
In particular,
$M^u(x',\sigma)$ is of codimension~$1$
iff~$\ind_\alpha+j=\ind_\beta+1$ as in~(1).
This way we obtain all terms in the second sum in~(2)
for~$\beta\prec_\sigma\alpha$,
and the very first term~$M_\alpha(\sigma_{1,\dots,k})$ in the first sum,
cf.~\SecondWTransFormel.

The remaining contributions to the codimension-$1$-stratum
arise as intersections of unstable cells with~$p^{-1}(\del\abs\sigma)$.
The only such intersections of codimension~$1$ in~$\overline{M^u_W(x,\sigma)}$
are of the form~$\overline{M^u_W(x,\sigma')}$
with~$\sigma'=\sigma_{0,\dots,\smash{\widehat j},\dots,k}$ for~$1\le j\le k$,
accounting for the rest of the first sum in~(2).
Note that~$j=0$ is impossible here,
because for a transversal intersection,
we had to assume that~$p(x)=\sigma_0\in\abs{\sigma'}$
in the proof of \SimplSmaleProp.

The preceding arguments show that the boundary of~$M_\alpha(\sigma)$
as a chain is given by~(2),
with constants~$a_{\alpha\beta}(\sigma_{0\dots j})$ as in~(1),
which might, however, still depend on~$\sigma$.
To see that this is not the case,
and to explain the sign factor~$(-1)^{k(j-1)}$,
we define~$a_{\alpha\beta}(\sigma_{0\dots j})$
to be the coefficient of~$M_\beta(\sigma_j)$
in the (chain) boundary of~$M_\alpha(\sigma_{0\dots j})$.
%For dimension reasons,
%we obtain \CWBaseProp~(1).
Suppose that we orient both~$M_\alpha(\sigma)$
and~$M_\beta(\sigma_{j\dots k})$
by placing an oriented base of~$\sigma_{j\dots k}$
to the right of oriented bases of~$M_\alpha(\sigma_{0\dots j})$
and~$M_\beta(\sigma_j)$, respectively.
By the arguments after~\FirstWTransFormel\ in the proof of \SimplSmaleProp,
the coefficient of~$M_\beta(\sigma_{j\dots k})$
in~$\del M_\alpha(\sigma)$
is still~$a_{\alpha\beta}(\sigma_{0\dots j})$
with respect to these orientations.
However,
since we actually have to fit the oriented base of~$\sigma_{j\dots k}$
to the left of the bases of~$T^uX|_{C_\alpha}$ and~$T^uX|_{C_\beta}$
according to the conventions fixed at the beginning of this subsection
and before \CWBaseProp,
these new orientations differ from the actual ones
by factors of~$(-1)^{(k-j)\ind_\alpha}$ and~$(-1)^{(k-j)\ind_\beta}$,
respectively.
By~(1), $\ind_\alpha-\ind_\beta=1-j$,
so we get the sign factor~$(-1)^{(k-j)(j-1)}=(-1)^{k(j-1)}$ as claimed.

Finally, because~$\partial^2=0$,
$$\align
	0&=\partial^2M_\alpha(\sigma)
\allowdisplaybreak
	&=\sum_j(-1)^j\partial M_\alpha(\sigma_{0\dots\smash{\widehat j}\dots k})\\
	&\qquad
		+\sum_j\sum_\beta(-1)^{k(j-1)}
			a_{\alpha\beta}(\sigma_{0\dots j})
			\,\partial M_\beta(\sigma_{j\dots k})
\allowdisplaybreak
	&=\sum_{i<j}\sum_\beta(-1)^j\,(-1)^{(k-1)(i-1)}
			a_{\alpha\beta}(\sigma_{0\dots i})
			\,M_\beta(\sigma_{i\dots\smash{\widehat j}\dots k})\\
	&\qquad
		+\sum_{i>j}\sum_\beta(-1)^j\,(-1)^{(k-1)i}
			a_{\alpha\beta}(\sigma_{0\dots\smash{\widehat j}\dots i})
			\,M_\beta(\sigma_{i\dots k})
\allowdisplaybreak
	&\qquad
		+\sum_{i\ge j}\sum_\beta(-1)^{k(j-1)}\,(-1)^{i-j}
			a_{\alpha\beta}(\sigma_{0\dots j})
			\,M_\beta(\sigma_{j\dots\widehat i\dots k})\\
	&\qquad
		+\sum_{i\ge j}\sum_{\beta,\gamma}
			(-1)^{k(j-1)}\,(-1)^{(k-j)(i-j-1)}
			a_{\alpha\beta}(\sigma_{0\dots j})
			\,a_{\beta\gamma}(\sigma_{j\dots i})
			\,M_\gamma(\sigma_{i\dots k})
\allowdisplaybreak
	&=\sum_j\sum_\beta(-1)^{(k-1)j}
		\biggl(\sum_{i\le j}(-1)^i
			a_{\alpha\beta}(\sigma_{0\dots\widehat i\dots j})\\
	&\kern8em
		\sum_{i\le j}\sum_\gamma
			(-1)^{j(i-1)}
			a_{\alpha\gamma}(\sigma_{0\dots i})
			\,a_{\gamma\beta}(\sigma_{i\dots j})\biggr)
		\,M_\beta(\sigma_{j\dots k})\;,
\endalign$$	
which implies~(3).
\qed\enddemo

\Abschnitt\GoodCoverSubsection Simplicial complexes and good coverings

In the previous subsections,
we constructed a CW complex on~$M$ over a simplicial complex~$S$ on~$B$
and gave an adapted basis~$M_\alpha(\sigma)$ for the associated
chain complex.
Each chain~$M_\alpha(\sigma)$
lived over a simplex~$\sigma\in S$ of~$B$.
For the following constructions however,
we need families of cells and chains over subsets of~$B$,
such that we can integrate over the fibres of these cells and chains
locally on~$B$.
We therefore modify the constructions of the previous subsections.
We will need a sufficiently fine simplicial complex,
in the following sense.

\Remark\SimplFineRem
Let~$\pi_1$, $\pi_2\colon B\times B\to B$ denote the two product projections,
and let~$\eps>0$ be defined as in \SimplifyingAssumptions.
Then there exists a neighbourhood~$U_{\Delta B}$ of the diagonal~$\Delta B$
in~$B\times B$ and a
diffeomorphism~$F\colon(\pi_2^*M)|_{U_{\Delta B}}\to(\pi_1^*M)|_{U_{\Delta B}}$
of fibre-bundles
extending~$\id_M\colon(\pi_2^*M)|_{\Delta B}\to(\pi_1^*M)|_{\Delta B}$,
such that
\roster
\item
$\abs{h\circ F_{q_1,q_2}-h}<{\eps^2\over2}$
for all~$(q_1,q_2)\in U_{\Delta B}$,
and
\item
$F_{q_1,q_2}\(\exp_c^X(z^s,z^u)\)
=\exp^X_{F_{q_1,q_2}c}\(F_{q_1,q_2,*c}z^s,F_{q_1,q_2,*c}z^u\)$
for~$(q_1,q_2)$ as above, $c\in\widehat p^{-1}(q_2)$,
and~$z^{s,u}\in T^{s,u}_cX$.
\endroster
We may choose the smooth ordered simplicial complex~$S$ so fine that
	$$\abs\sigma\times\abs\sigma\subset U_{\Delta B}
		\Rtext{for all~$\sigma\in S$.}
	\Formel\SimplFineFormel$$
\endremark

For~$\sigma\in S$,
let~$\inn\sigma$ denote the interior of~$\abs\sigma$ as a submanifold of~$M$,
and set
	$$U(\sigma)
	=\bigcup_{\textstyle{\sigma'\in S
			\atop\sigma\text{ is a face of }\sigma'}}
		\inn\sigma'
	\Formel\USigmaDef$$
for each~$\sigma\in S$.
We assume that~$S$ is so fine that the closure~$\overline{U(\sigma)}$
is contractible for all~$\sigma\in S$.
In particular,
	$$\Cal U(S)
	=\bigl\{\,U(\sigma)\bigm|\sigma\in S_0\,\bigr\}
	\Formel\GoodCoverDef$$
is a good open cover of~$B$.
%such that for~$\sigma^{(0)}<\dots<\sigma^{(k)}\in S_0$,
%we have
%	$$U\(\sigma^{(0)}\)\cap\cdots\cap U\(\sigma^{(k)}\)
%	=\cases
%		U(\sigma)	&\vtop{\hsize=2.2in\parindent=0pt
%					if there exists~$\sigma\in S_k$
%					with~$\sigma_j=\sigma^{(j)}$
%					for~$j=0$, \dots, $k$, and}\\
%	\noalign{\medskip}
%		\emptyset	&\text{otherwise.}
%	\endcases$$
%
Let us define
	$$\overline B(\sigma)
	=\overline{U(\sigma)}\times\abs\sigma
		\Mand
	\overline B
	=\bigcup_{\sigma\in S}\overline B(\sigma)
	\;\subset\;B\times B\;,
	\Formel\BqSigmaDef$$
then~$\overline B(\sigma)$ is a manifold with corners for all~$\sigma$.
Note that by~\SimplFineFormel,
	$$\overline B
	=\bigcup_{\sigma\in S}\abs\sigma\times\abs\sigma
	\;\subset\;U_{\Delta B}\;.$$


\midinsert
\psset{unit=0.05\hsize,fillstyle=solid}
$$\pspicture(-2,0)(4,4)
	\psline(-1,0)(2,0)
	\psdots(-1,0)(0,0)(1,0)(2,0)
	\uput[180](-1,0){$\scriptstyle S\text{ on }B$}
	\psline[linewidth=0.2pt]{->}(0.5,1.5)(0.5,0.5)
	\psline(-1,2)(0,2)
	\psline(-1,2.25)(1,2.25)
	\psline(0,2.5)(2,2.5)
	\psline(1,2.75)(2,2.75)
	\psdots(-1,2)(0,2.25)(1,2.5)(2,2.75)
	\uput[180](-1,2.25){$\scriptstyle\Cal U(S)=U(S_0)$}
	\psline(-1,3.5)(0,3.5)
	\psline(0,3.75)(1,3.75)
	\psline(1,4)(2,4)
	\uput[180](-1,3.75){$\scriptstyle U(S_1)$}
\endpspicture
\pspicture(-4,0)(3,4)
	\psline(-1,0)(2,0)
	\psdots(-1,0)(0,0)(1,0)(2,0)
	\psline[linewidth=0.2pt]{->}(0.5,0.8)(0.5,0.2)
	\uput[0](0.5,0.5){$\scriptstyle\pi_1$}
	\psline(-2,1)(-2,4)
	\psdots(-2,1)(-2,2)(-2,3)(-2,4)
	\psline[linewidth=0.2pt]{->}(-1.2,2.5)(-1.8,2.5)
	\uput[90](-1.5,2.5){$\scriptstyle\pi_2$}
	\pspolygon[linewidth=1pt,fillcolor=lightgray](-1,1)(0,1)(0,2)(-1,2)
	\pspolygon[linewidth=1pt,fillcolor=lightgray](0,2)(1,2)(1,3)(0,3)
	\pspolygon[linewidth=1pt,fillcolor=lightgray](1,3)(2,3)(2,4)(1,4)
%	\psline[linewidth=1pt](-1,1)(0,1)
%	\psline[linewidth=1pt](-1,2)(1,2)
%	\psline[linewidth=1pt](0,3)(2,3)
%	\psline[linewidth=1pt](1,4)(2,4)
	\uput[0](2,3.5){$\scriptstyle\overline B$}
\endpspicture$$
\centerline{\NeueFigur\BqFig
\BqFig. A one-dimensional simplical complex~$S$,
the cover~$\Cal U(S)$, and~$\overline B$.}
\endinsert


%For~$\sigma\in S_k$,
%we have a natural diffeomorphism
%	$$\id_{U(\sigma)}\times\sigma
%	\colon U(\sigma)\times\Delta_k
%	\to\overline B(\sigma)\;.$$
%If we assume \?what?,
%these diffeomorphisms combine to give a homeomorphism
%	$$\abs{U(S)}\simeq\overline B\;.$$
%
%The nerve~$U(S)$ of this good cover~$\Cal U$
%is a simplicial manifold whose manifold of $k$-simplices
%is by definition the disjoint union
%	$$U(S_k)=\coprod_{\sigma\in S_k}U(\sigma)\;,$$
%and for~$0\le i_0<\dots<i_j\le k$,
%the face map~$F_{i_0,\dots,i_j}\colon U(S_k)\to U(S_j)$
%is the union
%of the inclusions~$U(\sigma)\subset U(\sigma_{i_0\dots i_j})$
%for all~$\sigma\in S_k$.
%
%The geometric realisation~$\abs{U(S)}$ of the simplicial manifold~$U(S)$
%is by definition given as
%	$$\abs{U(S)}
%	=\biggl(\coprod_k\coprod_{\sigma\in S_k}U(\sigma)\times\Delta_k\biggr)
%		/\sim\;,$$
%where the equivalence relation~``$\sim$'' is generated by
%identifying
%	$$U(\sigma)\times\Delta_j
%	\subset U(\sigma_{i_0\dots i_j})\times\Delta_j
%		\Mtext{with}
%	U(\sigma)\times f_{i_0\dots i_j}(\Delta_j)
%	\subset U(\sigma)\times\Delta_k$$
%in the natural way
%for all $k$-simplices~$\sigma\in S_k$
%and all~$0\le i_0<\dots<i_j\le k$.
%
Consider the bundle~$\overline M=(\pi_1^*M)|_{\overline B}\to\overline B$
with its fibre-wise Morse function~$\overline h=\pi_1^*h$.
Then
	$$\overline M(\sigma)
	=(\pi_1^*M)|_{\overline B(\sigma)}
	\to\overline{U(\sigma)}
	\Formel\MqSigmaDef$$
is a bundle with fibre~$\abs\sigma\times X$.
For each~$\sigma$, each~$\alpha\in P(\sigma)$,
and each~$q_1\in\overline{U(\sigma)}$,
we define a submanifold
$$\split
	\overline M_\alpha(\sigma)|_{q_1}
	&=F_{q_1,\cdot}\(\{q_1\}\times M_\alpha(\sigma)\)\\
	&=\bigr\{\,F_{q_1,q_2}x\bigm|x\in M_\alpha(\sigma)
		\text{, and }q_2=p(x)\,\bigr\}
	\;\subset\;\overline M(\sigma)|_{q_1}
\endsplit\Formel\MqasConstruction$$
with conical singularities.
We still assume that the bundle~$T^uX|_C$ is oriented.

\Proposition\GCCWProp.
The submanifolds~$\overline M_\alpha(\sigma)|_{q_1}
\subset\overline M(\sigma)|_{q_1}$ with conical singularities
form a bundle~$\overline M_\alpha(\sigma)\to\overline{U(\sigma)}$,
such that
\roster
\item
for each~$\sigma\in S_k$,
the restrictions of the~$\overline M_\alpha(\sigma')$
to~$\overline{U(\sigma)}$ for all~$\alpha$ and all faces~$\sigma'$ of~$\sigma$
form a family of fibre-wise CW-complexes on~$\overline M(\sigma)\to
\overline{U(\sigma)}$;
\item
let~$a_{\alpha\beta}(\sigma)$ be the coefficients of \CWBaseProp,
then the boundary of~$\overline M_\alpha(\sigma)$
as a fibre-wise chain is given by
$$%\split
	\partial\overline M_\alpha(\sigma)%&
	=\sum_j(-1)^j\overline M_\alpha
		(\sigma_{0\dots\smash{\widehat j}\dots k})
			|_{\overline{U(\sigma)}}%\\
	%&\qquad
		+\sum_j\sum_\beta(-1)^{k(j-1)}
			a_{\alpha\beta}(\sigma_{0\dots j})
			\,\overline M_\beta(\sigma_{j\dots k})
				|_{\overline{U(\sigma)}}\;;
%\endsplit
$$
\item
we have
	$$\overline M_\alpha(\sigma)
		\cap\bigl\{\,(q_1,q_2,x)\in\overline M
			\bigm|h(x)>h_\alpha(q_1)-\eps^2\,\bigr\}
	=\(\pi_1^*D_{\sqrt 2\,\eps}T^uX\)|_{\overline B(\sigma)}\;.$$
\endroster

\Proof
From the construction~\MqasConstruction,
it follows that~$\overline M_\alpha(\sigma)\to \overline{U(\sigma)}$
is a bundle with fibre diffeomorphic to the~$M_\alpha(\sigma)$
of \CWBaseProp.
Also, the~$\overline M_\alpha(\sigma_{i_0\dots i_j})|_{\overline{U(\sigma)}}$
form a fibre-wise CW-complex on~$\overline M(\sigma)\to \overline{U(\sigma)}$
with fibre diffeomorphic to the CW-complex on~$p^{-1}(\abs\sigma)$
formed by the~$M_\alpha(\sigma_{i_0\dots i_j})$
for all~$\alpha$ and all~$0\le i_0<\dots<i_j\le k$,
so we get~(1).
Moreover,
from \CWBaseProp~(2),
we get~(2).
Finally,
(3) follows from \WDynProp\ by the specifications in \SimplFineRem\
and~\MqasConstruction. %\?explain?
\qed\enddemo

We want to state a dual version of \CWBaseProp\ and \GCCWProp.
Let~$F\to M$ be a flat vector bundle.
We no longer assume that the bundle~$T^uX\to C$ is oriented,
and set~$V_\alpha=o(T^uX)\otimes F|_{C_\alpha}\to B$
over any simplex~$\abs\sigma$.
We define~$V\to B$ as in \MorseBundleDef,
so~$V=\bigoplus_\alpha V_\alpha$.

Let~$\overline F=(\pi_1^*F)|_{\overline B}\to\overline M$
be the flat bundle over~$\overline M$ induced by~$F$,
and let~$\overline F(\sigma)=\overline F|_{\overline M(\sigma)}$.
We have natural pull-back maps
	$$\pi_1^*(\sigma)\colon\Omega^*\(p^{-1}(\overline{U(\sigma)});F\)
	\to\Omega^*\(\overline M(\sigma);\overline F\)\;,$$
and we define integration maps by integrating
over the fibres of~$\overline M_\alpha(\sigma)/\overline{U(\sigma)}$,
$$\split
	I_\alpha(\sigma)
	&=\int_{\overline M_\alpha(\sigma)/\overline{U(\sigma)}}
		\mathord{}\circ\pi_1^*(\sigma)
	\,\colon\;\Omega^*\(p^{-1}(\overline{U(\sigma)});F\)
	\to\Omega^*\(\overline{U(\sigma)};V_\alpha\)\\
		\Land
	I(\sigma)
	&=\bigoplus_\alpha I_\alpha(\sigma)
	\,\colon\;\Omega^*\(p^{-1}(\overline{U(\sigma)});F\)
	\to\Omega^*\(\overline{U(\sigma)};V\)\;.
\endsplit\Formel\IalphaDef$$

\Corollary\GCIntCor.
For all~$k$ and all~$\sigma\in S_k$,
there exist $\nabla^V$-parallel
endomorphisms~$a(\sigma)\in\Gamma\(\End^{1-k}V|_{\overline{U(\sigma)}}\)$
such that we have
$$\align\split
	\nabla^V\circ I(\sigma)
	&=(-1)^k\,I(\sigma)\circ\nabla^F%\\
%	&\qquad
		+\sum_j(-1)^j\,I(\sigma_{0\dots\widehat{j}\dots k})
			|_{\overline{U(\sigma)}}\\
	&\qquad
		+\sum_j(-1)^{k(j-1)}\,a(\sigma_{0\dots j})
			\circ I(\sigma_{j\dots k})|_{\overline{U(\sigma)}}\;,%\\
%	&\;\colon\;\Omega^*\(M|_{\overline{U(\sigma)}};F\)
%		\to\Omega^*\(\overline{U(\sigma)};V\)\;,
\endsplit\tag1\\
\split\Land
	0
	&=\sum_j(-1)^j\,a(\sigma_{0\dots\widehat{j}\dots k})
			|_{\overline{U(\sigma)}}\\
	&\qquad
		+\sum_j(-1)^{k(j-1)}\,a(\sigma_{0\dots j})
			\circ a(\sigma_{j\dots k})|_{\overline{U(\sigma)}}\;.%\\
%	&\in\Gamma\(\End^{1-k}V|_{\overline{U(\sigma)}}\)\;.
\endsplit\tag2\endalign$$
Suppose that~$v\in V_\beta|_{\overline{U(\sigma)}}$,
then
	$$a(\sigma%_{0\dots\smash{\widehat j}\dots k}
		)\,v
	\in\bigoplus_{\beta\prec_\sigma\alpha}V_\alpha\;.\tag3$$
Suppose that~$\omega\in\Omega^*\(\overline{U(\sigma)};F\)$
has support in~$h^{-1}(h_\alpha-\eps^2,\infty)
\subset M|_{U_\sigma}$,
then
	$$I_\alpha(\sigma)\,\omega
	=\cases\displaystyle
		\int_{D_{2\eps}T^uX|_{C_\alpha(\sigma)}/\overline{U(\sigma)}}\omega
			&\text{if~$\sigma\in S_0$, and}\\
		0	&\text{otherwise.}
	\endcases\tag4$$

Note that~(4) implies in particular that
if~$\supp\omega\subset h^{-1}(h_\alpha-\eps^2,\infty)$
and~$\beta\prec_\sigma\alpha$,
then~$I_\beta(\sigma)\,\omega=0$.

\Remark\ComplexWhiteheadRem
For each simplex~$\sigma\in S$,
the endomorphisms~$a(\sigma')$ for all faces~$\sigma'$ of~$\sigma$
give rise to a system~$e_\sigma$ of higher homotopies in the sense of Igusa,
just as in \IgusaSignRem.
Suppose that~$R$ is an associative ring with unit,
with a fixed complex representation~$\rho\colon\End(\C^r)$ for some~$r\ge 1$,
and that~$G$ is a subgroup of~$\GL_s(R)$ for some~$s\ge 1$,
such that the flat bundle~$F$ has structure
group~$\rho(G)\subset\GL_{rs}(\C)$
(if~$T^uX\to C$ is not orientable, we assume that~$-1\in G$).
Then the coefficients~$a_{\alpha\beta}(\sigma)\colon V_\beta\to V_\alpha$
of~$a(\sigma)$ naturally take values in the matrices~$M_s(R)$ over~$R$,
and the structure group of~$V$
is a subgroup of the group of $G$-monomial matrices
(matrices having exactly one non-zero entree in
each row or column, which is an element of~$G$).
Then the~$e_\sigma$ above
form a $G$-monomial functor from the category of simplices of~$S$
to the Whitehead space~$\Wh(R,G)$.
\endremark

	
\demo{Proof\/ \rm of \GCIntCor}
Consider the families of cochain-complexes
associated to the bundles of fibre-wise CW-complex
on~$\overline M(\sigma)\to \overline{U(\sigma)}$
constructed in \GCCWProp,
with values in~$\overline F$.
Since we now have to take the possible orientations of~$T^uX$ into account,
each cell~$\overline M_\alpha(\sigma')$
contributes a copy~$V_\alpha(\sigma')$ of~$V_\alpha|_{\overline{U(\sigma)}}\to \overline{U(\sigma)}$
to the family of complexes over~$\overline{U(\sigma)}$,
where~$\sigma'$ is a face of~$\sigma$.
For any fibre-wise cochain~$s$,
let~$s_\alpha(\sigma')$ denote the component of~$s$ in~$V_\alpha(\sigma')$.
By \CWBaseProp\ and \GCCWProp,
the coboundary map of this complex takes the form
	$$(\delta s)_\alpha(\sigma')
	=\sum_j(-1)^js_\alpha(\sigma'_{0\dots\smash{\widehat j}\dots k})
		+\sum_j\sum_\beta(-1)^{k(j-1)}
			a_{\alpha\beta}(\sigma'_{0\dots j})
			\,s_\beta(\sigma'_{j\dots k})
	\Formel\DualCWFormel$$
if~$\sigma'\in S_k$ is a face of~$\sigma\in S$,
where the~$a_{\alpha\beta}(\sigma'_{0\dots j})\colon V_\beta\to V_\alpha$
are parallel homomorphisms satisfying \CWBaseProp~(1) and~(3).

Let us define
	$$a(\sigma')
	=\bigoplus_{\alpha,\beta}a_{\alpha\beta}(\sigma')
	\,\colon\;V|_{U(\sigma')}\to V|_{U(\sigma')}\;,$$
then~(1) follows from \DualCWFormel\ and Stokes' theorem,
while~(2) and~(3) follow from \CWBaseProp~(1) and~(3).

If~$\supp\omega\subset h^{-1}\(h_\alpha-\eps^2,\infty\)$,
then by~\GCCWProp~(3) and~\IalphaDef,
	$$I_\alpha(\sigma),\omega
%	=\int_{\(\pi_1^*D_{\sqrt 2\,\eps}T^uX\)|_{\overline B(\sigma)}
%			/\overline{U(\sigma)}}
%		\pi_1^*(\sigma)\,\omega
	=\int_{\(\pi_1^*D_{2\eps}T^uX\)|_{\overline B(\sigma)}
			/\overline{U(\sigma)}}
		\pi_1^*(\sigma)\,\omega\;,$$
and~$\pi_1^*(\sigma)\omega$ vanishes in the directions of~$\abs\sigma$,
so this integral vanishes unless~$\sigma\in S_0$.
\qed\enddemo



\Abschnitt\ComplexPatchSubsection A ``mixed'' superconnection

Starting from the endomorphisms~$a(\sigma)$ of the previous subsection,
we construct a family of compatible
flat superconnections~$\nabla^{\overline V}+a'(\sigma,\emptyset)$
on~$V|_{\abs\sigma}$,
where~$a'(\sigma,\emptyset)\in\Omega^*(\abs\sigma,\End_+V)$
is an $\End V$-valued differential form of total degree~$1$.
We also construct a family of compatible
``integration maps''~$I'(\sigma,\emptyset)\colon\Omega^*(M;F)
\to\Omega^*(\abs\sigma;V)$ of total degree zero
that satisfy~$I'(\sigma,\emptyset)\circ\nabla^F
=(\nabla^{\overline V}+a'(\sigma,\emptyset))\circ I'(\sigma,\emptyset)$.
Using a partition of unity,
we pull~$\nabla^{\overline V}+a'(\emptyset)$ and~$I'(\emptyset)$ back
to a flat superconnection~$A'$ on~$V\to B$ of total degree one
and an integration map~$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$
satisfying~$I\circ\nabla^F=A'\circ I$.

If~$\sigma'=\sigma_{i_0\dots i_j}$ is a face of~$\sigma\in S_k$,
set
	$$\abs{\sigma\setminus\sigma'}
	=\abs{\sigma_{i_j,i_j+1,\dots,k}}\;,
	\Formel\RelSimplexDef$$
so~$\abs{\sigma\setminus\sigma'}$ is the simplex spanned by all
vertices of~$\sigma$ that are ``higher than or equal to''
all vertices of~$\sigma'$.
For convenience,
we introduce the {\em empty simplex\/}~$\emptyset$ of formal dimension~$-1$,
which we regard  as a face~$\emptyset=\sigma_\emptyset$ of all~$\sigma\in S$,
so that we may write~$\abs{\sigma}=\abs{\sigma\setminus\emptyset}$.

Let~$a(\sigma)\in\Gamma\(\End^{1-k}V|_{\overline{U(\sigma)}}\)$
be the parallel endomorphisms constructed in \GCIntCor.

\Proposition\SimplPatchProp.
For each~$0\le k\le l+1$ and each (possibly empty) face~$\sigma'\in S_{k-1}$
of~$\sigma\in S_l$,
there exist an~$\End V$-valued
differential form~$a'(\sigma,\sigma')$ on~$\abs{\sigma\setminus\sigma'}$
of total degree~$1-k$,
such that
\roster
\item
if~$\sigma$ is a face of~$\tau$,
then
	$$a'(\tau,\sigma')|_{\abs{\sigma\setminus\sigma'}}
	=a'(\sigma,\sigma')\;;$$
\item
the form~$a'(\sigma,\sigma)$ on~$\abs{\sigma_l}$ vanishes;
\item
if~$\sigma\in S_l$,
then
	$$a'(\sigma,\sigma_{0,\dots,l-1})|_{\abs{\sigma_l}}
	=a(\sigma)|_{\abs{\sigma_l}}\;;$$
\item
we have~$0=\bigl(\nabla^V+a'(\sigma,\emptyset)\bigr)^2$,
and for~$\sigma'\in S_k$ with~$k\ge 1$,
$$\align
	0
	&=(-1)^k\,\Bigl[\nabla^V,a'(\sigma,\sigma')\Bigr]
	+\sum_{j=0}^{k-1}(-1)^j
			\,a'(\sigma,\sigma'_{0,\dots,\smash{\widehat j},\dots,k-1})
	+(-1)^k\,a(\sigma')\\
	&\qquad
	+\sum_{j=0}^{k-1}(-1)^{k(j-1)}
		\,a(\sigma,\sigma_{0\dots j})\,a(\sigma,\sigma_{j,\dots,k-1})
	+a'(\sigma,\sigma')\,a'(\sigma,\emptyset)\;;\\
\endalign$$
\item
the component~$a'_{\alpha\beta}(\sigma,\sigma')$
in~$\Omega^*(\abs{\sigma\setminus\sigma'};
\Hom\(V^\beta,V^\alpha\)\bigr)$
vanishes unless~$\beta\prec_\sigma\alpha$.
\endroster


\Remark\SimplPatchRem
If~$a'(\sigma,\sigma')$ was a function,
\SimplPatchProp~(1) would mean that~$a'(\sigma,\sigma')$
is the restriction to~$\abs{\sigma\setminus\sigma'}$
of a piece-wise smooth function~$a'(\sigma')$
on the union of all~$\abs{\sigma\setminus\sigma'}$
for all~$\sigma$ having~$\sigma'$ as a face.
Note however that this union is not a differentiable manifold
(not even a manifold with corners).
Moreover, we are dealing with differential forms.
Suppose that~$\sigma$ is a codimension-$1$-face
of two simplices~$\tau$ and of~$\tau'$.
Then in local coordinates,
those coefficients of~$a'(\tau,\sigma')$ and~$a'(\tau',\sigma')$
that contain the direction normal to~$\abs\sigma$
may jump at~$\abs\sigma$,
so that~$a'(\sigma')$ would not even be well-defined along~$\abs\sigma$.
We will nevertheless write~$a'(\sigma')$
instead of~$a'(\sigma,\sigma')$ as long as no confusion about~$\sigma$ arises.

If we introduce the notation
	$$a(\sigma_{0,\dots,k-1,\punkt})
	=a'(\sigma_{0,\dots,k-1})
		\Mand
	a(\sigma_{\punkt})=a'(\emptyset)$$
then~(4) above without the term involving~$\nabla^{\overline V}$
looks precisely as \GCIntCor~(2),
where the index~``$k$'' has been replaced by~``$\punkt$''.
The placement of~``$\punkt$'' to the right of~$k-1$ in the index above
is compatible with the fact
that we may only evaluate~$a'(\sigma_{0,\dots,k-1})$
on the set of points ``behind''~$\sigma'_{k-1}$ in~$\sigma$.
In fact,
these two analogies with the coefficients~$a(\sigma)$ of \GCIntCor\
were the motivation for our construction of~$a'(\sigma')$.

While we could regard the~$a(\sigma)$
of \GCIntCor\ as coefficients of a purely combinatorial superconnection,
(or as a ``system of higher homotopies'' as in.~\IBuch, chapter~3),
we now have a mixed superconnection which is partly combinatorial
and partly differential.
Although the forms~$a'(\sigma')$ do not pull back
from endomorphisms defined on~$B$,
we can pull them back to~$B$ using a section of~$\pi_1\colon\overline B\to B$
constructed from a partition of unity,
to obtain a differential superconnection on~$V\to B$.
\endremark


\demo{Proof\/ \rm of \SimplPatchProp}
We define the forms~$a'(\sigma,\sigma')$
on~$\abs{\sigma\setminus\sigma'}$ for~$\sigma\in S_l$
by induction over~$l$.
For all~$\sigma\in S_0$,
the endomorphisms~$a'(\sigma,\sigma)=0$
and~$a'(\sigma,\emptyset)=a(\sigma)$
are determined by~(2) and~(3),
and~(4) holds on~$\sigma$ by \GCIntCor~(2).

Now fix~$\sigma\in S_l$ for~$l>0$
and suppose that all~$a(\tau,\sigma')$
have been constructed on all faces~$\tau$ of~$\sigma$,
such that~(1)--(5) are satisfied.
If~$\sigma'=\sigma_{i_0,\dots,i_{k-1}}\in S_{k-1}$
is a face of~$\sigma$ with~$i_{k-1}>k-1$,
then the simplex~$\tau=\sigma_{i_0,\dots,i_{k-1},i_{k-1}+1,\dots,l}$
is a proper face of~$\sigma$,
and~$\abs{\sigma\setminus\sigma'}=\abs{\tau\setminus\sigma'}$.
In particular,
the form~$a'(\sigma,\sigma')=a'(\tau,\sigma')$
is already determined by~(1) and induction,
and~(4) holds for~$a'(\sigma,\sigma')$.

Now fix~$\sigma'=\sigma_{0,\dots,k-1}$ for~$1\le k\le l+1$,
then the restriction of~$a'(\sigma,\sigma')$ to each of the
sets~$\abs{\sigma_{k-1,\dots,\smash{\widehat j},\dots,l}}$
with~$j\ge k$ is already determined by the induction hypothesis
applied to~$\tau=\sigma_{0,\dots,\smash{\widehat j},\dots,l}$
and~$\sigma'$.
To summarise,
the restriction of~$a'(\sigma,\sigma')$ to the set
	$$\partial\abs{\sigma_{k-1,\dots,l}}
		\setminus\inn\sigma_{k,\dots,l}
	\Formel\GCAGivenDomain$$
is already defined.
Similarly,
the restriction of~$a'(\sigma,\emptyset)$
to~$\partial\abs\sigma$ is already known.

We will construct~$a'(\sigma,\sigma')$
by induction on~$l-k$.
First,
we have~$a'(\sigma,\sigma)=0$ by~(2),
and by~(3), (4) reduces to the trivial equation
	$$a'(\sigma,\sigma_{0,\dots,l-1})-a(\sigma)
	=a(\sigma)-a(\sigma)=0\;.$$
Moreover, (5) follows from \GCIntCor~(3).

Now fix~$\sigma'=\sigma_{0,\dots,k-1}$ for~$1\le k\le l$,
and suppose that~$a'(\sigma,\sigma_{0\dots k})$
has already been constructed
on~$\abs{\sigma_{k\dots l}}\subset\abs{\sigma_{k-1,\dots,l}}
=\abs{\sigma\setminus\sigma'}$.
Writing~$a'(\punkt)$ instead of~$a'(\sigma,\punkt)$
as in \SimplPatchRem,
we define
	$$a''(\sigma')
	\in\Omega^*\(\abs{\sigma_{k\dots l}}
		;\End V\)$$
by rewriting~(4) as
$$\multline
	(-1)^{k+1}\,a''(\sigma')
	=\sum_{j=0}^{k-1}(-1)^j
		\,a'(\sigma_{0\dots\smash{\widehat j}\dots k})
	+(-1)^{k+1}\,a(\sigma_{0\dots k})\\
\split
	&\qquad
	+\sum_{j=1}^k(-1)^{(k+1)(j-1)}
		\,a(\sigma_{0\dots j})\,a'(\sigma_{j\dots k})\\
	&\qquad
	+(-1)^{k+1}\,\(\nabla^{V}+a(\sigma_0)\)
		\,a'(\sigma_{0\dots k})
	+a'(\sigma_{0\dots k})\,\(\nabla^{V}+a'(\emptyset)\)
\endsplit\endmultline\tag\FormelNummer\GCSSCADef$$
Then~$a''(\sigma')$ clearly satisfies~(5).
For~$j>k$,
the induction hypothesis
applied to the face~$\sigma_{0\dots\smash{\widehat j}\dots l}$
shows that~$a'(\sigma')$ satisfies the same equation
on~$\abs{\sigma_{k\dots\smash{\widehat j}\dots l}}$,
thus~$a''(\sigma')$ coincides with~$a'(\sigma')$
on
	$$\partial\abs{\sigma_{k\dots l}}
		\setminus\inn\sigma_{k+1,\dots,l}\;,
	\Formel\GCACoincDomain$$
see \ADomainFig.


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\ADomainFig. The domains of definition of~$a'(\sigma')$ and~$a''(\sigma')$.}
\endinsert

Because the analogue of~\GCSSCADef\
holds for~$a'(\sigma_{0\dots k})$
on~$\abs{\sigma_{k+1,\dots,l}}$,
and because the superconnections~$\nabla^{\overline V}+a(\sigma_0)$
and~$\nabla^{\overline V}+a'(\emptyset)$ are flat,
we can rewrite~\GCSSCADef\ for~$a''(\sigma')$
as
$$\align
	&\kern-2em
	(-1)^{k+1}a''(\sigma')\\
	&=\sum_{j=0}^{k-1}(-1)^ja'(\sigma_{0\dots\smash{\widehat j}\dots k})
		+(-1)^{k+1}\,a(\sigma_{0\dots k})%\\
%	&\qquad
	+\sum_{j=1}^k(-1)^{(k+1)(j-1)}
		\,a(\sigma_{0\dots j})\,a'(\sigma_{j\dots k})
\allowdisplaybreak
	&\qquad
	-\(\nabla^{V}+a(\sigma_0)\)\,
		\Biggl(\sum_{j=0}^k(-1)^j\,
			a'(\sigma_{0\dots \smash{\widehat j}\dots k+1})
			+(-1)^k\,a(\sigma_{0\dots k+1})\\
	&\kern11em
			+\sum_{j=1}^{k+1}(-1)^{k(j-1)}\,a(\sigma_{0\dots j})
				\,a'(\sigma_{j\dots k+1})\Biggr)
\allowdisplaybreak
	&\qquad
	+(-1)^k\,\Biggl(\sum_{j=0}^k(-1)^j\,
			a'(\sigma_{0\dots \smash{\widehat j}\dots k+1})
			+(-1)^k\,a(\sigma_{0\dots k+1})\\
	&\kern8em
			+\sum_{j=1}^{k+1}(-1)^{k(j-1)}\,a(\sigma_{0\dots j})
				\,a'(\sigma_{j\dots k+1})\Biggr)
		\,\(\nabla^{V}+a'(\emptyset)\)\;.
\endalign$$
We now apply~(4) inductively as well as \GCIntCor~(2)
to the last two terms on the right hand side,
and find
$$\align
	&\kern-2em
	(-1)^{k+1}a'(i_0\dots i_{k-1},\punkt)\\
	&=\sum_{j=0}^{k-1}(-1)^ja'(\sigma_{0\dots\smash{\widehat j}\dots k})
		+(-1)^{k+1}\,a(\sigma_{0\dots k})%\\
%	&\qquad
	+\sum_{j=1}^k(-1)^{(k+1)(j-1)}
		\,a(\sigma_{0\dots j})\,a'(\sigma_{j\dots k})
\allowdisplaybreak
	&\qquad
	-\sum_{j=0}^k(-1)^{j+k}
		\Biggl(\sum_{j'=0}^{j-1}(-1)^{j'}
			\,a'(\sigma_{0\dots\smash{\widehat{j'}}\dots
				\smash{\widehat j}\dots k+1})\\
	&\kern10em
		-\sum_{j'=j+1}^{k+1}(-1)^{j'}
		\,a'(\sigma_{0\dots\smash{\widehat j}\dots
				\smash{\widehat{j'}}\dots k+1})%\\
%	&\kern10em
		+(-1)^{k+1}\,a(\sigma_{0\dots\smash{\widehat j}\dots k+1})
\allowdisplaybreak
	&\kern10em
		+\sum_{j'=1}^{j-1}(-1)^{(k+1)(j'-1)}\,a(\sigma_{0\dots j'})
			\,a'(\sigma_{j'\dots\smash{\widehat j}
				\dots k+1})\\
	&\kern10em
		+\sum_{j'=j+1}^{k+1}(-1)^{(k+1)j'}
			\,a(\sigma_{0\dots\smash{\widehat j}\dots j'})
			\,a'(\sigma_{j'\dots k+1})\Biggr)
\allowdisplaybreak
	&\qquad
		-a(\sigma_0)\,a'(\sigma_{1\dots k+1})
		+a(\sigma_{0\dots k+1})\,a'(\emptyset)\\
	&\qquad
		-\sum_{j'=0}^{k+1}(-1)^{j'}
			\,a(\sigma_{0\dots\smash{\widehat{j'}}\dots k+1})%\\
%	&\qquad
		-\sum_{j'=1}^{k+1}(-1)^{(k+1)(j'-1)}\,a(\sigma_{0\dots j'})
			\,a(\sigma_{j'\dots k+1})
\allowdisplaybreak
	&\qquad-\sum_{j=1}^{k+1}(-1)^{(k+1)(j-1)}\Biggl(
		\sum_{j'=0}^j(-1)^{j'}
			\,a(\sigma_{0\dots\smash{\widehat{j'}}\dots j})\\
	&\kern13em
		+\sum_{j'=1}^j(-1)^{j(j'-1)}\,a(\sigma_{0\dots j'})
			\,a(\sigma_{j'\dots j})\Biggr)
		\,a'(\sigma_{j\dots k+1})
\allowdisplaybreak
	&\qquad-\sum_{j=1}^{k+1}(-1)^{kj}\,a(\sigma_{0\dots j})\,\Biggl(
		\sum_{j'=j}^{k+1}(-1)^{j'-j}
			\,a'_(\sigma_{j,\dots\smash{\widehat{j'}}\dots k+1})%\\
%	&\kern13em
		+(-1)^{k-j}\,a(\sigma_{j\dots k+1})\\
	&\kern13em
		+\sum_{j'=j}^{k+1}(-1)^{(k-j)(j'-j-1)}\,a(\sigma_{j\dots j'})
			\,a'(\sigma_{j'\dots k+1})\Biggr)\\
	&=(-1)^{k+1}\,a'(\sigma_{0\dots k-1})
	=(-1)^{k+1}\,a'(\sigma')\;.
\endalign$$
Note that all terms in the intermediate step
except~$(-1)^{k+1}\,a'(\sigma_{0\dots k-1})$
cancel by their signs.
In particular,
we can now extend~$a'(\sigma')$
to~$\partial\abs{\sigma_{0,\dots,k-1}}$
by~$a''(\sigma')$.
We choose an arbitrary smooth continuation~$a'(\sigma,\sigma')$
of~$a'(\sigma')|_{\partial\abs{\sigma_{0,\dots,k-1}}}$
to~$\abs{\sigma_{0,\dots,k-1}}$
that is of total degree~$1-k$,
cf.\ \IBuch, Lemma~2.2.3.
By \CWPrecRemark~(2),
the relation~$\prec_\sigma$ refines on the corresponding relations
for the faces of~$\sigma$,
so we can choose~$a'(\sigma,\sigma')$ such that~(5) still holds.

In the last step,
we define~$a''(\emptyset)$ on~$\abs\sigma$ by
	$$\nabla^{V}+a''(\emptyset)
	=\(\id+a'(\sigma_0)\)^{-1}
		\,\Bigl(\nabla^{V}+a(\sigma_0)\Bigr)
		\,\(\id+a'(\sigma_0)\)\;.
	\Formel\SimplPatchOneFormel$$
As above,
we automatically get~$a''(\emptyset)=a'(\emptyset)$
on~$\partial\abs\sigma\setminus\inn\sigma_{1\dots l}$,
and a similar calculation as above shows~$a''(\emptyset)=a'(\emptyset)$
on the whole boundary of~$\abs\sigma$,
so~$a'(\emptyset)$ naturally extends to~$\abs\sigma$.
Moreover,
since~$a(\sigma_0)$ is a parallel endomorphism with~$a(\sigma_0)^2=0$,
the superconnection~$\nabla^V+a(\sigma_0)$  on~$\abs\sigma$ is flat,
and thus the superconnection~$\nabla^{\overline V}+a'(\emptyset)$
on~$\abs\sigma$ is also flat,
which means that~(4) is satisfied also for~$a'(\sigma,\emptyset)$.
\qed
\enddemo

Given the coefficients~$a'(\sigma,\sigma')$
of a ``mixed'' superconnection,
we now have to construct the coefficients~$I'(\sigma,\sigma')$
of a ``mixed'' chain homomorphism
in such a way that the analogue of \GCIntCor~(1) holds.
Let~$I(\sigma)$ be the integration maps of \GCIntCor.


\Proposition\SimplPatchPropZwei.
For each~$0\le k\le l+1$ and each (possibly empty)
face~$\sigma'\in S_{k-1}$ of~$\sigma\in S_l$,
there exists a map
	$$I'(\sigma,\sigma')
	\colon\Omega^*(M;F)
	\to\Omega^*\(\abs{\sigma\setminus\sigma'};V\)$$
of total degree~$-k$,
such that
\roster
\item
if~$\sigma$ is a face of~$\tau$,
then
	$$I'(\sigma,\sigma')\,\omega
	=\(I'(\tau,\sigma')\,\omega\)|_{\abs{\sigma\setminus\sigma'}}
		\Rtext{for all~$\omega\in\Omega^*(M;F)$;}$$
\item
the map~$I'(\sigma,\sigma)$ vanishes;
\item
if~$\sigma\in S_l$,
then
	$$\(I'(\sigma,\sigma_{0,\dots,l-1})\,\omega\)|_{\abs{\sigma_l}}
	=\(I(\sigma)\,\omega\)|_{\abs{\sigma_l}}
		\Rtext{for all~$\omega\in\Omega^*(M;F)$;}$$
\item
we have~$I'(\sigma,\emptyset)\circ\nabla^F=\nabla^V\circ I'(\sigma,\emptyset)$,
and for~$\sigma'\in S_k$ with~$k\ge 1$,
$$\align
	I'(\sigma,\sigma')\circ\nabla^F
	&=(-1)^k\nabla^V\circ I'(\sigma,\sigma')\\
	&\qquad
	+\sum_{j=0}^{k-1}(-1)^j
		\,I'(\sigma,\sigma'_{0,\dots,\smash{\widehat j},\dots,k-1})
	+(-1)^k\, I(\sigma')\\
	&\qquad
	+\sum_{j=0}^{k-1}(-1)^{k(j-1)}
		\,a(\sigma_{0,\dots,j})\,I'(\sigma,\sigma_{j,\dots,k-1})
	+a'(\sigma,\sigma_{0,\dots,k-1})\,I'(\sigma,\emptyset)\;;
\endalign$$
\item
for each face~$\sigma''\in S_j$ of~$\sigma$
and~$\alpha$, $\beta\in P(\sigma)$,
there exist differential forms
	$$b_{\alpha\beta}(\sigma,\sigma',\sigma'')
	\in\Omega^{\ind_\beta-\ind_\alpha+j-k}
		(\abs{\sigma\setminus\sigma'})\;,$$
with~$b_{\alpha\beta}(\sigma,\sigma',\sigma'')=0$
unless~$\beta\prec_\sigma\alpha$ or~$\beta=\alpha$,
such that
	$$I'_\alpha(\sigma,\sigma')
	=\sum_\beta\sum_{\sigma''}
		b_{\alpha\beta}(\sigma,\sigma',\sigma'')
		\,I_\beta(\sigma'')$$
for the component~$I'_\alpha(\sigma,\sigma')$
of~$I(\sigma,\sigma')$ with values
in~$\Omega^*\bigl(\overline B(\sigma,\sigma');\overline V_\alpha\bigr)$;
\item
if\/~$\omega\in\Omega^*(M,F)$
has\/~$\supp(\omega)\subset h^{-1}\(h_\alpha-\eps^2,\infty\)$,
then on any~$\abs{\sigma\setminus\sigma'}$,
we have
	$$I'_\alpha(\sigma,\sigma')\,\omega
	=\cases\displaystyle
		\int_{D_{2\eps}T^uX|_{C_\alpha(|\sigma\setminus\sigma'|)}
				/\abs{\sigma\setminus\sigma'}}\omega
			&\text{if~$\sigma'=\emptyset$, and}\\
		0	&\text{otherwise.}
	\endcases$$
\endroster

Note that the analogue of \SimplPatchRem\
applies to the maps~$I'(\sigma,\sigma')$ as well:
We may regard~$I'(\sigma,\sigma')$ as a restriction of
a map~$I'(\sigma')$ from~$\Omega^*\(M|_{U(\sigma')};F\)$
to the $\overline V$-valued forms on the union
of all~$\overline B(\sigma,\sigma')$.
Moreover,
if we introduce the notation~$I(\sigma'_{0,\dots,k-1,\punkt})=I'(\sigma')$
for~$\sigma'\in S_{k-1}$,
then~(4) above is related to \GCIntCor~(1)
just as \SimplPatchProp~(4) is related to \GCIntCor~(2).
Finally,
properties~(5) and~(6) will be crucial
when we prove \MainTheorem\ in \WittenKapitel.

\Proof
We basically repeat the proof of \SimplPatchProp.
For all~$\sigma\in S_0$,
we set~$I'(\sigma,\sigma)=0$ and~$I'(\sigma,\emptyset)=I(\sigma)$.
By \GCIntCor, this agrees with~(2)--(6) above.

Fix~$\sigma\in S_l$ and assume that all~$I'(\tau,\sigma')$
have already been defined for all faces~$\tau$ of~$\sigma$
and~$\sigma'$ of~$\tau$.
Then by the same argument as in the previous proof,
for any face~$\sigma'=\sigma_{i_0,\dots,i_{k-1}}$ of~$\sigma$
with~$i_{k-1}>k-1$,
the map~$I'(\sigma,\sigma')$ is already inductively defined by~(1),
and~(4)--(6) hold for~$I'(\sigma,\sigma')$
on~$\abs{\sigma\setminus\sigma'}$.
Note also that~$I'(\sigma,\sigma)=0$ by~(2),
which is compatible with~(3)--(6).

Now fix~$\sigma'=\sigma_{0,\dots,k-1}$ for~$0<k\le l$
and assume that~$I'(\sigma,\sigma_{0,\dots,k})$ has already been defined.
Then~$I'(\sigma,\sigma')$ is already known
on~$\partial\abs{\sigma_{k-1\dots l}}
\setminus\inn\sigma_{k\dots l}$,
cf.~\GCAGivenDomain.
Writing~$I'(\punkt)$ shorthand for~$I'(\sigma,\punkt)$,
we define~$I''(\sigma')\colon\Omega^*(M;F)
\to\Omega^*(\abs{\sigma_{k\dots l}};V)$ by
$$\multline
	(-1)^{k+1}\,I''(\sigma')
	=\sum_{j=0}^{k-1}(-1)^j
		\,I'(\sigma_{0\dots\smash{\widehat j}\dots k})
	+(-1)^{k+1}\, I(\sigma_{0\dots k})\\
\split
	&\qquad
	+\sum_{j=1}^k(-1)^{(k+1)(j-1)}
		\,a(\sigma_{0\dots j})\,I'(\sigma_{j\dots k})\\
	&\qquad
	+(-1)^{k+1}\,\(\nabla^V+a(\sigma_0)\)
		\,I'(\sigma_{0\dots k})
	+I'(\sigma_{0\dots k})\,\nabla^F\;,
\endsplit\endmultline\tag\FormelNummer\GCSSCIDef$$
cf.~\GCSSCADef.

Note that since~(5) holds for~$I'(\sigma_{0\dots k})$
by the construction in the previous step,
we have
$$\multline
	(-1)^{k+1}\,\nabla^V\,I'_\alpha(\sigma_{0\dots k})
		+I'_\alpha(\sigma_{0\dots k})\,\nabla^F\\
	=\sum_{\beta}\sum_{\textstyle{\sigma''\in S\atop\sigma''
			\text{ is a face of }\sigma}}
		\biggl((-1)^{k+1}\,d
			b_{\alpha\beta}(\sigma,\sigma_{0\dots k},\sigma'')
			\, I_\beta(\sigma'')\\
		+b_{\alpha\beta}(\sigma,\sigma_{0\dots k},\sigma'')
			\, \Bigl((-1)^{j+\ind_\beta-\ind_\alpha}
					\,\nabla^V\,I_\beta(\sigma'')
				+I_\beta(\sigma'')\,\nabla^F\Bigr)\biggr)\;.
\endmultline\tag\FormelNummer\PropThreeFormula$$
By \GCIntCor, \GCSSCIDef\ and~\PropThreeFormula,
we see that~$I''(\sigma')$ satisfies~(5),
where the degrees of the coefficient forms
can be calculated using the fact that~$I''(\sigma')$
is of total degree~$-k$.

As in the proof of \SimplPatchProp,
$I'(\sigma')$ and~$I''(\sigma')$
coincide on~$\partial\abs{\sigma_{k\dots l}}
\setminus\inn\sigma_{k+1\dots l}$,
cf.~\GCACoincDomain.
In order to check
that~$I'(\sigma')=I''(\sigma')$
on~$\abs{\sigma_{k+1\dots l}}$,
we proceed by computations similar to those in the previous proof.
In fact, comparing~\GCSSCADef\ with its analogue~\GCSSCIDef,
we find that we can re-use the computations in the previous proof
after making the following changes.
\roster
\item
First, replace each occurrence of~$\nabla^V+a'(\emptyset)$
on the right hand side of a summand by~$\nabla^F$.
\item
Then replace the rightmost occurrence of~$a'(\punkt)$ or~$a(\punkt)$
by~$I'(\punkt)$ or~$ I(\punkt)$, respectively.
\endroster
Then all calculations remain valid and show that
we can use~$I''(\sigma')$ to extend~$I'(\sigma')$
to~$\partial\abs{\sigma_{k-1\dots l}}$.
We now choose smooth continuations of the
coefficients~$b_{\alpha\beta}(\sigma,\sigma',\sigma'')$ in~(5)
to~$\partial\abs{\sigma_{k-1\dots l}}$,
thus extending~$I'(\sigma')$ to~$\abs{\sigma\setminus\sigma'}$.

In order to determine~$I'(\emptyset)$,
we may rewrite~(4) as
	$$I''(\emptyset)
	=\(\id+a'(\sigma_0)\)^{-1}
		\,\Bigl(\(\nabla^V+a(\sigma_0)\)\circ I'(\sigma_0)
			+I'(\sigma_0)\circ\nabla^F
			+ I(\sigma_0)\Bigr)\;.
	\Formel\SimplPatchTwoFormel$$
As before, we can show that~$I''(\emptyset)=I'(\emptyset)$
on~$\partial\abs\sigma$,
so we can set~$I'(\emptyset)=I''(\emptyset)$ on~$\abs\sigma$.
By the argument following~\PropThreeFormula,
property~(5) then also holds for~$I'(\emptyset)$.
Moreover, using \GCIntCor, \SimplPatchOneFormel\ and~(4),
we finally find that
$$\align
	I'(\emptyset)\circ\nabla^F
	&=\(\id+a'(\sigma_0)\)^{-1}
		\,\Bigl(\(\nabla^V+a(\sigma_0)\)
				\circ I'(\sigma_0)
			+ I(\sigma_0)\Bigr)\circ\nabla^F\\
	&=\(\id+a'(\sigma_0)\)^{-1}
		\,\(\nabla^V+a(\sigma_0)\)\\
	&\qquad
		\Bigl(\(\id+a'(\sigma_0)\)\,I'(\emptyset)
			- I(\sigma_0)
			-\(\nabla^V+a(\sigma_0)\)\,I'(\sigma_0)
			+ I(\sigma_0)\Bigr)\\
	&=\(\nabla^V+a'(\emptyset)\)\,I'(\emptyset)\;.
\endalign$$

We have now proved \SimplPatchPropZwei~(1)--(5).
Claim~(6) follows from~(5) by \GCIntCor~(4),
where we use that for~$\sigma'\in S_{k-1}$ and~$\sigma''\in S_j$,
$b_{\alpha\alpha}(\sigma,\sigma',\sigma'')=0$
unless~$j\ge k$.
\qed\enddemo


\Abschnitt\SmoothingSubsection A smoothing construction

We note that by \SimplPatchProp~(4) and \SimplPatchPropZwei~(4),
we obtain a superconnection~$\nabla^V+a'(\sigma,\emptyset)$
on~$\Omega^*(\abs\sigma;V)$ and a cochain-map from~$\Omega^*(M;F)$
to~$\Omega^*(\abs\sigma;V)$.
By \SimplPatchRem,
these objects do not patch together to yield a smooth flat superconnection
and a smooth cochain-map over all of~$B$.
There are several possible ways out.

For example,
one can define an algebra of ``$S$-piece-wise smooth'' differential forms,
given as families~$\alpha=\(\alpha_\sigma\)_{\sigma\in S}$
with~$\alpha_\sigma\in\Omega^*(\abs\sigma)$,
such that~$\alpha_\sigma|_\tau=\alpha_\tau$ whenever~$\tau$
is a face of~$\sigma$.
This means that near a simplex~$\sigma\in S$,
the components of~$\alpha$ tangential to~$\abs\sigma$ are continuous,
while the remaining components are only bounded with all their derivatives.
One can show that it is possible to do analysis in such a setup.
E.g., the equation~$\(\nabla^V+a'(\punkt,\emptyset)\)^2=0$
is still meaningful.
It is possible to define an $S$-piece-wise smooth torsion form on~$B$.

We will instead give a smoothing construction
that leads to a smooth superconnection~$A'$ on~$V$.
Choose a partition of unity~$\{\,\phy_\sigma\colon M\to[0,1]|\sigma\in S_0\,\}$
subordinate to the open covering~$\Cal U(S)$ of \GoodCoverDef,
i.e., with~$\supp(\phy_\sigma)\subset U_\sigma$ for all~$\sigma\in S_0$.
If~$\sigma\in S_k$,
then all these functions vanish on~$\abs\sigma$
except~$\phy_{\sigma_0}$, \dots, $\phy_{\sigma_k}$,
and of course~$\phy_{\sigma_0}+\cdots+\phy_{\sigma_k}=1$ on~$\abs\sigma$.
If we interpret~$\phy_{\sigma_0}$, \dots, $\phy_{\sigma_k}$
as barycentric coordinates on~$\Delta_k$,
we obtain a smooth map~$\phy\colon B\to B$
by setting
	$$\bar\phy(b)
	=\sigma\(\phy_{\sigma_0}(b),\phy_{\sigma_1}(b),\dots\)
	\in\abs\sigma
		\Rtext{for~$b\in\inn\sigma$.}
	\Formel\PartUnFormel$$
Note that~$\bar\phy$ maps a small neighbourhood
	$$U'(\sigma)
	=\bigl\{\,b\in B\bigm|b\notin\supp(\phy_{\sigma'})
		\text{ unless~$\sigma'$ is a vertex of }\sigma\,\bigr\}$$
of each simplex~$\sigma\in S$ to~$\abs\sigma$.
%Let~$\bar\phy=\operatorname{graph}\phy\colon B\to\overline B\subset B\times B$
%denote the graph of this map,
%cf.~\PhyqFig.
%This implies that the following constructions make sense
%and give a smooth superconnection and a smooth integration map.

\midinsert
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	\psline(-1,1)(-0.8,1)
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	\uput[90](-1,1){$\scriptstyle\phy_{\punkt}$}
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%	\psline[linewidth=1pt](0,3)(2,3)
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\centerline{\NeueFigur\PhyqFig
\PhyqFig. A partition of unity
and the corresponding section of~$\pi_1^*$.}
\endinsert

\Definition\PatchSuperconnectionDefinition
Define a superconnection~$A'=\nabla^V+a'$
on~$V\to B$ with~$a'\in\Omega^*(B,\End^*_+V)$,
and a map~$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$,
such that restricted to~$\abs\sigma$,
$$\align
	A'&=\nabla^V+\bar\phy^*a'(\sigma,\emptyset)\;,\\
		\Land
	I&=\sum_{\sigma''\text{ is a face of }\sigma}
		\bar\phy^*b(\sigma,\emptyset,\sigma'')\cdot I(\sigma'')\;,
\endalign$$
where the coefficients~$b(\sigma,\emptyset,\sigma'')$
are given in \SimplPatchPropZwei~(5).
\enddefinition

Let~$h^V$ denote the endomorphism of~$V\to B$ induced by~$h|_C$,
then we will call~$(V,A',h^V)$ a {\em family Thom-Smale complex\/}
for~$p\colon M\to B$ and~$F\to M$.
The complexes~$\Omega^*(M;F)$ and~$\Omega^*(B;V)$
are filtered by~$F^*\Omega^*(M;F)$ and~$F^*\Omega^*(B;V)$
according to the horizontal degree,
see \HorDegFiltDef.
Recall that~$\(E_k^{*,*}(M;F),d_k\)$ and~$\(E_k^{*,*}(B;V),d_k\)$
denote the accociated spectral sequences,
and that~$E_*^{*,*}(M;F)$ is the Leray spectral sequence in de Rham theory
with coefficients in~$F\to M$.


\Theorem\PatchThm.
The superconnection~$A'=\nabla^V+a'$ is flat and of total degree one,
and~$I$ %$\colon\(\Omega^*(M;F),\nabla^F\)\to\(\Omega^*(B;V),A'\)$
is a cochain map of total degree zero.
The endomorphism~$a'$ decomposes locally over~$\abs\sigma$
into components~$a'_{\alpha\beta}\in\Omega^*\(B,\Hom(V_\beta,V_\alpha)\)$
with
	$$a'_{\alpha\beta}=0
		\Rtext{unless~$\beta\prec_\sigma\alpha$.}\tag1$$
Locally on~$\abs\sigma$,
the map~$I$ is an $\Omega^*(B;\End V)$-linear combination
of the maps~$I_\alpha(\sigma')$ of \IalphaDef,
with~$\sigma'$ a face of~$\sigma$.
If~$\omega\in\Omega^*(M;F)$ has support
in~$h^{-1}(h_\alpha-\eps^2,\infty)$
locally on~$\overline{U(\sigma)}\subset B$,
then the component~$I_\alpha\omega$ of~$I\,\omega$
in~$\Omega^*\(\overline{U(\sigma)};V_\alpha\)$ is given by
	$$I_\alpha\omega
	=\int_{D_{2\eps}T^uX|_{C_\alpha(\sigma)}/\overline{U(\sigma)}}\omega
	\quad\in\Omega^*\(\overline{U(\sigma)};V_\alpha\)\;.\tag2$$
The map~$I$ commutes with the action of~$\Omega^*(B)$,
maps~$F^p\Omega^*(M;F)$ to~$F^p\Omega^*(B;V)$ for all integers~$p\ge 0$,
and induces cochain homomorphisms
	$$I_k^{p,q}
	\colon\(E_k^{*,*}(M;F),d_k\)
	\to\(E_k^{*,*}(B;V),d_k\)\tag3$$
for~$k$, $p$, $q\ge 0$
that are isomorphisms for~$k\ge 1$.
In particular,
$I\colon\(\Omega^*(M;F),\nabla^F\)\to\(\Omega^*(B;V),A'\)$
is a quasi-isomorphism.


\Proof
We know that near a simplex~$\sigma\in S$,
the components of~$a'(\punkt,\emptyset)$ and~$I'(\punkt,\emptyset)$
that are tangential to~$\abs\sigma$ are continuous,
together with all their higher derivatives in tangential directions.
On the other hand,
since~$\bar\phy$ is smooth and maps an open neighbourhood of~$\abs\sigma$
to~$\abs\sigma$,
pulling back by~$\bar\phy$ kills all components and all higher derivatives
of~$a'(\punkt,\emptyset)$ and~$I'(\punkt,\emptyset)$
that are not purely tangential to~$\abs\sigma$.
This implies that~$A'$ and~$I$ are smooth.

It is immediate from~\SimplPatchProp\
that~$A'$ is a flat superconnection of total degree~$1$,
and that~(1) holds.

Furthermore,
\SimplPatchTwoFormel\ and \PatchSuperconnectionDefinition\ imply that
for all~$\sigma$,
there exists a map
	$$I'_\sigma
	=\sum_{\sigma''\text{ is a face of }\sigma}
		\bar\phy^*b(\sigma,\sigma_0,\sigma'')\,I(\sigma'')\;,$$
such that
$$\align
	I
	&=\(\id+\bar\phy^*a'(\sigma_0)\)^{-1}
		\,\Bigl(\(\nabla^V+a(\sigma_0)\)\circ I'_\sigma
			+I'_\sigma\circ\nabla^F+I(\sigma_0)\Bigr)\;,
	\Formel\SmoothIHomotopyFormel\\
		\Land
	I\circ\nabla^F
	&=\(\nabla^V+\bar\phy^*a'(\emptyset)\)\circ I
	=A'\circ I\;.
\endalign$$
From the construction and \SimplPatchPropZwei~(5) and~(6),
we conclude that~$I|_{\abs\sigma}$
is indeed an $\Omega^*(B;\End V)$-linear combination
of the maps~$I_\alpha(\sigma')$ of \IalphaDef,
and that~(2) holds.
Since the maps~$I_\alpha(\sigma')$ are defined by integration
over the fibres of certain smooth families
of submanifolds with conical singularities,
these maps clearly commute with~$\Omega^*(B)$,
and so does~$I$.

As an $\Omega^*(B)$-homomorphism,
$I$ maps~$F^p\Omega^*(M;F)$ to~$F^p\Omega^*(B;V)$ for all~$p$,
so it induces cochain homomorphisms of spectral sequences as claimed.
It suffices to show that~$I_1^{*,*}$ is an isomorphism,
then all~$I_k^{*,*}$ with~$k\ge 1$ are also isomorphisms,
and~$I$ is thus a quasi-isomorphism.
By~\SmoothIHomotopyFormel, %~\SimplPatchOneFormel\ and~\SimplPatchTwoFormel,
for each~$\sigma\in S_k$, % and each~$0\le j\le k$,
we have a diagramme of cochain maps
$$\psset{linewidth=0.5pt,nodesep=8pt}
\matrix\format\l&\l&\l\\
	\bigl(\Omega^*(M;F),\Rnode{MF}{\nabla^F}\bigr)
	&\kern4em&
	\bigl(\Rnode{O0}{\Omega^*}(\abs\sigma %{\sigma\setminus\sigma_j}
		;\Rnode{V0}{V})
		,A'\bigr)\\
	&&\vphantom{\Biggm|}\\
	&&\bigl(\Rnode{Oi}{\Omega^*}(\abs\sigma %{\sigma\setminus\sigma_j}
		;\Rnode{Vi}{V})
		,\nabla^V+a(\sigma,\sigma_0)\bigr)\;
\endmatrix
\ncline{->}{MF}{O0}\Aput{\scriptstyle I}
\ncline{->}{MF}{Oi}\Bput{\scriptstyle I(\sigma_0)}
\ncline{->}{V0}{Vi}
	\Aput{\scriptstyle\id+\bar\phy^*a'(\sigma,\sigma_0)}
	\Bput{\rotateleft{\cong}}$$
that commutes up to the cochain homotopy~$I'_\sigma$.
Note that by classical Morse-theory,
the diagonal arrow induces an isomorphism from~$H^*(X;F)$
to~$H^*\(V,a(\sigma_0)\)$ for each fibre~$X$ over~$\abs\sigma$.

Passing to~$E_1^{p,q}$,
the map
	$$\(\nabla^V+\bar\phy^*a(\sigma_j)\)\,I'_\sigma
	+I'_\sigma\,\nabla^F
	\colon\Omega^*(M;F)\to\Omega^*\(\abs\sigma;V\)$$
induces~$0$,
because in the grading induced by the filtrations in \HorDegFiltDef,
it splits into a fibre-wise
homotopy %~$a(i)\,I(i,\punkt)+I(i,\punkt)\,\nabla^F|_{TX}$
and a part %~$\nabla^{\overline V}\,I(i,\punkt)+I(i,\punkt)\,\nabla^F|_{p^*TB}$
that increases the horizontal degree.
In particular,
the diagramme
$$\psset{linewidth=0.5pt,nodesep=8pt}
\matrix\format\l&\l&\l\\
	\Omega^p\bigl(B;H^q(M/B;\Rnode{HMF}{F)}\bigr)
	&\kern5em&
	\Rnode{OH0}{\Omega^p}\bigl(\abs\sigma %{\sigma\setminus\sigma_j}
		;\Rnode{HV0}{H^q}(V,a'_0)\bigr)\\
	&&\vphantom{\Biggm|}\\
	&&\Rnode{OHi}{\Omega^p}\bigl(\abs\sigma %{\sigma\setminus\sigma_j}
		;\Rnode{HVi}{H^q}\bigl(V,a(\sigma,\sigma_0)\dg0\bigr)\bigr)\;
\endmatrix
\ncline{->}{HMF}{OH0}\Aput{\scriptstyle
	I_1^{p,q}}
\ncline{<-}{OHi}{HMF}\aput[0pt]{:U}{\sim}
\ncline{->}{HV0}{HVi}\aput[0pt]{:U}{\sim}$$
commutes,
where~$a'_0$ and~$a(\sigma,\sigma_0)\dg0$ denote the components
of horizontal degree~$0$.
It follows that~$I_1^{p,q}$ is an isomorphism for all~$p$, $q$,
which finishes the proof of \PatchThm.
\qed\enddemo

\Remark\PatchWittenRem
As we will see in \WittenKapitel,
the Witten deformation will give another proof of the fact
that~$I_1^{*,*}$ is a quasi-isomorphism.
\endremark

\Remark\GCHorDegreeRemark
Let~$\sigma'\in S_k$ be a face of~$\sigma\in S_l$.
It is clear from the proof of~\SimplPatchProp\
that we can always choose
	$$a'(\sigma,\sigma')
	\in\bigoplus_{j=0}^{l-k-1}
		\Omega^j\(\abs{\sigma\setminus\sigma'};
			\End^{1-j-k}_+V\)\;.$$
However,
this means that~$A'$ may have components
of horizontal degree up to~$\dim X+1$.
If we assume that the dimension of the fibres is higher
than the dimension of the base%
---which we are forced to assume when we apply the main theorem of~\Iframed\
in order to construct~$h$---%
then~$A'$ can have components in any possible horizontal degree.
\endremark

The construction so far has involved many choices.
We had to pick 
a diffeomorphism~$F$
of the bundles~$\pi_1^*M$ and~$\pi_2^*M$
over a neighbourhood~$U$ of~$\Delta B$ in~$B\times B$,
a sufficiently fine smooth ordered simplical complex~$S$ on~$B$,
a continuous vector field~$W$
that over each simpex of~$S$ is smooth and satisfies the Smale transversality
condition,
and finally,
we had to choose smooth continuations of the coefficients
of~$a'(\sigma')$ and~$I'(\sigma')$ from the boundary of certain simplices
to their interiors.
Still, the superconnections~$A'$ and integration maps~$I$
of \PatchSuperconnectionDefinition\ coming from different choices
are not completely unrelated.

By \PatchThm,
any two pairs of complexes and integration maps
	$$I_i\colon\(\Omega^*(M;F),\nabla^F\)
	\to\(\Omega^*(B;V_i),A'_i\)$$
for~$i=0$, $1$ are related by a quasi-isomorphism.
One possible right inverse for~$I_i$
is the inverse of the restriction of~$I_i$ to the space of
very small eigenvalues under the Witten deformation,
cf.\ \SmallevSubsection.
However,
we can make the relation of~$I_0$, $A'_0$ and~$I_1$, $A'_1$
much more concrete.

\Proposition\RelativityProp.
Assume that~$A'_0$ and~$A'_1$ are two superconnections
and~$I_0$, $I_1$ are integration maps for these superconnections,
all constructed using the methods of this chapter.
Then the same methods allow us to construct a superconnection~$\overline A'$
on the pull-back~$V\times[0,1]$ of~$V$ to~$B\times[0,1]$,
together with an integration
map~$\overline I\colon\Omega^*(M\times[0,1];F\times[0,1])
\to\Omega^*(B\times[0,1];V\times[0,1])$,
such that \PatchThm\ holds for~$\overline A'$ and~$\overline I$,
and~$\overline A'|_{B\times\{s\}}=A'_s$
and~$\overline I|_{B\times\{s\}}=I_s$ for~$s=0$, $1$.

\Proof
One has to check that all choices made so far on~$B\times\{0,1\}$
can be extended to the interior of~$\overline B=B\times[0,1]$,
still satisfying all required properties.
For the diffeomorphism~$F$ of \SimplFineRem,
this is clear because it is just a perturbation
of~$\id_{\pi_1^*M}|_{\Delta B}$.
Suppose that~$\overline F$ extends~$F$
on a neighbourhood~$U_{\Delta\overline B}$
of the diagonal in~$\overline B\times\overline B$,
then we can construct a smooth ordered simplicial complex~$\overline S$
as fine as~\SimplFineFormel\ that restricts to the given complexes
on~$B\times\{0\}$ and~$B\times\{1\}$.

In the next step,
we extend the vertical gradient field~$-\nabla\TX h$
and the cut-off function~$\phy_M$ over~$B\times\{0,1\}$
that were used in~\WConstruction\
smoothly to~$B\times[0,1]$,
thus defining a preliminary vector field~$W$ on~$M\times[0,1]$.
Now the constructions of \SimplSmaleProp, \SimplPatchProp,
and \SimplPatchPropZwei\
are all by induction over the degree of simplices,
so we can keep every choice we made over~$B\times\{0,1\}$.
For \PatchSuperconnectionDefinition,
it suffices to find a partition of unity on~$\overline B$
that restricts to the given partitions on~$B\times\{0\}$ and~$B\times\{1\}$.
This finishes the proof of our proposition.
\qed\enddemo

We will need this proposition in \TeilZwei, % \WhKapitel,
to show that a certain ``classifying''
map from~$B$ to the Whitehead space is well-defined up to homotopy.
\RelativityProp\ can also be used to compare
the torsion forms~$T\(A'_s,g^V,h^V\)$ of \TorsionKapitel\ for~$s=0$, $1$
coming from superconnections~$A'_s$ related as above.
Of course,
a formula relating~$T\(A'_0,g^V,h^V\)$ and~$T\(A'_1,g^V,h^V\)$
will ultimately follow from \MainTheorem.

\enddocument

example.tex
%auto-ignore
\input morse.def

\document

\Kapitel\BeispielKapitel=5 Exotic smooth fibre bundles

We calculate the analytic torsion form
of a sphere bundle~$p_\gamma\colon M_\gamma\to S^{4k}$
that is fibre-wise homeomorphic to a trivial bundle,
but not diffeomorphic.
We then use this bundle to construct ``exotic'' bundle structures
on a quite general class of compact fibre bundles.

The construction of the sphere bundle~$p_\gamma$ in~\HatcherSubsection\
goes back to Hatcher,
and was kindly explained to us by Igusa,
see also~\IBuch, Section~6.
The computation of~$\Cal T(M_\gamma,S^{4k};\C)$ 
in~\HatcherMorseSubsection\
and the extension to bundles with arbitrary fibres in~\GenFibreSubsection\
are still closely related to Igusa's work.

In~\HatcherMoreSubsection,
we use Hatcher's example to exhibit a large class of fibre bundles~$M\to B$
where we can construct infinitely many bundles
that are homeomorphic but not diffeomorphic as bundles to~$M\to B$,
see \HatcherCorollary.
In~\HomeoDiffeoSubsection\ we finally prove a refinement of \ApplThm,
see \HatcherTheorem.
We will also relate our results to~\DWW.


\Abschnitt\HatcherSubsection Hatcher's construction

We will construct a bundle~$N_\gamma\to S^{4k}$,
with fibre
	$$D^{m+n}
	=(S^m\times D^n)\cup_{S^m\times D^{n-1}}(D^{m+1}\times D^{n-1})\;,
	\Formel\HatcherFaserFormel$$
see \HatcherFaserFig.
Here, $\gamma\in\ker J$, see~\JHomoFormel\ below.
The bundle~$M_\gamma$ will then be given by the fibre-wise double of~$N_\gamma$.
Over the ``southern'' hemisphere~$D^{4k}_-\subset S^{4k}$ of the base sphere,
we take a fixed embedding of~$D^{n-1}$
into a fixed hemisphere of~$S^{n-1}=\del D^n$ in~\HatcherFaserFormel,
thus obtaining a fixed trivialisation of~$D^{m+n}\times D^{4k}_-$.

\midinsert
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%	\uput[180](-2.3,0.2){$\scriptstyle D^n$}
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\centerline{\NeueFigur\HatcherFaserFig
\HatcherFaserFig. A single fibre of~$N_\gamma$.}
\endinsert

We will now describe the construction
over the ``northern'' hemisphere~$D^{4k}_+\subset S^{4k}$.
For fixed~$k$ and~$m$,
we have the $J$-homomorphism,
which is given as
	$$J
	\colon\pi_{4k-1}\(O(m)\)
	\to\pi_{4k-1}\(C^0((S^m,x_0),(S^m,x_0))\)
	\cong\pi_{4k-1+m}(S^m,x_0)\;,
	\Formel\JHomoFormel$$
where~$x_0\in S^m$ denotes the north pole.
The groups~$\pi_{4k-1}\(O(m)\)$ and~$\pi_{4k-1+m}(S^m,x_0)$
are independent of~$m$ for~$m$ sufficiently large,
and~$\pi_{4k-1+m}(S^m,x_0)$ is finite,
while~$\pi_{4k-1}\(O(m)\)$ is infinite cyclic.
Thus,
pick~$0\ne\gamma\in\ker J$,
which is represented by a map~$\gamma\colon S^{4k-1}\to O(m)$
that extends continuously
to~$\gamma\colon D^{4k}_+\to C^0((S^m,x_0),(S^m,x_0))$.

For~$n$ large enough,
using transversality,
we can approximate the map
	$$D^{4k}_+\times(S^m,x_0)
		\buildrel\gamma\over\longrightarrow(S^m,x_0)
		\into\(S^m\times D^{n-1},(x_0,0)\)$$
by a smooth family
of embeddings~$\widetilde\gamma_q\colon(S^m,x_0)\into\(S^m\times D^{n-1},(x_0,0)\)$
for~$q\in D^{4k}_+$,
which equals
	$$(qr,x)\mapsto\(\gamma_q(x),0\)\Rtext{for~$q\in S^{4k-1}=\del D^{4k}_+$,
		$\textstyle r\in\bigl[{1\over2},1\bigr]$,
		and }x\in S^m\;.$$
The normal bundle~$\Cal N_q\to S^m$ of~$\widetilde\gamma_q$
is stably trivial for all~$q\in D^{4k}_+$,
because the tangent bundles of~$S^m$
and~$S^m\times D^{n-1}$ are stably trivial.
Making~$n$ even larger if necessary,
we may assume that the~$\Cal N_q$ are trivial.
Then the bundle~$\Cal N\to D^{4k}_+\times S^m$ is of course also trivial.
Fixing a trivialisation of~$\Cal N$ and
using the normal exponential map,
we can extend~$\widetilde\gamma$ to a family of embeddings
	$$\widetilde\gamma_q\colon S^m\times D^{n-1}\to S^m\times D^{n-1}
		\Rtext{for }q\in D^{4k}_+\;.$$
We may assume that there exists
a map~$\gamma'\colon S^{4k-1}\times S^m\to O(n-1)$
such that for all~$q\in S^{4k-1}=\del D^{4k}_+$
and all~$r\in\bigl[{1\over2},1\bigr]$,
the embedding~$\tilde\gamma_{qr}$ becomes
	$$\widetilde\gamma_{qr}(x,y)=\(\gamma_q(x),\gamma'_{q,x}(y)\)\;.$$

\Proposition\HatcherFramingProp.
The family of embeddings~$\widetilde\gamma_q\colon S^m\times D^{n-1}
\to S^m\times D^{n-1}$ for~$q\in D^{4k}_+$
is isotopic to a family of embeddings
that restricts over~$S^{4k-1}=\del D^{4k}_+$ to
	$$(x,y)\mapsto\(\gamma_q(x),(\gamma_q^{-1}\oplus\id_{\R^{n-1-m}})(y)\)
		\Rtext{for all }q\in S^{4k-1}\;.$$

With this result,
we finally construct~$N_\gamma$.
We still identify~$D^{n-1}$ with a fixed hemisphere of~$S^{n-1}=\del D^n$.
For~$q\in D^{4k}_+$,
we construct the fibre
	$$D^{m+n}
	=(S^m\times D^n)\cup_{S^m\times D^{n-1}}(D^{m+1}\times D^{n-1})$$
as in~\HatcherFaserFormel,
identifying
	$$(x,y)\in S^m\times D^{n-1}\subset\del(D^{m+1}\times D^{n-1})
		\Mtext{with}
	\widetilde\gamma_q(x,y)\in S^m\times D^{n-1}\subset\del(S^m\times D^n)\;.$$
Then we can glue~$N_\gamma|_{D^{4k}_+}$ with~$N_\gamma|_{D^{4k}_-}$ along~$S^{4k-1}$
using the identity on~$S^m\times D^n$
and identifying
	$$(q,x,y)\in N_\gamma|_{D^{4k}_+}
		\Mtext{with}
	\bigl(q,\gamma_q(x),(\gamma_q^{-1}\oplus\id_{\R^{n-1-m}})(y)\bigr)
	\in N_\gamma|_{D^{4k}_-}\;.$$

%\?Bildhcen?

Finally,
we construct the sphere bundle
	$$p\colon M_\gamma=N_\gamma\cup_{\id_{\del N_\gamma}}N_\gamma\to S^{4k}\;.
	\Formel\HatcherConstruction$$

\demo{Proof\/ \rm of \HatcherFramingProp}
Consider the family of rotations~$S^{4k-1}\to O(2m)$ given by
	$$q\mapsto\(\gamma_q\oplus\gamma_q^{-1}\)\colon S^{4k-1}\to O(2m)\;.$$
By a well-known trick from $K$-theory,
this family is homotopic to the trivial family~$q\mapsto\id_{\R^{2m}}$:
apply the Cartan projection~$GL(2m,\R)\to O(2m)$ to the homotopy
	$$(q,t)
	\mapsto\pmatrix1&t\gamma_q\\0&1\endpmatrix
		\,\pmatrix1&0\\-t\gamma_q^{-1}&1\endpmatrix
		\,\pmatrix1&t\gamma_q\\0&1\endpmatrix
		\,\pmatrix1-t&-t\\t&1-t\endpmatrix
	\Formel\KHomotopyFormel$$
in~$GL(2m,\R)$,
where~$t\in[0,1]$.
In particular,
there exists a map~$G\colon D^{4k}_+\to O(m+n)$
that for all~$q\in S^{4k-1}$ and all~$r\in\bigl[{1\over2},1\bigr]$
takes the form
	$$G_{rq}=\(\gamma_q\oplus\gamma_q^{-1}\oplus\id_{\R^{n-m}}\)\;.$$

The map~$\widetilde\gamma_q\colon S^m\times D^{n-1}\to S^m\times D^{n-1}$
is determined by its 1-jet
	$$D\widetilde\gamma_q|_{S^m\times\{0\}}
	\colon TS^m\times\R^{n-1}\to TS^m\times TD^{n-1}\;.$$
We add the trivial line~$\R$ on both sides,
and identify~$TS^m\times\R$ with~$\R^{m+1}$
using the standard embedding~$S^m\hookrightarrow\R^{m+1}$.
Then~$D\widetilde\gamma_q|_{S^m\times\{0\}}$ acts on~$TS^m\times\R$
by the matrix~$\gamma_q\oplus\id_\R\in O(m+1)$ for~$q\in S^{4k-1}$.
This implies that~$G_q^{-1}\circ D\widetilde\gamma_q|_{S^m\times\{0\}}$
acts trivially on~$TS^m\times\R$ for~$q\in S^{4k-1}$.

Because the Stiefel manifold~$O(m+n+1)/O(m+1)$ is $4k$-connected
for~$n$ sufficiently large,
we can deform~$G_q^{-1}\circ D\widetilde\gamma_q$ on~$D^{4k}_+$
relative to~$S^{4k-1}$
such that~$TS^m\times\R\cong\R^{m+1}$ is trivially mapped to itself.
Finally,
we can deform the action on the factor~$\R^{n-1}$ to the trivial one
over~$D^{4k}_+$.
This shows that the map~$G_q^{-1}\circ D\widetilde\gamma_q|_{S^m\times\{0\}}$
can be deformed to the identity,
relative to $S^{4k-1}\times TS^m\times\R$.
Then~$D\widetilde\gamma_q|_{S^m\times\{0\}}$ can be deformed to~$G_q$
relative to $S^{4k-1}\times TS^m\times\R$.
In particular,
the embedding~$\widetilde\gamma_q$ is isotopic to an embedding
that takes the form given in the proposition.
%that takes the form~$(x,y)
%\mapsto\(\gamma_q(x),(\gamma_q^{-1}\oplus\id_{\R^{m-n}})(y)\)$
%for~$q\in S^{4k-1}$.
\qed
\enddemo

In the remainder of this chapter,
we will use the example above to construct families
of bundles which are pairwise homeomorphic as bundles,
but not diffeomorphic in one the following two senses.

\Definition\BdlHomeoDef
Two fibre bundles~$p\colon M\to B$ and~$p'\colon M'\to B$ are
{\em homeomorphic (diffeomorphic) as bundles\/}
iff there exists a homeomorphism (diffeomorphism)~$\Phi$ such that
the left diagramme below commutes.
$$\CD
	M@>\Phi>\botsim>M'	&\kern1in&	M@>\Phi>\botsim>M'\\
	@VpVV@VVp'V				@VpVV@VVp'V\\
	B@>\id>>B		&&		B@>\Psi>\botsim>B
\endCD$$
We say that~$p$ and~$p'$ are {\em homeomorphic (diffeomorphic) as maps\/}
iff there exist homeomorphisms (diffeomorphisms)~$\Phi$ and~$\Psi$
such that the right diagramme above commutes.
\enddefinition


\Proposition\HatcherTrivialProp.
The bundle~$p\colon M_\gamma\to S^{4k}$ is homeomorphic as a bundle
to a trivial sphere bundle.

\Proof
%The double of~$S^{4k}\times S^m\times D^n$
%in~$M_\gamma\to S^{4k}$ forms an open trivial subbundle.
%In particular,
%we can pick two nowhere equal sections~$s_+$, $s_-$
%of the bundle~$p\colon M_\gamma\to S^{4k}$.
%The bundle~$M_\gamma\setminus s_-\to S^{4k}$ is a disc bundle.
%Because the vertical tangent bundle restricted to~$s_+$ is trivial,
%this disc bundle is homeomorphic to the trivial disc bundle
%by Alexander's trick.
%Since~$M_\gamma$ is the fibre-wise one-point compactification of~$M_\gamma\setminus s_-$,
%$M_\gamma$ is homeomorphic to a trivial sphere bundle.
It is well known that the diffeomorphism group of~$D^{m+n}$
with the $C^0$-topology retracts to~$O(m+n)$,
so~$M_\gamma\to S^{4k}$ is homeomorphic to some sphere bundle
with structure group~$O(m+n)$.
Our claim then follows because
the bundle~$p\colon M_\gamma\to S^{4k}$ admits sections~$C_1$, \dots, $C_6$,
and the restriction of~$TX$ to each of these sections is trivial.
\qed\enddemo

\Remark\HatcherStabilityRemark
In the construction above,
we did not care at all about the dimension of the fibre of~$p\colon M_\gamma\to S^{4k}$.
It should be clear that there are some numbers~$p$, $q$
such that one can always keep the dimension of the fibre at most~$pk+q$,
and it is interesting to know explicite values of~$p$ and~$q$.
This question is adressed in~\IBuch,
see Theorem~6.2.2 and Corollary~6.5.8.
\endremark


\Abschnitt\HatcherMorseSubsection The higher torsion invariants
of Hatcher's sphere bundle

We calculate Bismut-Lott's analytic torsion form and Igusa's higher
Franz-Reidemeister torsion of~$p\colon M_\gamma\to S^{4k}$
by specifying a fibre-wise Morse function on~$M_\gamma$.

We first construct a fibre-wise Morse function~$h\colon N_\gamma\to\R$
such that~$dh$ is positive on the outward normal vector to~$\del N_\gamma$.
We start with a fixed Morse function on~$S^m$,
having exactly two critical points:
a minimum of~$-3$ at the south pole,
and a maximum of~$-2$ at the north pole.
Adding a small positive function on~$D^n$
whose only critical point is a minimum~$0$ at the origin,
we obtain a Morse function on the trivial bundle~$S^m\times D^n\times S^{4k}\to S^{4k}$.

We then choose a Morse function on~$D^{m+1}\times D^{n-1}$
with only one critical point at~$(0,0)$ of value~$-1$,
such that~$D^{m+1}\times\{0\}$ is the descending cell,
and~$\{0\}\times D^{n-1}$ is the ascending cell.
Clearly,
these fibre-wise Morse functions can be patched together
to give a fibre-wise Morse function~$h\colon N_\gamma\to\R$
with three leaves of fibre-wise critical points of indices~$0$, $m$
and~$m+1$,
see \HatcherMorseFig.
We may assume that~$h^{-1}(0)=\del N_\gamma$.

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\centerline{\NeueFigur\HatcherMorseFig
\HatcherMorseFig. The negative gradient flow of the fibre-wise Morse function~$h$ on~$N_\gamma$.}
\endinsert

We finally get a Morse function on~$M_\gamma=N_\gamma\cup_{\id_{\del N_\gamma}}N_\gamma$
by taking~$h$ on one copy of~$N_\gamma$ and~$-h$ on the other.
Assuming that~$n\ge m+3$,
this new Morse function has six fibre-wise critical leaves~$C_1$, \dots, $C_6$
of indices~$0$, $m$, $m+1$, $n-1$, $n$ and~$m+n$,
with  values~$-3$, $-2$, $-1$, $1$, $2$ and~$3$, respectively.
In particular,
no critical leaf can ``cross over'' another critical leaf
of higher or equal index.
For this reason,
the ``simplicial superconnection''~$a$ of \CWBaseProp\
has no terms of higher degree,
i.e., $a_{\alpha\beta}(\sigma)=0$ unless~$\sigma\in S_0$.

Letting~$F=M_\gamma\times\C$ be trivial,
the bundle~$V=S^{4k}\times\C^6$ is also trivial.
We can arrange the superconnection~$A'$ of \PatchThm\
to equal~$a_0+\nabla^V$,
with
	$$a_0
	=\left(\smallmatrix
		0\\&0\\&1&0\\&&&0\\&&&1&0\\&&&&&0\endsmallmatrix\right)
	\in\Gamma(\End V)$$
parallel with respect to the trivial connection~$\nabla^V$.
Thus
	$$\(\Omega^*(S^{4k};V),A'\)
	=\(\Omega^*(S^{4k}),d\)\otimes(\C^6,a_0)$$
is a product,
and the bundle~$H=H^*(M_\gamma/S^{4k};\C)\to S^{4k}$ is trivial, too.
%so the Leray spectral sequence of the bundle~$M_\gamma\to S^{4k}$ with trivial coefficients
%degenerates at~$E_2=E_\infty$.

We also find that~$T \(A',g^V,h^V\)\in\R\subset\Omega^*(S^{4k})$
is homogeneous of degree~$0$ (it actually vanishes).
However,
the term
	$$\hat p_*{}^0\!J\((-1)^{\ind_h}\,(T^sX-T^uX)\)$$
in \MainTheorem\ does not vanish.
The critical leaves~$C_1$, $C_2$, $C_5$ and~$C_6$
lie within the trivial bundle
	$$\((S^m\times D^n)\cup_{\id_{S^m\times D^{n-1}}}(S^m\times D^n)\)
		\times S^{4k}
	\to S^{4k}$$
and come with trivial bundles~$T^sX$, $T^uX$,
so they do not contribute.
The bundles~$TX|_{C_3}$ and~$TX|_{C_4}$ are trivial,
because they arise from trivial bundles over~$D^{4k}_+$ and~$D^{4k}_-$
using the clutching function~$q\mapsto(\gamma_q\oplus\gamma_q^{-1}\oplus\id)$,
which is homotopic to the identity by~\KHomotopyFormel.
However,
the bundles~$T^{s,u}X|_{C_3}$ and~$T^{s,u}X|_{C_4}$
are not trivial because~$\gamma_q$ was chosen non-trivial.
The bundle~$T^sX|_{C_4}$ is isomorphic to~$T^uX|_{C_3}$,
and has clutching function~$\gamma^{-1}$.
If~$u$ is a generator of~$H^{4k}(S^{4k};\Z)\subset H^{4k}\dR(S^{4k})$,
if~$\gamma$ is the $n_\gamma$-th multiple of a generator of~$\ker J$,
and if~$0\ne b_k\in\Q$ is the coefficient discussed in~\IBuch, Remark~6.4.3,
then
	$$p_j\(T^uX|_{C_3}\)=p_j\(T^sX|_{C_4}\)
	=\cases
		n_\gamma\,b_k\,u	&\text{if~$j=k$, and}\\
		0			&\text{otherwise.}
	\endcases
	\Formel\HatcherPontrijaginFormel$$
By triviality of~$TX|_{C_3}$ and~$TX|_{C_4}$
and multiplicativity of the total Pontrijagin class,
	$$p_j\(T^uX|_{C_4}\)=p_j\(T^sX|_{C_3}\)=-p_j\(T^uX|_{C_3}\)
	=\cases
		-n_\gamma\,b_k\,u	&\text{if~$j=k$, and}\\
		0	&\text{otherwise,}
	\endcases
	\Formel\HatcherPontrijaginFormelTwo$$

Because the two flat bundles~$F\to M_\gamma$ and~$H\to S^{4k}$ are trivial,
we have a higher analytic torsion
class~$\Cal T (M_\gamma/S^{4k};\C)\in\Omega^{\ge2}(S^{4k})$
as in \AnalyticTorsionClassDef.
By \MainTheorem\ and \MainCorollary,
we get a parallel result to~\IBuch, Theorem~6.4.2.
Note that passing from Pontrijagin to Chern classes
in the definition of~${}^0\!J$ gives an extra factor~$2^{-k}$.
%\?Konstanten?

\Theorem\HatcherExpThm.
The analytic torsion class of the bundle~$M_\gamma\to S^{4k}$ above is given by
$$\align
	\Cal T (M_\gamma/S^{4k};\C)
	&=(-1)^{m+1}\,{}^0\!J\(T^sX|_{C_3}\)
		+(-1)^m\,{}^0\!J\(T^uX|_{C_3}\)\\
	&\qquad
		+(-1)^{n-1}\,{}^0\!J\(T^sX|_{C_4}\)
		+(-1)^n\,{}^0\!J\(T^uX|_{C_4}\)\\
	&=\cases
		(-1)^m\,2^{2-k}\,b_k\,\zeta'(-2k)\,u\in\Omega^{4k}\(S^{4k}\)
			&\text{if~$n+m$ is odd, and}\\
		0	&\text{if~$n+m$ is even.}\\
	\endcases
\endalign$$

In particular,
neither the sphere bundle~$M_\gamma\to S^{4k}$ nor the disc bundle~$N_\gamma\to S^{4k}$
are diffeomorphic to a trivial bundle.
Thus we recover a result by B\"okstedt~\Boekstedt,
cf.~\IBuch, Chapter~6.

Note on the other hand
that using Igusa's framing principle (Theorem~6.1.1 in~\IBuch),
the higher Franz-Reidemeister torsion of~\IBuch\
equals
	$$\tau (M_\gamma/S^{4k};\C)
	=(-1)^m\,2\,{}^0\!J\(T^uX|_{C_3}\)
		+(-1)^n\,2\,{}^0\!J\(T^uX|_{C_4}\)
	=\Cal T (M_\gamma/S^{4k};\C)\;.$$
This is consistent with \TeilZweiTheorem,
by which
	$$\Cal T (M_\gamma/S^{4k};\C)
	-\tau (M_\gamma/S^{4k};\C)
	=\int_{M_\gamma/S^{4k}}e(TX)\,{}^0\!J(TX)=0\;,$$
because the bundle~$TX|_C$ is trivial by construction.

\Remark\HatcherNonSmaleRem
Note that the superconnection~$A'=\nabla^V+a_0$ on~$V\to S^{4k}$
looks like a superconnection of a fibre-wise Morse function
with a fibre-wise gradient field satisfying the Smale transversality condition,
cf.~\BGzwei, section~5.
On the other hand,
if there was a fibre-wise Smale gradient field for~$h$,
we would have
$$\alignat2
	T^sX|_{C_2}&\cong T^sX|_{C_3}\oplus\R\;,&
	T^uX|_{C_2}\oplus\R&\cong T^uX|_{C_3}\;,\\
	T^sX|_{C_4}&\cong T^sX|_{C_5}\oplus\R\;,&
		\Mand
	T^uX|_{C_4}\oplus\R&\cong T^uX|_{C_5}
\endalignat$$
by~\BGzwei, section~7.10,
contradicting~\HatcherPontrijaginFormel, \HatcherPontrijaginFormelTwo.
This shows that the considerations in \ComplexKapitel\
are indeed needed to deal with this example.
\endremark

\Remark\HatcherHalfRemark
Consider once more the $(m+n)$-disc bundle~$N_\gamma\to S^{4k}$ above.
If~$m+n$ is odd,
the boundary~$\del N_\gamma$ becomes a trivial smooth sphere bundle,
possibly after replacing~$N_\gamma$ by~$N_\gamma\times D^{2l}\to S^{4k}$ for some~$l\ge 0$.
This means that instead of doubling~$N_\gamma$,
we could glue~$N_\gamma$ and a trivial disc bundle
to obtain another sphere bundle~$\overline N_\gamma\to S^{4k}$ with a fibre-wise Morse function
with only four fibre-wise critical points,
namely~$C_1$, $C_2$ and~$C_3$ on~$N_\gamma$ and a maximum on the trivial disc bundle.
Then clearly
	$$\Cal T (\overline N_\gamma/S^{4k};\C)
	={1\over 2}\,\Cal T (M_\gamma/S^{4k};\C)
	=(-1)^m\,2^{1-k}\,b_k\,\zeta'(-2k)\,u\in\Omega^{4k}\(S^{4k}\)\;.$$
In order to keep this chapter reasonably self-contained,
we will stick to the slightly more complicated bundle~$p\colon M_\gamma\to S^{4k}$
constructed above.
\endremark


\Abschnitt\GenFibreSubsection Exotic bundles
over~$S^{4k}$ with general fibres

We extend Hatcher's example to generate examples of bundles
with arbitrary fibre.
This involves a very simple kind of a vertical gluing formula.
A more general kind of vertical gluing will be discussed in~\TeilZwei.
The following construction is closely related to Corollary~6.5.8 in~\IBuch.
We need it as an intermediate step
for the general construction in the following subsection.

Fix~$k>0$
and let~$X$ be a compact manifold
of dimension~$\dim X=2l-1$,
such that~$l$ is sufficiently large with respect to~$k$.
Let~$p_X\colon X\times S^{4k}\to S^{4k}$ be the trivial bundle.
We fix a Morse function on~$X$,
which gives rise to a trivial fibre-wise
Morse function~$h_X\colon X\times S^{4k}\to\R$.
Any flat vector bundle~$F_X\to X$ of rank~$r$ with a parallel metric
pulls back to a flat bundle~$F_X\to X\times S^{4k}$.
%and we clearly have
%	$$\Cal T \((M\times S^{4k})/S^{4k};F_X\)
%	=T \((M\times S^{4k})/S^{4k};F_X,h_X\)
%	=0\in H\dR^{\even,\ge2}(S^{4k})\;.$$

Let~$p_\gamma\colon M_\gamma\to S^{4k}$
be the bundle with fibre~$S^{2l-1}$ constructed in \HatcherSubsection.
In particular,
there exists a fibre-wise Morse function~$h_\gamma\colon M_\gamma\to\R$
that has six critical points on each fibre.
Let~$S^{4k}\cong C_6\subset M_\gamma$ be the fibre-wise maximum,
then~$C_6$ has a trivial neighbourhood~$U\cong C_0\times D^{2l-1}$
in~$M_\gamma$.
%where~$h_\gamma$ takes the same form as in \SimplifyingAssumptions.

We can now replace a small neighbourhood of one of the local minima of~$h_X$
by~$M_\gamma\setminus U$,
obtaining a new bundle~$p'_\gamma\colon M'_\gamma\to S^{4k}$ with fibre~$X$
that is still homeomorphic as a bundle to~$p_X\colon X\times S^{4k}\to S^{4k}$.
By gluing the bundle~$F_X\to X\times S^{4k}$
with the trivial bundle of rank~$r$ over~$M_\gamma$,
we get a flat vector bundle~$F'_\gamma\to M'_\gamma$,
and~$F'_\gamma\to M'_\gamma$ is the pull back of~$F$
by the homeomorphism~$M'_\gamma\to X\times S^{4k}$ above.
In particular,
the fibre-wise cohomology of~$M'_\gamma\to S^{4k}$ is still
a trivial bundle over~$S^{4k}$ with fibre~$H^*(X;F_X)$.

%\?Figur?
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%	\psellipse[linestyle=dashed](6,3)(1,0.4)
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%	  \psellipse(6,3)(1,0.4)\endpsclip
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%	\rput{30}(5.6,1.5){\psellipse(0.7,0.2)}
%	\psellipse(6,4)(1,0.4)
%	\psarc(6,4){1}{0}{180}
%\EndFigur\ExpFigurZwei The bundle $p'_\gamma\colon M'_\gamma\to S^{4k}$



\Proposition\HatcherGenFibreProp{(cf. Corollary~6.5.8 in~\IBuch)}.
If~$F_X\to X$ is trivial,
we have
	$$\Cal T (M'_\gamma/S^{4k};\C)
	=2^{2-k}\,j\,b_k\,\zeta'(-2k)\,u
		\quad\in\quad H^{\even,\ge2}(B;\R)\;.$$

Thus again,
the higher analytic torsion is able to detect
the ``exotic'' differentiable structure of the homeomorphically
trivial bundle~$p'_\gamma\colon M'_\gamma\to S^{4k}$.

In~\TeilZwei,
we will exhibit a spectral sequence argument
which gives rise to a gluing formula for higher torsion invariants.
In particular,
we will find that
	$$\Cal T (M'_\gamma/S^{4k};F'_\gamma)
	=2^{2-k}\,j\,b_k\,\zeta'(-2k)\,u\,\rk F_X
		\quad\in\quad H^{\even,\ge2}(B;\R)\;.$$
Note that in the argument below,
there are no spectral sequence contributions by~\GenFibreCohFormel.

\Proof
After subtracting a suitable constant from~$h_\gamma$,
gluing with~$h_X$,
and smoothing along the common boundary,
we obtain a new fibre-wise Morse function~$h'_\gamma\colon M'_\gamma\to\R$.
Let~$C'_\gamma\subset M'_\gamma$ denote the critical points of~$h'_\gamma$,
then
	$$C'_\gamma=C^{\prime+}_\gamma\mathbin{\dot\cup}C^{\prime-}_\gamma\;,$$
where~$C^{\prime-}_\gamma$
are the five remaining critical levels of~$h_\gamma$,
and~$C^{\prime+}_\gamma$ are the remaining critical points of~$h_X$.
This induces a splitting
	$$V'_\gamma=V^{\prime+}_\gamma\oplus V^{\prime-}_\gamma\to B\;.
	\Formel\GenFibreSplitting$$
Since the descending cells of the critical levels~$C^{\prime+}_\gamma$
can meet the ascending cells
of~$C^{\prime-}_\gamma$
in any possible way,
we cannot conclude any more that the superconnection~$A'=\nabla^{V'_\gamma}+a'$
of \PatchSuperconnectionDefinition\
on the family Thom-Smale complex~$\Omega^*(S^{4k};V)$
still takes the form~$A'=\nabla^{V'_\gamma}+a_0$
with~$a_0\in\Omega^0(B,\End V)$.
We may however conclude that
	$$a'
	=\pmatrix b'&d'\\0&c'\endpmatrix$$
with respect to~\GenFibreSplitting,
such that~$b'\in\Omega^0(B;\End V^{\prime +}_\gamma)$
and~$c'\in\Omega^0(B;\End V^{\prime-}_\gamma)$
are both of horizontal degree~$0$
and parallel with respect to~$\nabla^{V'_\gamma}$.

We note that the fibre-wise cohomologies of~$(V^{\prime+}_\gamma,b')$
and~$(V^{\prime-}_\gamma,c')$ are given by
	$$H^j\(V^{\prime+}_\gamma,b'\)
	=\cases
		0		&j=0\;,\\
		H^j(X;\C)	&j>0\;,
	\endcases
		\Mand
	H^j\(V^{\prime-}_\gamma,c'\)
	=\cases
		\C		&j=0\;,\\
		0		&j>0\;,
	\endcases$$
because~$H^*\(V^{\prime-}_\gamma,c'\)$ is the cohomology of a disc.
This implies in particular that
	$$H^*\(V'_\gamma,a'_0\)
	=H^*(X;\C)
	=H^j\(V^{\prime+}_\gamma,b'\)\oplus H^j\(V^{\prime-}_\gamma,c'\)\;,
	\Formel\GenFibreCohFormel$$
so the spectral sequence associated with the
filtration~$V'\gamma\supset V^{\prime +}_\gamma\supset 0$
degenerates already at~$E_1=E_\infty$.

We now need the analytic analogue of Igusa's splitting principle
(\IBuch, Lemma~5.7.8).
Multiplying the (parallel) metric~$g^V$
restricted to~$V^{\prime-}_\gamma$
% the remaining five critical points of~$h_\gamma$
by a small positive constant~$T^{-2}$
has the same effect on~$T(A',g^V,h^V)$ as conjugating~$a'$
by~$\left(\smallmatrix 1&0\\0&T\endsmallmatrix\right)$.
In the limit~$T\to\infty$,
only terms in~$\Omega^0(S^{4k};\End V)$ remain,
giving a torsion form in~$\Omega^0(S^{4k})$.
The class~$T (M'_\gamma/S^{4k};\C,h)\in H^{\even,\ge 2}(S^{4k})$
is unaffected by this deformation for all~$T>0$
according to \MorseTorsionCor.
Because the rank of the fibre-wise cohomology
does not jump in the limit~$T\to\infty$ by~\GenFibreCohFormel,
we find that
	$$T (M'_\gamma/S^{4k};\C,h)
	=T \(\nabla^{V^{\prime+}_\gamma}+b',
			g^{V^{\prime+}_\gamma}\)\dg{\ge2}
		+T \(\nabla^{V^{\prime-}_\gamma}+c',
			g^{V^{\prime-}_\gamma}\)\dg{\ge2}
	=0\;.$$

Because the bundles~$T^sX$ and~$T^uX$ are trivial along~$C^{\prime+}_\gamma$,
and the $\^0J$-classes of~$(T^sX-T^uX)|_{C^{\prime-}_\gamma}$
are the same as in \HatcherExpThm,
\MainTheorem\ implies that
	$$\Cal T (M'_\gamma/S^{4k};\C)
	=2^{2-k}\,j\,b_k\,\zeta'(-2k)\,u\;.\qed
	\Formel\HatcherGenFibreFormel$$
\enddemo


\Abschnitt\HatcherMoreSubsection Exotic structures on general fibre bundles

We show that we can alter the differentiable structure on a quite general
fibre bundle by ``horizontally'' gluing with the bundle~$M'_\gamma$
of the previous subsection.
In the following subsection,
we use the bundles obtained this way to prove \ApplThm.

%This shows that the higher analytic torsion
%gives rise to non-trivial cohomology classes of~$\Diff_0(X)$
%equipped with the $C^\infty$-topology,
%that do not arise as cohomology classes of~$\Homeo_0(X)$
%in a stable range,
%for all odd-dimensional, compact manifolds~$X$,
%cf.~\IBuch, Corollary~6.5.8.
%Note that~$\Diff_0(X)$ and~$\Homeo_0(X)$ here denote
%the connected components of the identity.
%We now combine the ``vertical'' glueing of \HatcherGeneralSphereExp\
%with the ``horizontal'' glueing of \GenFibreSubsection.

Let~$p=p_0\colon M_0\to B$ be a smooth proper submersion of compact manifolds,
and assume that the base is orientable with~$\dim B=4k$,
and that~$\dim X=\dim M_0-\dim B=2l-1$,
such that~$l$ is sufficiently large with respect to~$k$.
Let~$F\to M_0$ be a unitarily flat vector bundle.
If~$H=H^*(M_0/B;F)\to B$ admits a parallel metric,
the higher torsion~$\Cal T (M_0/B;F)\in H^{\even,\ge2}\dR(B)$
is well-defined.
Otherwise,
we fix an arbitrary metric~$g^H$ on~$H$
and regard the class
	$$\Cal T (M_0/B;F,g^H)
	=\Bigl(\Cal T \(T^HM_0,g\TX,\nabla^F,g^F\)
		+\chslo\(\nabla^H,g^H,g^H_{L_2}\)\Bigr)\dg{\ge 2}$$
in~$\Omega^\even(B)/d\Omega^\odd(B)$,
which is independent of the choices of~$T^HM_0$, $g\TX$
and the parallel metric~$g^F$,
but depends on~$g^H$.

Let~$X_0=p^{-1}(b_0)\subset M_0$ be a fixed fibre of~$p$ for a fixed~$b_0\in B$,
and let~$F_0$ be the restriction of~$F$ to~$X_0$.
Let~$M'_\gamma\to S^{4k}$ be constructed as in the previous subsection
with fibre~$X=X_0$.
We may assume that the bundles~$M_0\to B$ and~$M'_\gamma\to S^{4k}$
are trivialised in small discs~$D^{4k}_0$ around~$b_0\in B$
and~$D^{4k}_-\subset S^{4k}$,
and we pick the horizontal subbundles~$T^HM_0$ and~$T^HM'_\gamma$
according to that trivialisation.
Fixing a metric~$g^X$ on~$X$ and~$g^H_0$ on~$H(X;F)$,
we define~$g\TX$ and~$g^H_0$ over~$D^{4k}_0$ and~$D^{4k}_-$
using the trivialisation above together with the given~$g^X$ and~$g^H_0$.
Then the data defining the classes~$\Cal T \(M_0/B;F,g^H\)$
and~$\Cal T \(M'_\gamma/S^{4k};E_i\)$
coincide on~$\del D^{4k}_0$ and~$\del D^{4k}_-$,
which allows us to glue~$M_0|_{B\setminus D^{4k}_0}$
and~$M'_\gamma|_{S^{4k}\setminus D^{4k}_+}$
along their common boundary,
obtaining a new bundle
	$$p_1\colon M_1\to B_1\;,$$
where~$B_1$ is naturally diffeomorphic to~$B$.
In other words,
we replace the restriction of~$M_0$ to~$D^{4k}_0$
by the restriction~$M'_\gamma$ to~$D^{4k}_+$.

%\?Bildchen?

Then~$p_1\colon M_1\to B_1\cong B$
is homeomorphic to~$p_0\colon M_0\to B$ as a bundle
by \HatcherTrivialProp.
Because the bundle~$F|_X$ is diffeomorphic and isometric
to~$E_j|_{p^{-1}(b_j)}$,
we can also identify the flat vector bundles~$F$ and~$E_j$
along~$M|_{\del D}$,
obtaining a new flat vector bundle~$F_1\to M_1$.

We note that all secondary data~$T^HM$, $g\TX$, $g^F$ and~$g^H_0$
has carefully been chosen so as to be compatible with the gluing process.
We now restrict our attention to the trivial bundle~$F=M_0\times\C$.
\HatcherGenFibreProp\ implies that
$$\multline
	\Cal T \(T^HM_1,g^{TX_1},d,g^{\roman{triv}}\)
		+\chslo\(\nabla^H,g^H_{0,L_2},g^H_{1,L_2}\)\\
	=\Cal T \(T^HM_0,g^{TX_0},d,g^{\roman{triv}}\)
		+2^{2-k}\,b_k\,\zeta'(-2k)\,u
	\quad\in\quad\Omega^*(B)/d\Omega^*(B)\;,
\endmultline\tag\FormelNummer\HatcherGenFormFormel$$
where~$g^H_{0,L_2}$ and~$g^H_{1,L_2}$ are the $L_2$-metrics
on~$H\cong H^*(M_0/B)\cong H^*(M_1/B)$,
and where~$u$ represents a generator of~$H^{4k}(B;\Z)\subset H^{4k}\dR(B)$.
Note that both~$\chslo\(\nabla^H,g^H_{0,L_2},g^H_{1,L_2}\)$
and~$u$ are given by smooth forms supported on~$D_0^{4k}\subset B$,
which are well-defined modulo differentials of forms supported on~$D_0^{4k}$.

Because the bundles~$p_1$ and~$p_0$ are naturally homeomorphic,
we can identify the fibre-wise cohomology bundles~$H(M_1/B;\C)\to B_1\cong B$
and~$H(M_0/B;\C)\to B$ as flat vector bundles,
and this identification is canonical
on~$B\subset D_0^{4k}=B_1\subset D^{4k}_+$.
In particular,
the metric~$g^H$ can be regarded as a metric on~$H(M_1/B;\C)$ as well.
Passing to classes,
equation~\HatcherGenFormFormel\ becomes
	$$\Cal T \(M_1/B;\C,g^H\)
	=\Cal T \(M_0/B;\C,g^H\)
		+2^{2-k}\,b_k\,\zeta'(-2k)\,u
	\quad\in\quad\Omega^*(B)/d\Omega^*(B)\;.$$
Iterating this process gives bundles~$p_j\colon M_j\to B$ for all~$j\ge 0$.
If we repeat the constructions above with~$\gamma$ replaced by~$\gamma^{-1}$
in~\HatcherConstruction,
we also obtain bundles~$p_j\colon M_j\to B$ for all~$j<0$,
such that
	$$\Cal T \(M_j/B;\C,g^H\)
	=\Cal T \(M_0/B;\C,g^H\)
		+2^{2-k}j\,b_k\,\zeta'(-2k)\,u
	\quad\in\quad\Omega^*(B)/d\Omega^*(B)
	\Formel\HatcherGenExpFormel$$
for all~$j\in\Z$.
Let us summarize our examples so far.


\Corollary\HatcherCorollary.
Let~$p\colon M\to B$ be a smooth fibre bundle
with compact fibre~$X$ of dimension~$2l-1$
and an orientable, $4k$-dimensional, compact base space~$B$,
such that~$l$ is sufficiently large with respect to~$k$,
and let~$g^H$ be a fixed metric on the fibre-wise cohomology~$H(M/B;\C)\to B$.
Then
there exist infinitely many bundles~$p_j\colon M_j\to B$ for~$j\in$, $j>0$,
with fibres diffeomorphic to~$X$,
that are all homeomorphic to~$p=p_0$ as bundles,
but with pairwise different values
of
	$$\int_B\Cal T \(M_j/B;\C,g^H\)\;.$$

\Remark\HatcherCorRemark
The result above is related to Proposition~6.5.7 in~\IBuch\
for trivial bundles over arbitrary compact orientable base spaces.
Indeed,
if the bundle~$H(M/B;\C)\to B$ is trivial,
then we could have stated \HatcherCorollary\ using
Igusa's higher Franz-Reidemeister torsion~$\tau(M_j/B;\C)$.
However,
Igusa's~$\tau(M_j/B;\C)$ has so far only been defined
if~$H(M/B;\C)\to B$ is trivial,
so we use~$\Cal T (M_j/B;F,g^H)$ to get a more general statement.
Since we used \MainTheorem\ to relate the higher analytic torsion
of Hatcher's sphere bundles to~$\tau(M_\gamma/B,\C)$,
\HatcherCorollary\ is in fact an application
of the methods developped in this paper.
\endremark


\Remark\HatcherCorRemZwei
At this point,
it is natural to ask for those cohomology classes on~$B$
that can be realised as differences of higher torsion classes
of fibre bundles that are homeomorphic to a given~$M\to B$ as bundles.
\HatcherCorollary\ only realises certain multiples
of the fundamental class as such differences.
However,
the gluing techniques used in the proof immediately give a slightly
more general result,
which applies in particular to the trivial bundle~$M\times B\to B$.

Let~$p\colon M\to B$ be a smooth fibre bundle
with compact fibre~$X$ of dimension~$2l-1$
and base space~$B$.
Assume that~$B$ contains a properly immersed oriented submanifold~$A$
of codimension~$4k>0$ with trivial normal bundle,
which has a lift~$N\subset M$,
such that~$l$ is sufficiently large with respect to~$k$,
and the vertical tangent bundle~$TX|_N$ is trivial.
Let~$\alpha\in H^{4k}(B,\R)$ be the Poincar\'e dual of~$A$.
Let~$g^H$ be a fixed metric on the fibre-wise cohomology~$H(M/B;\C)\to B$.
Then
there exist infinitely many bundles~$p_j\colon M_j\to B$ for~$j\in$, $j>0$,
with fibres diffeomorphic to~$X$,
that are all homeomorphic to~$p=p_0$ as bundles,
but with
	$$\Cal T \(M_1/B;\C,g^H\)
	=\Cal T \(M_0/B;\C,g^H\)
		+2^{2-k}\,b_k\,\zeta'(-2k)\,\alpha
	\quad\in\quad\Omega^*(B)/d\Omega^*(B)\;.
	\Formel\HatcherCorRemZweiFormel$$
Clearly, equation~\HatcherGenExpFormel\
is contained here as a special case where~$A$ is a single point.

To proof this claim,
we may assume that~$N$ is embedded in~$M$ by transversality.
Then let~$U$ denote a tubular neighbourhood of~$N$,
fix~$p\in N$, and let~$D_p$ be a fibre of the bundle~$U\to N$.
We may assume that~$U$ fibres
over a $\dim B$-dimensional submanifold~$V\subset M$ with fibre~$D^{2l-1}$,
where~$V$ is transversal to the fibres of~$M\to B$.
By our assumptions, the bundle~$U\to V$ is trivial,
and we have a projection
	$$\pi\colon V\to D^{4k}
		\Mtext{with}
	U=\pi^*\(D^{2l-1}\times D^{4k}\)\to V\;.$$
Thus,
we can replace~$U$ by the pullback
of the bundle~$M_\gamma|_{D^{4k}_+}\to  D^{4k}$
with a neighbourhood of the critical level~$C_1$ removed
as in \GenFibreSubsection.
Then~\HatcherCorRemZweiFormel\ follows by the same arguments
as~\HatcherGenFibreFormel\ and~\HatcherGenExpFormel.
\endremark




\Abschnitt\HomeoDiffeoSubsection Homeomorphism classes
and diffeomorphism classes of submersions,
and a proof of Theorem 0.2

In the last section,
we have exhibited a method to construct an infinite family of
pairwise homeomorphic fibre-bundles with pairwise different
higher analytic torsion classes,
starting from an arbitrary fibre-bundle with compact fibre
and compact orientable base of the right dimensions.
To prove \ApplThm,
we have to show that an infinite subfamily has diffeomorphic
total spaces
and that these bundles are pairwise not diffeomorphic as bundles,
which is again a consequence of \HatcherCorollary\ above.
In \HatcherTheorem,
we give a necessary condition for a diagramme
$$\CD
	M@>\Phi>\botsim>M\\
	@Vp_iVV@VVp_jV\\
	B@>\Psi>\botsim>B
\endCD$$
as in \BdlHomeoDef\ to commute,
where~$\Phi$ and~$\Psi$ are now diffeomorphisms.
This proves \ApplThm\ and gives the refinement mentioned in the introduction.

Therefore,
let~$p\colon M\to B$ be a smooth fibre bundle
with compact fibre~$X$ of dimension~$2l-1$
and an orientable, $4k$-dimensional, compact base space~$B$,
such that~$l$ is sufficiently large with respect to~$k$.
Let~$F\to M$ be a unitarily flat vector bundle,
and let~$p_j\colon M_j\to B$ and~$F_j\to M_j$ be constructed
as in \HatcherCorollary.
%Let~$p_j\colon M_j\to B$ be constructed as in \HatcherCorollary.
We do not require that the pair-wise isomorphic flat vector
bundles~$H_j=H^*(M_j/B;\C)\to B$ carry a parallel metric.

A careful investigation of the construction in \HatcherMoreSubsection\ shows
that~$M_j$ arises from~$M$ by removing several subsets~$D_i$ diffeomorphic
to~$D^{4k}\times D^{2l-1}$ and gluing them back in a different way,
such that before and after the gluing,
the fibres of~$p$ respectively~$p_j$ intersect~$D^{4k}\times D^{2l-1}$
either in~$\{b\}\times D^{2l-1}$ for some~$b\in D^{4k}$ or not at all.
After sliding the sets~$D_i$ around within~$M$,
we may assume that all~$D_i$ are contained in a larger
set~$\overline D\subset M$ diffeomorphic to~$D^{4k}\times D^{2l-1}$,
such that each inclusion~$D_i\into\overline D$
projects to the identity on~$D^{4k}$.
If we denote the corresponding subset of~$M_j$ by~$\overline D_j$,
then~$\overline D_j$ is still diffeomorphic to~$D^{4k}\times D^{2l-1}$,
where the fibre of~$p_j$ intersect~$D^{4k}\times D^{2l-1}$
either in~$\{b\}\times D^{2l-1}$ for some~$b\in D^{4k}$ or not at all.

We construct~$M_j$ for~$j<0$ similarly
by glueing in~$\abs j$ copies of another exotic sphere bundle
where we start with the inverse of~$\gamma$ in~$\pi_{4k-1}\(O(m)\)$.
Then~\HatcherGenExpFormel\ still holds.

\Proposition\HatcherDiffeoProp.
There exists~$j_0\in\Z$
such that there exists a diffeomorphism
	$$\Theta_j\colon M\to M_j
		\Mtext{with}
	\Theta_j|_{M\setminus\overline D}
	=\id_{M\setminus\overline D}
	\colon M\setminus\overline D\to M_j\setminus\overline D_j$$
for all~$j\in j_0\Z$.
In particular,
$\Theta_j$ lifts to a parallel map
	$$\overline\Theta_j\colon F\to F_j
		\Mtext{with}
	\overline\Theta_j|_{F|_{M\setminus\overline D}}
	=\id_{F|_{M\setminus\overline D}}
	\colon F|_{M\setminus\overline D}\to
		F_j|_{M_j\setminus\overline D_j}\;.$$

\Proof
By the constructions
in~Subsections~\NummerVon\GenFibreSubsection\
and~\NummerVon\HatcherMoreSubsection,
there exists a possibly exotic $(4k+2l-1)$-dimensional sphere~$\Sigma$
such that
	$$M_j=M\#\underbrace{\Sigma\#\cdots\#\Sigma}_{j\text{ times}}$$
for all~$j>0$,
where the connected sum operation takes places inside~$\overline D$.
For~$j<0$,
we have to take the connected sum with $\abs j$ copies
of another possibly exotic sphere~$\Sigma'$.
Because the group of exotic spheres is finite in each dimension,
the first claim now follows.

The second claim is then clear
because~$F$ and~$F_j$ are up to isomorphism
the unique extensions
of~$F|_{M\setminus\overline D}$ to~$M$ and~$M_j$,
respectively,
thus~$\Theta_j^*F_j$ and~$F$ are isomorphic as flat bundles
and can thus be identified.
\qed\enddemo

It is an immediate consequence of \HatcherCorollary\
that the maps~$p$ and~$p_j\circ\Theta_j$ differ somewhere on~$\overline D$
for~$j\ne 0$.
%at least,
%if the bundle~$H\to B$ admits a parallel metric.
On the other hand,
recall that by \HatcherTrivialProp,
there always exists a homeomorphism~$\Theta'_j\colon M\to M_j$
supported on~$\overline D$
such that~$p=p_j\circ\Theta'_j$.
In the remainder of this chapter,
we want to generalise and strengthen this observation.
But first of all,
we need some facts on Chern-Simons classes.

Let~$(F,\nabla^F)\to M$ be a flat vector bundle,
and let~$\Diff(M;F)$ denote the group of fibre preserving,
fibre-wise linear, parallel diffeomorphisms~$\overline\Phi\colon F\to F$.
An element~$\Phi\in\Diff(M;F)$
is a parallel lift of a diffeomorphism of~$M$,
which by an abuse of notation will again be named~$\Phi$.
If~$M$ is oriented,
let~$\Diff^+(M;F)\subset\Diff(M;F)$ denote the subgroup
of lifts of orientation preserving diffeomorphisms of~$M$.

Let~$\overline F=F\times[0,1]$ denote the trivial extension of~$F$
to~$\overline M=M\times[0,1]$.
For~$\Phi\in\Diff(M;F)$,
the bundle~$F$ naturally extends to a flat vector bundle
	$$T_\Phi F=\overline F/\sim
		\quad\to\quad
	T_\Phi M=\overline M/\sim$$
over the mapping torus~$T_\Phi M$,
where~``$\sim$'' is generated by~$(v,1)\sim(\Phi(v),0)$ for all~$v\in F$,
respectively~$(p,1)\sim(\Phi(p),0)$ for all~$p\in M$.
If~$g^F$ is any metric on~$F$,
let~$g^F_t$ be a family of metrics that interpolates
between~$g^F_0=g^F$ and~$g^F_1=\Phi^*g^F$.
Let~$g^{\overline F}$ denote the corresponding metric on~$\overline F$,
then~$g^{\overline F}$ descends to a metric~$g^{T_\Phi F}$ on~$T_\Phi F$.

Now we assume in addition that~$M$ is oriented
and that~$\Phi$ preserves the orientation.
By the definition of the Chern-Simons class~$\chslo$,
we have
	$$\int_M\chslo\(\nabla^F,g^F,\Phi^*g^F\)
	=\int_{M\times[0,1]}
		\cho\Bigl(\nabla^{\overline F},g^{\overline F}\Bigr)
	=\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]\;,
		\Formel\ChsloHomoFormel$$
independent of the choice of~$g^F$.

\Proposition\ChsloHomoProp.
Assume that~$M$ is oriented,
then the map
	$$\Diff^+(M;F)\to\R
		\Mtext{with}
	\Phi\mapsto\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]$$
is a group homomorphism.
\parr
Assume in addition that~$\dim M>0$.
Then~$\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]$
depends only on the homotopy type of the action of~$\Phi$ on~$M$.
If~$F$ carries a parallel metric
or if~$\Phi^k\sim\id_M$ for some~$k\ne 0$,
then~$\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]=0$.

To see that the invariant~$\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]$
is nontrivial,
one may consider a surface~$M$ of genus~$\ge 2$
and a diffeomorphism~$\Phi$ such that~$T_\Phi M=\Bbb H^3/\Gamma$
becomes hyperbolic with fundamental group~$\Gamma\subset SL(2,\C)$.
We then choose~$F\to M$ and a lift~$\Phi$ to~$F$
such that~$T_\Phi F$ becomes the flat bundle
associated to the adjoint representation of~$SL(2,\C)$ restricted to~$\Gamma$.
In this case, $\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]$
is a nonzero multiple of the hyperbolic volume of~$T_\Phi M$.


\Proof
The map~$\chslo(M;F)$ is a homomorphism by~\ChsloHomoFormel,
because for~$\Phi$, $\Phi'\in\Diff^+(M;F)$,
we have
	$$\chslo\(\nabla^F,g^F,(\Phi\circ\Phi')^*g^F\)
	=\chslo\(\nabla^F,g^F,\Phi^*g^F\)
		+\Phi^*\chslo\(\nabla^F,g^F,(\Phi')^*g^F\)$$
modulo exact forms.

We now assume that~$\dim M>0$.
Let the action of~$\Phi$ on~$M$ be homotopic to the identity,
then~$T_\Phi M$ is diffeomorphic to~$M\times S^1$.
Because flat vector bundles on~$M\times S^1$ are given by representations
of~$\pi_1(M)\times\Z$,
the bundle~$T_\Phi F$ decomposes into parallel subbundles of the form
	$$\pi_1^*F_M\otimes\pi_2^*F_{S^1}\;,$$
where~$F_M\to M$ and~$F_{S^1}\to S^1$ are flat vector bundles.
By Proposition~1.13 in~\BL,
we have
	$$\cho\Bigl(\nabla^{\pi_1^*F_M\otimes\pi_2^*F_{S^1}}\Bigr)
	=\pi_1^*\cho\(\nabla^{F_M}\)\,\rk F_{S^1}
		+\rk F_M\,\pi_2^*\cho\(\nabla^{F_{S^1}}\)\;,$$
thus in particular
	$$\cho\Bigl(\nabla^{\pi_1^*F_M\otimes\pi_2^*F_{S^1}}\Bigr)
		\bigl[M\times S^1\bigr]
	=0\;.$$
If~$\Phi^k$ is homotopic to the identity,
the homomorphism property implies that
	$$k\,\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]
	=\cho\(\nabla^{T_{\Phi^k}F}\)[T_{\Phi^k}M]=0\;.$$

Assume that~$g^F$ is a parallel metric,
then~$\Phi^*g^F$ is also parallel,
and we may interpolate by a family~$g^F_t$ of metrics
that are parallel on~$M\times\{t\}$ for each~$t$.
Then~$\cho\(\nabla^{T_\Phi F},g^{T_\Phi F}\)\in \Omega^1\(T_\Phi M\)$,
in particular
	$$\cho\(\nabla^{T_\Phi F}\)[T_\Phi M]=0\;.\quad\qed$$
\enddemo



We now want state the main application of
Subsections~\NummerVon\GenFibreSubsection\
and~\NummerVon\HatcherMoreSubsection,
which allows us to give many examples of manifolds~$M$, $B$
that allow infinitely many smooth maps~$M\to B$
that are pair-wise conjugate by homeomorphisms,
but not by diffeomorphisms.
Whenever the dimensions match and~$B$ is compact and orientable,
we can construct infinitely many proper submersions~$M\to B$
that are homeomorphic but not diffeomorphic as bundles.
Let us write~$q_j=p_{jj_0}\circ\Theta_{jj_0}$,
where~$j_0>0$ and~$\Theta_{jj_0}$ are given by \HatcherDiffeoProp.

\Theorem\HatcherTheorem.
Let~$p\colon M\to B$ be a proper submersion.
%and let~$(F,\nabla^F)\to M$ be a vector bundle with a parallel metric~$g^F$.
Assume that~$B$ is compact, orientable, and of dimension~$4k>0$.
Let the bundles~$\(q_j\colon M\to B\)_{j\in\Z}$ with~$q_0=p$
be constructed as above.
Assume that for~$i$, $j\in\Z$ with~$i\ne j$,
there exists diffeomorphisms~$\Phi\in\Diff(M)$ and~$\Psi\in\Diff^+(B)$
such that the diagramme
$$\CD
	M@>\Phi>>M\\
	@Vq_iVV@VVq_jV\\
	B@>\Psi>>B
\endCD$$
commutes.
Then~$\Psi\in\Diff^+(B;H)$ for~$H=H^*(M/B;\C)\to B$,
and
	$$\cho\(\nabla^{T_\Psi H}\)[T_\Psi B]
	=2^{2-k}\,(j-i)\,j_0\,b_k\,\zeta'(-2k)\;.$$
In particular,
this is only possible if~$H\to B$ does not carry a parallel metric,
and if~$\Psi$ does not represent a torsion element
in the mapping class group.

\demo{Proof\/ \rm of \ApplThm}
By \HatcherTrivialProp,
the maps~$q_j$ are pairwise homeomorphic as bundles,
in the sense of \BdlHomeoDef.
In particular,
they are also homeomorphic as maps from~$M$ to~$B$.
\ChsloHomoProp\ and \HatcherTheorem\ have the following two easy consequences.
\roster
\item
If the bundle~$H=H^*(M/B;\C)$ carries a parallel metric,
then the maps~$q_j$ are not pairwise conjugate by diffeomorphisms,
in particular,
they are not pairwise isomorphic as smooth fibre bundles.
\item
If the bundle~$H$ does not admit a parallel metric,
then the maps~$q_j$ are not pairwise isomorphic as smooth fibre bundles,
because~$\id_B$ clearly is a torsion element in the mapping class group.
\endroster
This proves \ApplThm.
\qed\enddemo


\demo{Proof\/ \rm of \HatcherTheorem}
We know from \HatcherTrivialProp\ and the construction in
Subsections~\NummerVon\GenFibreSubsection\
and~\NummerVon\HatcherMoreSubsection\
that the fibre bundles~$q_i$, $q_j\colon M\to B$ are homeomorphic to~$M\to B$
by a homeomorphism that is supported on a contractible subset of~$M$.
To avoid confusion,
we will write~$q_i$ and~$q_j$ instead of~$M/B$ whenever necessary.
In particular,
the fibrewise cohomologies~$H^*(q_i;F)$ and~$H^*(q_j;F)\to B$
can be identified with the original~$H\to B$.
By functoriality of the fibre-wise cohomology,
the diffeomorphism~$\Psi$ lifts to~$\Diff^+(B;H)$.

If~$g^H_0$ is any metric on~$H$,
the class
	$$\Cal T \(T^HM,g\TX,d,g^{\roman{triv}}\)
		-\chslo\(\nabla^H,g^H_{L_2},g^H_0\)
	\quad\in\quad\Omega^\even(B)/d\Omega^\odd(B)$$
is independent of the choices of~$T^HM$ and~$g\TX$
by \FVCor\ and \MVBTorsionThm.
Since we assume that~$\dim B>0$,
we obtain an invariant
	$$\Cal T \(M/B,\C;g^H_0\)
	=\int_B\Bigl(\Cal T \(T^HM,g\TX,d,g^{\roman{triv}}\)
		-\chslo\(\nabla^H,g^H_{L_2},g^H_0\)\Bigr)\;,$$
which depends on the choice of~$g^H_0$ by
	$$\Cal T \(M/B,\C;g^H_1\)-\Cal T \(M/B,\C;g^H_0\)
	=-\int_B\chslo\(\nabla^H,g^H_0,g^H_1\)\;.
	\Formel\HatcherThmFormelEins$$

By naturality of torsion,
we find that
	$$\Cal T \(q_j,\C;g^H_0\)
	=\Cal T \(q_i,\C;\Psi^*g^H_0\)\;.
	\Formel\HatcherThmFormelZwei$$
On the other hand,
it follows from~\HatcherGenExpFormel\ that
	$$\Cal T \(q_j,\C,g^H_0\)
		-\Cal T \(q_i,\C,g^H_0\)
	=2^{2-k}(j-i)\,j_0\,b_k\,\zeta'(-2k)\,u\;,
	\Formel\HatcherThmFormelDrei$$
where~$u$ is the positive generator of~$H^{4k}(B;\Z)$.
The formula in the theorem follows
from~\ChsloHomoFormel\
and \HatcherThmFormelEins--\HatcherThmFormelDrei.
The remaining conclusions are a consequence of \ChsloHomoProp.
\qed\enddemo


\Abschnitt\ExampleLottSubsection
Hatcher's example and Lott's secondary $K$-theory

We reformulate \HatcherCorollary\ in the language of Lott's
secondary $K$-theory~$\Kqo_R$ of~\LottSec,
cf. \MainLottSubsection.
We will see that the different maps~$p_j\colon M\to B$
constructed in Subsections~\NummerVon\HatcherMoreSubsection\
and~\NummerVon\HomeoDiffeoSubsection\
give rise to different pushdown maps~$p_{j!}\colon\Kqo_R(M)\to\Kdo_R(B)$,
at least if~$\pi_1(B)$ is finite.
We also discuss this fact in the light of Conjecture~1 in~\LottSec,
which relates the pushdown~$p_!$ to a Becker-Gottlieb transfer.

Thus,
let~$p\colon M\to B$ be a smooth fibre bundle with compact fibres,
and assume that~$B$ is compact oriented with~$\dim B=4k$
and the dimension of the fibre is odd and sufficiently large.
Let~$\(q_i\colon M\to B\)_{i\in\Z}$ with~$q_0=p$
denote the maps considered in \HatcherTheorem.
We take~$R$ to be one of~$\Z$, $\R$, $\C$ for simplicity
and let~$\rho\colon R\to\C$ be the inclusion.
Then by~\OurPshriekFormel\ and \MainLottThm,
	$$q_{i!}\bigl[M\times R,g^{\triv},0\bigr]
	=q_{0!}\bigl[M\times R,g^{\triv},0\bigr]
	+[0,0,i\,u]
	\Formel\LottHatcherElt$$
for some nonzero~$u\in H^{4k}(S^{4k},\R)$.
By \LottSplitProp\ below,
the elements of~$\Kqo_R(B)$ given by~\LottHatcherElt\ are pairwise different,
at least if~$\pi_1(B)$ is finite.

In the following,
we let~$B$ be a smooth manifold with finite fundamental group.
Then we claim that there exists a well-defined map
	$$\chdo\colon\Kdo\to\Omega^{\even,\ge2}(B)/d\Omega^\odd(B)\;.$$
To define~$\chdo$ on a generator~$(F,g^F,\eta)$,
we choose a parallel metric~$g^F_0$ on~$F_\C$,
which is possible because~$B$ is simply connected,
and we set
	$$\chdo(F,g^F,\eta)
	=\chslo\(\nabla^F,g^F_0,g^F\)\dg{\ge2}-\eta\dg{\ge2}\;.
	\Formel\LottChdoDef$$
This is independent of~$g^F_0$
because for two parallel metrics~$g^F_0$, $g^F_1$ on~$F$,
we have~$\chslo(\nabla^F,g^F_0,g^F_1)\in\R\subset\Omega^0(B)$
modulo exact forms.

It is easy to check that~\LottChdoDef\ complies with~\LottVariationFormel\
due to the properties of~$\chslo$.
We check that~\LottChdoDef\ is compatible
with the $\Kdo_R$-relations~\KdachRelations.
Regard a short exact sequence as in~\LottShortSeqEins.
By~\LottVariationFormel,
we may choose parallel metrics~$g^{F_i}$ on the bundles~$F_{i,\C}$.
Then the associated Bismut-Lott torsion of~\LottShortSeqZwei\ vanishes
in degree~$\ge 2$,
so~\LottChdoDef\ is compatible with~\KdachRelations\
and the map~$\chdo$ is well-defined.

\Proposition\LottSplitProp.
Assume that~$\pi_1(B)$ is finite,
then
	$$\Kqo_R(B)\cong\Kqo_R(\pt)\oplus H^{\even,\ge2}(B,\R)\;,$$
and the combined map
$$\CD
	H^{\even,\ge2}(B,\R)@>>>\Kqo_R(B)@>\chdo>>H^{\even,\ge2}(B,\R)
\endCD$$
is multiplication by~$-1$.

\Proof
Because~$B$ is simply connected,
the sequence~\LottExactSeq\ becomes
	$$\CD H^\even(B,\R)@>>>\Kqo_R(B)@>>>K^0_R(\pt)@>>>0\;.\endCD$$
Since any element of~$\Kqo_R(B)$ can be represented
as a sum of $\Kdo_R$-generators~$(F,g^F,\eta)$
with parallel~$g^F$ and~$\eta\in H^\even(B,\R)$,
the proposition follows from the fact that~$\chdo$ is well-defined.
\qed\enddemo

\Remark\LottBGRemark
Lott conjectured in~\LottSec\
that~$p_!$ is related to a Becker-Gottlieb transfer.
More precisely,
let~$\Cal F^*_R$ denote the homotopy fibre of the map
	$$\CD K^*(R)@>\rho_*>>K^*(\C)@>>>\prod\limits_j K(\R,2j+1)\endCD\;,$$
where~$K^*$ denotes the spectrum of algebraic $K$-theory,
and~$K(\R,2j-1)$ denotes the Eilenberg-McLane spectrum of odd real homology.
Here,
the left arrow is induced by the representation~$\rho\colon R\to\End(C^n)$,
and the right arrow is the Borel regulator map.
Then~$\Cal F^*_R$ is again a generalised cohomology theory,
and the Becker-Gottlieb transfer gives a homomorphism
	$$\tau_{\roman{BG}}^*\colon\Cal F^0_R(M)\to\Cal F^0_R(B)\;.
	\Formel\LottFTransfer$$
Lott conjectures the existence of a natural map~$\Kqo_R\to\Cal F^0_R$
that fits into commutative diagrammes
$$\CD
	H^{\even}(M;\R)@>>>\Kqo_R(M)@>>>K^0_R(M)\\
	@VVV@VVV@VVV\\
	H^{\even}(M;\R)@>>>\Cal F^0_R(M)@>>>K^0_{\roman{alg},R}(M)&\;,
\endCD\Formel\LottAxiomEins$$
cf.~\LottExactSeq, and
$$\CD
	\Kqo_R(M)@>>>\Cal F^0_R(M)\\
	@Vp_!VV@VV\tau_{\roman{BG}}^*V\\
	\Kqo_R(B)@>>>\Cal F^0_R(B)&\;.
\endCD\Formel\LottAxiomZwei$$

The Becker-Gottlieb transfer itself
is a stable map~$\tau_{\roman{BG}}\colon B^+\to M^+$,
which can be defined without referring to the smooth structure
of~$q\colon M\to B$,
see~\BeG.
This implies that~$\tau_{\roman{BG}}^*$ in~\LottFTransfer\
is the same map for all~$q_i$, $i\in\Z$,
whereas the maps~$q_{i!}\colon\Kqo_R(M)\to\Kqo_R(B)$
are pairwise different by \LottHatcherElt\ and \LottSplitProp\
if~$\pi_1(B)$ is finite.
We conclude that the $\Kqo_R$-pushdown contains more information
than the Becker-Gottlieb transfer~$\tau_{\roman{BG}}^*$ in~$\Cal F^0_R$.
Moreover,
if a natural map~$\Kqo_R\to\Cal F^0_R$ satisfying~\LottAxiomEins\
and~\LottAxiomZwei\ above exists,
then the element~$[0,0,u]\in\Kqo_R(B)$ of~\LottHatcherElt\
lies the kernel of~$\Kqo_R(B)\to\Cal F^0_R(B)$
for all closed oriented~$B$ of dimension~$4k$.
\endremark


\enddocument


\Remark\HatcherDWWRemark
We may use \ChsloHomoProp\ and \HatcherTheorem\
to construct examples for a phenomenon described in~\DWW.
We recall that the classes~$\cho(F)=\cho(\nabla^F)\in H^\odd(M;\R)$
as well as the Becker-Gottlieb transfer
	$$\trace_{\roman BG}\colon H^*(M;\R)\to H^*(B;\R)
		\Mtext{with}
	\alpha\mapsto\int_{M/B}e(TX)\,\alpha$$
can be defined without a differentiable structure on~$M$, $B$ or~$p$.
Thus,
let~$p\colon M\to B$ be a topological fibre bundle.
In~\DWW\ it was shown that the Bismut-Lott index theorem
holds for~$M\to B$ on a $K$-theoretical level,
iff the fibre-bundle~$M\to B$ admits a smooth structure.

Let us consider the maps~$p'_i$ and~$p'_j$

\?weiter?
\endremark


intro.tex
%auto-ignore
\input morse.def

\topmatter
\title Morse theory and higher torsion invariants I\endtitle
\author Sebastian Goette\endauthor
\address Mathematisches Institut der Uni T\"ubingen,
Auf der Morgenstelle~10, 72076 T\"ubingen, Germany\endaddress
\email sebastian.goette\@uni-tuebingen.de\endemail
\thanks Research at MSRI is supported in part by NSF-grant DMS-9701755\endthanks
\keywords analytic torsion forms, higher Franz-Reidemeister torsion,
Cheeger-M\"uller theorem, Whitehead space,
Witten deformation\endkeywords
\abstract
We compare the higher analytic torsion~$\Cal T$ of Bismut and Lott
of a fibre bundle~$p\colon M\to B$ equipped with a flat vector bundle~$F\to M$
and a fibre-wise Morse function~$h$ on~$M$
with a higher torsion~$T$ that is constructed in terms of a families
Thom-Smale complex associated to~$h$ and~$F$,
thereby extending previous joint work with Bismut.
Under additional conditions on~$F$,
the torsion~$T$ is related to
Igusa's higher Franz-Reidemeister torsion.
As an application,
we use the higher analytic torsion to detect infinite families
of smooth bundles~$p_i\colon M\to B$ with diffeomorphic fibres
that are homeomorphic but not diffeomorphic as bundles.
\endabstract
\subjclass\nofrills\rm 2000 \it Mathematics Subject Classification.\/
\rm Primary 58J52; Secondary 57R22, 57Q10\endsubjclass
\endtopmatter

\rightheadtext{Morse theory and higher torsion invariants I}


\document

This is the first of
two papers devoted to a comparison of Igusa's and Klein's
higher Franz-Reidemeister torsion~$\tau$ with
Bismut's and Lott's higher analytic torsion~$\Cal T$.
In this paper,
we evaluate the Bismut-Lott torsion form~$\Cal T$ for families
that carry a fibre-wise Morse function,
thus extending earlier work with Bismut (\BGzwei).
In the second part of the series,
we relate~$\tau$ and~$\Cal T$ for families with fibre-wise Morse functions
in those situations where both invariants are defined.
%In a tentative third part,
%we extend the results of the first two papers to general families,
%using Igusa's existence result for framed generalised Morse functions
%(\Iframed).
Let us recall the development of higher torsion invariants.

Franz and Reidemeister constructed a numerical invariant~$\tau_{\roman{FR}}$
of chain complexes in \Reidemeister, \Franz,
and used it to detect homeomorphism types
of homotopy-equivalent Lense-spaces.
The higher Franz-Reidemeister torsion~$\tau$
is an extension of~$\tau_{\roman{FR}}$
to families of manifolds~$p\colon M\to B$,
which takes values in the cohomology of~$B$.
It was first constructed by John Klein in~\KtorsionOnly\
using a variation of Waldhausen's $A$-theory.
Other describtions of~$\tau$ were later given by Igusa and Klein,
see~\IBuch\ for more detailled references.
In~\IK,
they computed~$\tau$ in the special case of circle bundles.

We will constantly refer to the construction of~$\tau$ in~\IBuch.
Given a family~$p\colon M\to B$ of smooth manifolds,
one first finds a function~$h\colon M\to\R$
that has only Morse-type and cubical singularities along each fibre of~$p$,
such that the unstable manifolds of the fibre-wise singularities
are trivialised in a compatible way.
By~\Iframed,
such a ``framed'' function always exists if~$\dim M>2\,\dim B$.
If~$\dim M\le 2\,\dim B$,
one may replace~$M$ by~$M\times\R P^{2N}$ 
for some sufficiently large~$N$.
Let~$F\to M$ be a unitarily flat complex vector bundle
that is fibre-wise acyclic,
then~$F$ and~$h$ give rise to a functor from the category
of generic small simplices on~$B$
to the simplicial Whitehead category~$\Wh^h\(M(\C),U\)$
associated to the infinite matrix ring~$M(\C)=\lim_{\to}M_n(\C)$
and the infinite unitary group~$U=\lim_{\to}U(n)$.
Now,
the $k$-th higher torsion~$\tau_k(M/B;F)$ is defined as the pull-back
of a certain cohomology class~$D_{2k}
\in H^{2k}\bigl(\Wh^h\(M(\C),U\),\R\bigr)$.

On the other hand,
Ray and Singer defined an analytic torsion~$\Cal T_{\roman{RS}}$
of unitarily flat complex vector bundles on compact manifolds in~\RS\
and conjectured that~$\Cal T_{\roman{RS}}=\tau_{\roman{FR}}$.
This conjecture was established independendly by Cheeger (\Cheeger)
and M\"uller (\Meins),
and further generalised by M\"uller (\Mzwei) to unimodular flat bundles,
and by Bismut and Zhang (\BZeins) to arbitrary flat complex vector bundles.

In~\BL,
Bismut and Lott constructed characteristic
differential forms~$\cho(\nabla^F,g^F)$ of flat vector bundles~$(F,\nabla^F)$
with arbitrary metrics~$g^F$,
which provide obstructions towards finding a parallel metric.
In fact,
the forms defined in~\BL\ have different coefficients,
but we prefer to state the results of~\BL\
in the so-called ``Chern normalisation''
that was introduced in~\BGzwei,
for reasons explained below.
Let~$p\colon M\to B$ be a family of compact manifolds
with a horizontal distribution~$T^HM\subset TM$ and a metric~$g\TX$
on the vertical tangent bundle~$TX$,
and let~$(F,\nabla^F)\to M$ be a flat vector bundle with metric~$g^F$.
Then Bismut and Lott defined a higher analytic torsion
form~$\Cal T (T^HM,g\TX,\nabla^F,g^F)$ on~$B$ such that
	$$d^B\Cal T \(T^HM,g\TX,\nabla^F,g^F\)
	=\int_{M/B}e\(\nabla\TX\)\,\cho\(\nabla^F,g^F\)
		-\cho\(\nabla^H,g^H\)\;,
	\tag*$$
where~$e\(\nabla\TX\)$ is the Euler form of the vertical tangent bundle
with its natural connection~$\nabla\TX$,
and~$H=H^*(M/B;F)\to B$ is the bundle of the fibre-wise cohomology of~$F$,
equipped with its natural connection~$\nabla^H$
and the $L_2$-metric~$g^H$ on harmonic forms.
They also showed that the component of~$\Cal T (T^HM,g\TX,\nabla^F,g^F)$
in degree~$0$ equals the Ray-Singer analytic torsion of the fibres.


Formula~$(*)$ above in particular implies a ``Grothendieck-Riemann-Roch''
theorem for flat vector bundles at a cohomological level.
In~\DWW,
Dwyer, Weiss and Williams proved such a theorem
at the level of algebraic $K$-theory.
They also suggested another higher torsion invariant.
We do not know wether their higher torsion is related to one of the above.

In~\LottSec,
Lott defined a secondary family index using the analytic torsion form,
which is a real analogue of the push-forward in Arakelov geometry.
Ma has studied the behaviour of~$\Cal T $ under iterated fibrations
in~\Ma.
As a consequence, Bunke showed in~\BuMa\ that Lott's
secondary index is indeed functorial.
One can use \MainTheorem\ below to evaluate Lott's index in certain cases.

If the family~$p\colon M\to B$ has compact structure group~$G$
and if~$F$ is induced from a $G$-equivariant flat vector bundle on
the typical fibre~$X$,
then the higher analytic torsion of Bismut and Lott
can be treated with equivariant methods,
see~\BL, Chapter~4.
In~\BuEins\ and~\BuZwei,
Bunke computed both the equivariant torsion
and the higher analytic torsion for compact structure groups in degree~$>0$
for fibre-wise acyclic, unitarily flat vector bundles
from the equivariant Euler characteristic of~$X$.
This was used in~\BuDrei\ to construct cohomology classes
of~$\Diff(S^{2n-1})$ both with the smooth and with the discrete topology.
Bunke's results also imply a relation
between higher analytic torsion for families with compact structure group
and equivariant torsion in the special case above,
which was proved in general in~\BGdrei.
Since this result depends on the Chern-normalisation
of the analytic torsion forms,
while all other results discussed here hold
for any possible choice of coefficients,
we have chosen to work with the particular normalisation~$\Cal T $.

In~\BGzwei,
we calculated~$\Cal T (T^HM,g\TX,\nabla^F,g^F)$
for families~$p\colon M\to B$ with a function~$h\colon M\to\R$
that is fibre-wise Morse,
such that there exists a fibre-wise gradient field for~$h$
that satisfies Smale's transversality condition on every fibre of~$p$.
In this setting,
the fibre-wise Thom-Smale complexes
form a flat $\Z$-graded vector bundle~$(V,\nabla^V)$ over~$B$,
the metric~$g^F$ defines a metric~$g^V$ on~$V$,
and the coboundary map~$v$ becomes a parallel section of~$\End V$.
In particular,
there is a flat superconnection~$A'=\nabla^V+v$ of total degree~$1$,
and we showed that~$\Cal T (T^HM,g\TX,\nabla^F,g^F)$
is related to the torsion form~$T (A',g^V)$ defined in~\BL, section~2.
If~$H$ still denotes the fibre-wise cohomology
and~$g^H_V$ denotes the metric induced by identifying~$H$
with a subbundle of~$V$ using finite-dimensional Hodge theory,
this torsion form satisfies
	$$d^BT \(A',g^V\)=\cho\(\nabla^V,g^V\)-\cho\(\nabla^H,g^H_V\)\;.$$ 
When~\BGzwei\ was written,
Igusa's framing principle of~\IBuch,
chapter~6,
was not yet available.
Thus in order to compare our result with the theory of Igusa and Klein,
we could only regard unitarily flat bundles~$F\to M$
such that the fibre-wise cohomology forms a trivial flat bundle over~$B$,
and only framed Morse functions~$h$,
i.e., Morse functions whose fibre-wise unstable manifolds are trivialised.
Moreover, we still needed the existence of a fibre-wise Smale gradient field,
in which case there always is a deformation of~$h$
to a self-indexing Morse function on~$M\to B$.
In this very special case,
both the higher Franz-Reidemeister torsion of Igusa and Klein
and the higher analytic torsion form vanish.

In the present paper,
we generalise the results of \BGzwei\
to families of compact manifolds~$M\to B$ that carry
a fibre-wise Morse function~$h\colon M\to\R$,
but we do not require the existence of a fibre-wise Smale gradient field.
Let~$\hat p\colon C\to B$ denote the covering of~$B$ by the
fibre-wise critical points of~$h$,
and let~$o(T^uX)\to C$ denote the orientation bundle
of the unstable manifolds along~$C$.
We still define a vector bundle
	$$V=\hat\pi_*\(F|_C\otimes o(T^uX)\)$$
of fibre-wise Thom-Smale complexes.
A generic fibre-wise gradient field will satisfy the Smale condition
everywhere outside a stratified subset of~$B$ of codimension~$1$.
Whenever one passes this subset,
the coboundary map~$v$ on the bundle~$V$
will change by an elementary isomorphism.
This elementary isomorphism will change whenever we pass the stratum
of codimension~$2$,
and the composition of the elementary isomorphisms that we encounter
when we go around the stratum of codimension~$2$
will be homotopic to the identity,
but not necessarily equal to it.
If we fix homotopies on the stratum of codimension~$2$,
then these will change when we pass the stratum of codimension~$3$,
and the process continues.
The algebraic structure arising from these coboundary maps,
elementary isomophisms, homotopies etc.\
is called a ``system of higher homotopies'' in~\IBuch.
We encounter such structures in \ComplexKapitel,
where we interpret them as ``simplicial superconnections'',
since they behave similar as classical superconnections,
except that the exterior differential~$d$
gets replaced by the simplicial coboundary operator~$\delta$.

In order to define an analytic torsion form for a family~$V$
of Thom-Smale complexes where the fibre-wise coboundary map~$v$ jumps
as described above,
we replace the superconnection~$\nabla^V+v$ of~\BGzwei\
by a flat superconnection~$A'$ of total degree~$1$ on~$V$
that may now contain terms of arbitrarily high degree
with respect to~$\Omega^*(B)$.
This unfortunately implies
that the analytic torsion form~$T(A',g^V)$ of Bismut-Lott is no longer defined.
However,
the Morse function~$h$ induces an endomorphism~$h^V$ of~$V$
by multiplication with~$h|_C$,
and we can construct an analytic torsion form~$T(A',g^V,h^V)$
that depends explicitly on~$h^V$,
such that still
	$$d^BT \(A',g^V,h^V\)
	=\cho\(\nabla^V,g^V\)-\cho\(\nabla^H,g^H_V\)\;.$$
In contrast to the situation in~\BGzwei,
the form~$T \(A',g^V,h^V\)$ can be non-trivial
even if the bundle~$F$ is fibre-wise acyclic and~$g^F$ is parallel.
For example,
it will follow from~\IBuch\ and our results in~\TeilZwei\
that~$T \(A',g^V,h^V\)$ restricts to the Borel regulator classes
on the Volodin $K$-theory space.

In order to state the main theorem of this paper,
let~$\chslo\(H,g^H_V,g^H_{L_2}\)\in\Omega^*(B)/d^B\Omega^*(B)$
be the natural Chern-Simons class satisfying
	$$d^B\chslo\(H,g^H_V,g^H_{L_2}\)
	=\cho\(H,g^H_{L_2}\)-\cho\(H,g^H_V\)\;.$$
Let~$\psi\(\nabla\TX,g\TX\)$ denote the Mathai-Quillen current
on the total space of~$TX\to M$,
such that
	$$d^M\Bigl(\(\nabla\TX h\)^*\psi\(\nabla\TX,g\TX\)\Bigr)
	=e\(TX,\nabla\TX\)-(-1)^{\ind_h}\,\delta_C\;,$$
where~$\delta_C$ denotes the current of integration over the critical set~$C$.
Finally,
let~$\^0J$ denote the additive genus defined in~\BGzwei,
cf.~\JDef\ below.

\Theorem\MainTheorem.
Modulo exact forms on~$B$,
$$\multline
	\Cal T \(T^HM,g\TX,\nabla^F,g^F\)
		-T \(A',g^V,h^V\)
		-\chslo\(\nabla^H,g^H_{L_2},g^H_V\)\\
	=-\int_{M/B}\cho\(\nabla^F,g^F\)
			\,\(\nabla\TX h\)^*\psi\(\nabla\TX,g\TX\)
		+\hat p_*\((-1)^{\ind_h}\,{}^0\!J(T^sX-T^uX)\)\,\rk F\;.
\endmultline$$

The formulation of \MainTheorem\ precisely equals Theorem~0.1
in~\BGzwei\ in the non-equivariant case,
only~$T \(A',g^V\)$ has been replaced by~$T \(A',g^V,h^V\)$.
In fact, a large portion of the proof of \MainTheorem\
is identical to the corresponding parts of~\BGzwei\
and has therefore been omitted.
However,
the computation of the limit of the analytic torsion
under the Witten deformation in~\BGzwei\
relies on Smale's transversality condition;
we give a replacement for this part of the proof in \WittenKapitel.

As an application of \MainTheorem,
we use the higher analytic torsion to detect homeomorphic,
but not diffeomorphic fibre bundles.
More precisely,
we say that two maps~$p$, $p'\colon M\to B$ are
{\em homeomorphic (diffeomorphic) as maps\/}
iff there exist homeomorphisms (diffeomorphisms)~$\Phi$ of~$M$
and~$\Psi$ of~$B$ such that the diagramme
$$\CD
	M@>\Phi>\botsim>M\\
	@VpVV@VVp'V\\
	B@>\Psi>\botsim>B
\endCD$$
commutes.
If~$p$ and~$p'$ are continuous (smooth) fibre bundles,
we say that~$p$ and~$p'$ are
{\em homeomorphic (diffeomorphic) as fibre bundles\/}
iff they are equivalent as continuous (smooth) maps
and we can take~$\Psi=\id_B$ in the diagramme above.
The following result generalises Igusa's Corollary~6.5.8
in~\IBuch\ for smooth bundles that are homeomorphic to trivial bundles.

\Theorem\ApplThm.
For each integer~$k>0$ there exists~$l_0>0$ such that the following holds.
Let~$p\colon M\to B$ be a smooth fibre bundle
with compact, connected, oriented base~$B$ of dimension~$4k$
and with compact, connected fibres of dimension~$2l-1$,
where~$l\ge l_0$.
Then there exist smooth fibre bundles
$p_j\colon M\to B$ for all~$j\in Z$ with~$p_0=p$,
which are all homeomorphic to~$p$ as fibre bundles
and have diffeomorphic fibres,
but which are pairwise not diffeomorphic as fibre bundles.
\parr
Moreover,
if the fibre-wise cohomology bundle~$H^*(M/B;\C)$ admits
a parallel metric with respect to the Gau\ss-Manin connection,
then the~$p_j$ are pairwise not diffeomorphic as maps.

We give a  more general ``quantitative'' version
in \HatcherTheorem\ below.
The bundles~$p_i$ are constructed by a gluing construction involving
homeomorphically trivial bundles with base~$S^{4k}$ and fibre~$S^{2l-1}$.
The construction of these bundles goes back to Hatcher
and is discussed in \BeispielKapitel.
Note that the second statement of \ApplThm\ can be proved using an extension
of Igusa's higher Franz-Reidemeister torsion that we will introduce
in~\TeilZwei.
Our proof of the first statement however relies on the fact
that the higher analytic torsion forms can be defined
even if the cohomology bundle does not admit a parallel metric.

In the second paper~\TeilZwei\ of this series,
we will show that the construction of the bundle~$(V,A',h^V)$ over~$B$
gives rise to a map~$\xi_{p,h,F}$ from~$B$
to Igusa's Whitehead space~$\Wh\(M_r(\C),GL(r,\C)\)$,
which is homotopic to a similar map constructed in Chapter~4 of~\IBuch.
%Here, $GL(\C)=\lim_{\to} GL_n(\C)=M(\C)^\times$
%denotes the stabilised general linear group.
Conversely, each such functor~$\xi_{p,h,F}$ up to homotopy
determines a $\Z$-graded flat vector bundle~$V\to B$
with a flat superconnection~$A'=\nabla^V+a'$ and an endomorphism~$h^V$.
Thus, $\xi_{p,h,F}$ is the analogue of a classifying map
for flat vector bundles with the extra structure described above.
The bundle~$V$ admits a parallel metric~$g^V$
that is adapted to~$h^V$ and~$A'$
and the fibre-wise cohomology~$H=H^*(V,a'_0)\to B$ vanishes
iff the image of~$\xi_{p,h,F}$
lies in the subcategory~$\Wh^h\(M_r(\C),U(r)\)
\subset\Wh\(M_r(\C),GL(r,\C)\)$.
In this case,
the torsion form~$T \(A',g^V,h^V\)$ becomes a cohomology class on~$B$,
which is the pull-back of a cohomology class~$T 
\in H^*\bigl(\Wh^h\(M_r(\C),U(r)\)\bigr)$.
This class~$T $ will be shown to
equal Igusa's torsion class~$\tau $.

We also construct a Whitehead space~$\Wh^u\bigl(M_r(\C),U(r)\)\bigr)$
that classifies complexes~$(V,A')$ as above
where both~$F$ and~$H$ carry parallel metrics.
Then~$T =\tau $
extends to a cohomology class of~$\Wh^u\bigl(M_r(\C),U(r)\)\bigr)$.
Igusa's general construction
gives a map~$\xi\colon B\to\Wh^u\(M_r(\C),U(r)\)$,
and the pull-back~$\xi^*T $
yields a new class~$\tau (M/B;F)\in H^*(B)$
for all families~$M\to B$ of smooth manifolds
with a unitarily flat vector bundles~$(F,\nabla^F,g^F)\to M$
with unitarily flat fibre-wise cohomology~$(H,\nabla^H,g^H)$%
---the existence of a fibre-wise Morse function is no longer required,
and compactness of the fibres can be relaxed as in~\IBuch.

In the case that the fibres of~$p\colon M\to B$ are compact
and the bundles~$F\to M$ and~$H\to F$ admit parallel metrics,
the form
	$$\Cal T (M/B;F)
	=\Bigl(\Cal T \(T^HM,g\TX,\nabla^F,g^F\)
		+\chslo\(\nabla^H,g^H,g^H_{L_2}\)\Bigr)\dg{\ge2}$$
of \AnalyticTorsionClassDef\ is closed,
and its cohomology classes in~$H^*(B)$ is independent
of~$T^HM$, $g\TX$, $g^F$, and~$g^H$.
Using Igusa's framing principle and \MainTheorem,
we will prove the following result.


\Theorem\TeilZweiTheorem(\TeilZwei).
Let~$M\to B$ be a family of compact manifolds,
and let~$(F,\nabla^F)\to M$ be
a flat complex vector bundle with a parallel Hermitian metric,
such that~$(H,\nabla^H)$ also admits a parallel metric.
Assume that there exists a function~$h\colon M\to\R$
that is fibre-wise Morse,
then
	$$\Cal T (M/B;F)
	=\tau (M/B;F)\dg{\ge 2}
		+\int_{M/B}e(TX)\;{}^0\!J(TX)\,\rk F
	\quad\in H^*(B)\;.$$

Note that the fibre-wise Morse function does not appear in the conclusion.
We conjecture
that \TeilZweiTheorem\ holds without the assumption
that such a function exists.
In fact,
by~\Iframed,
after taking the fibre-wise product with~$\R P^{2N}$ for~$N$ sufficiently large,
there exists a framed function~$h\colon M\to\R$
whose fibre-wise singularities are either of Morse type or cubical.
In order to treat the general case,
it thus remains to incorporate cubical singularities into
Theorems ~\NummerVon\MainTheorem\ and~\NummerVon\TeilZweiTheorem.

The present paper is organised as follows.
In \ComplexKapitel,
we construct a CW-structure on~$M$
over a fixed simplicial structure on~$B$,
such that the cells of~$M$ over each simplex of~$B$
give a decomposition of the fibres that is adapted to~$h$.
We then describe how to turn the dicrete coboundary operator
of the CW-complex into a smooth flat superconnection~$A'$ on the bundle~$V$.
We also construct a de-Rham or ``integration'' map~$I$
that gives a quasi-isomorphism
from the total de Rham complex~$\(\Omega^*(M;F),\nabla^F\)$
to~$\(\Omega^*(B;V);A'\)$
and has certain geometric properties
that are needed in \WittenKapitel.

In \TorsionKapitel,
we recall the definition of the characteristic form~$\ch\(\nabla^F,g^F\)$
and of the analytic torsion classes~$T \(A',g^V\)$
and~$\Cal T \(T^HM,g\TX,\nabla^F,g^F\)\in\Omega^*(B)/d^B\Omega^*(B)$
from~\BL\ and~\BGzwei.
We describe the behaviour of the superconnection~$A'$
and the endomorphism~$h^V$ on~$V$,
and use it to define
the torsion class~$T \(A',g^V,h^V\)\in\Omega^*(B)/d^B\Omega^*(B)$.
This definition already involves a finite-dimensional Witten-deformation
on~$V$,
which is similar to the Witten deformation used in the proof of \MainTheorem.
We also define the higher torsion classes~$T (M/B;F,h)$
and~$\Cal T (M/B;F)$
if~$F$ and~$H$ are unitarily flat.

In \MainKapitel,
we show that \MainTheorem\ is compatible with several known results
on higher analytic torsion.
We apply our result to study Lott's $\Kqo_R$-pushdown.
We then recall the outline of the proof of Theorem~0.1 in~\BGzwei,
stating all the intermediate results needed to prove it,
and noting that all but one of these intermediate steps
hold unchanged in our situation.

In \WittenKapitel,
we study the behaviour of the analytic torsion
form~$\Cal T \(T^HM,g\TX,\nabla^F,g^F\)$ under Witten deformation,
without using the Smale property.
Here,
we use all the properties of the superconnection~$A'$ on~$V$,
the map~$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$,
and the higher torsion class~$T \(A',g^V,h^V\)$
that were established in Sections~\NummerVon\ComplexKapitel\
and~\NummerVon\TorsionKapitel.

Finally,
in \BeispielKapitel,
we use Hatcher's example of a non-linear disc bundle
to construct a certain sphere bundle~$p\colon M\to B$.
Using the higher torsion invariants,
we see that~$p\colon M\to B$
is homeomorphic but not diffeomorphic to a trivial sphere bundle.
Thus we recover a result by B\"okstedt,
cf.~\IBuch, Chapter~6.
We then use Hatcher's example prove \ApplThm\ above.
We also discuss Lott's conjecture
that the $\Kqo_R$-pushdown is related to a Becker Gottlieb transfer.

We have not attempted to keep this paper self-contained,
because we rely heavily on the methods and results of~\BGzwei.
However,
at the beginnings of Sections~\NummerVon\ComplexKapitel\
and~\NummerVon\TorsionKapitel,
we recall all necessary constructions from~\BL\ and~\BGzwei,
so that at least the part of this paper that constructs~$T \(A',g^V,h^V\)$
is reasonably self-contained.

We are grateful to J.-M. Bismut, U. Bunke, and P. Teichner for
fruitful discussions and their interest in this project.
We are in great debt to K. Igusa for giving us some topological background
of his work~\IBuch\ on higher FR-torsion,
and for explaining Hatcher's example, cf.\ \BeispielKapitel.
We also thank the Mathematical Sciences and Research Institute at Berkeley
and Igusa's family for their hospitality.

\enddocument

main.tex
%auto-ignore
\input morse.def

\document

\Kapitel\MainKapitel=3 Higher analytic torsion and Morse functions

We recall our main result, \MainTheorem,
and show that it is compatible with other known results
on higher analytic torsion.
We also name a few conclusions in~\BGzwei\ that do not hold
without Smale's transversality condition.
Lott used the higher analytic torsion to define a pushdown
in a secondary $K$-theory in~\LottSec.
In \MainLottSubsection,
we apply \MainTheorem\ to simplify this pushdown in the presence
of a fibre-wise Morse function.
In \MainProofSubsection,
we adapt the proof of Theorem~0.1 in~\BGzwei\ to the current situation.
\WittenKapitel\ contains a new proof for the only intermediate result
of \BGzwei\ that is affected by dropping the Smale condition.


\Abschnitt\MainSubsection Statement of the main result

Let~$p\colon M\to B$ be a smooth fibre bundle with compact fibres,
let~$\(F,\nabla^F\)\to M$ be a flat complex vector bundle,
and assume that there exists a smooth function~$h\colon M\to\R$
such that~$h$ is a Morse function on each fibre of~$p$.

Let~$g^F$ be a Hermitian metric on~$F$.
Then we have an odd characteristic form~$\cho\(\nabla^F,g^F\)$
of \ChoDef\
that vanishes if~$g^F$ is parallel with respect to~$\nabla^F$.

Let~$T^HM\subset TM$ be a horizontal subbundle complementary
to the vertical tangent bundle~$TX$,
and let~$g\TX$ be vertical metric on~$TX$.
Let~$\nabla\TX$ be the natural connection on~$TX$ induced by these data.

Let~$H=H^*(M/B;F)\to B$ denote the bundle of the fibre-wise cohomology
with values in~$F$,
then~$H$ carries a natural Gau\ss-Manin connection~$\nabla^H$,
such that~$(H,\nabla^H)$ is the $E_1$-term in the Leray spectral sequence
in de Rham theory for~$p$ with coefficients in~$F$.
Moreover,
fibre-wise Hodge theory gives an $L_2$-metric~$g^H$ on~$H$
as in~\BGzwei.
Let~$\cho\(\nabla^H,g^H\)$ be the corresponding odd characteristic form.

Let~$\Cal T \(T^HM,g^{TX},\nabla^F,g^F\)\in\Omega^*(B)/d\Omega^*(B)$
be the analytic torsion form of Definitions~\NummerVon\TanDef\
and \NummerVon\ChernNormDef\
in the Chern normalization of~\BGzwei,
such that
	$$d^B\Cal T \(T^HM,g^{TX},\nabla^F,g^F\)
	=\int_{M_g/B}e\(\nabla\TX\)\,\cho\(\nabla^F,g^F\)
		-\cho\(\nabla^H,g^H\)\;,
	\Formel\MainTanTransFormel$$
as explained in \TanThm\ and \ChernNormCor.

Let~$h\colon M\to\R$ be a smooth function
that is Morse on each fibre of~$p$,
and let~$\nabla\TX h$ be the fibre-wise gradient of~$h$ with respect to~$g\TX$.
Let~$C\subset M$ denote the set of fibre-wise critical points of~$h$,
then~$\hat p=p|_C\colon C\to B$ is a finite covering of~$B$.
We denote the stable and unstable part
of the vertical tangent bundle~$TX|_C$ of~$C$
by~$T^sX$, $T^uX\to C$.

Let~$V=(p|_C)_*o(T^uX)\otimes F|_C\to B$ be the bundle
constructed in \MorseBundleDef,
equipped with a flat superconnection~$A'$ constructed as in \PatchThm.
The component~$a_0$ of~$A'$ of horizontal degree~$0$ in~$\Omega^*(B)$
defines a cochain complex on~$V$.
Let~$h^V$ be the endomorphism of~$V$ induced by~$h|_C$.
Let~$g^V$ denote the Hermitian metric on~$V$ induced by~$g^F$,
and let~$g^H_V$ denote the $L_2$-metric on~$H$
that is constructed as the restriction of~$g^V$
to the $(a_0+a_0^*)$-harmonic elements of~$V$.
Then we have the
Chern-Simons class~$\chslo\(\nabla^H,g^H_{L_2},g^H_V\)
\in\Omega^*(B)/d^B\Omega^*(B)$ of Definitions~\NummerVon\FslDef\
and \NummerVon\ChernNormDef,
which satisfies
	$$d^B\chslo\(\nabla^H,g^H_{L_2},g^H_V\)
	=\cho\(\nabla^H,g^H_V\)-\cho\(\nabla^H,g^H_{L_2}\)\;.
	\Formel\MainChsloTransFormel$$

Let~$T \(A',g^V,h^V\)\in\Omega^*(B)/d^B\Omega^*(B)$
be the torsion form of Definitions~\NummerVon\MVBTorsionDef\
and \NummerVon\ChernNormDef,
such that
	$$d^BT \(A'g^V,h^V\)
	=\cho\(\nabla^V,g^V\)
		-\cho\(\nabla^H,g^H_V\)
	\quad\in\Omega^*(B)_h\;.
	\Formel\MainTfTransFormel$$


Let~$\delta_0$ be the $\pi^*o(TX)$-valued current
on the total space of the vertical tangent bundle~$\pi\colon TX\to M$,
given by integration over the zero section~$M$ of~$TX$.
Let~$\Omega^*(TX)_0$ denote currents that are smooth
away from the zero section of~$TX$,
and whose wave front set is contained in~$T^*X|_M\subset T^*TX$.
Let~$\psi\(\nabla\TX,g^{TX}\)$ denote the $\pi^*o(TX)$-valued
Mathai-Quillen current on~$T^*X$ of~\MQ, see also~\BZeins,
such that
	$$d\TX\psi\(\nabla\TX,g^{TX}\)
	=\pi^*e\(\nabla\TX\)-\delta_0
	\quad\in\Omega^*(TX)_0\;.
	\Formel\MainPsiTransFormel$$

Let~$\delta_C$ be the $o(TX)$-valued current on~$M$,
given by integration over~$C$.
Then the wave front sets of~$\delta_C$\
and~$\(\nabla\TX h\)^*\psi\(\nabla\TX,g^{TX}\)$
are contained in~$T^*X|_C$,
and we have
	$$d^M\Bigl(\(\nabla\TX h\)^*\psi\(\nabla\TX,g^{TX}\)\Bigr)
	=e\(\nabla\TX\)-(-1)^{\ind_h}\,\delta_C\;.
	\Formel\MainDhPsiTransFormel$$

Finally,
let~${}^0\!J$ denote the additive genus introduced in~\BGzwei.
Let~$\zeta$ denote the Riemann $\zeta$-function,
and for a real vector bundle~$E\to M$,
let~$\ch(E)\dg{4k}$ denote the component of the Chern character of~$E$
in~$H^{4k}(M)$.
Then
	$${}^0\!J(E)
	={1\over2}\,\sum_{k=1}^\infty\zeta'(-2k)\,\ch(E)\dg{4k}\;.
	\Formel\JDef$$

For convenience,
we recall that \MainTheorem\
states that
$$\multline
	\Cal T \(T^HM,g\TX,\nabla^F,g^F\)
		-T \(A',g^V,h^V\)
		-\chslo\(\nabla^H,g^H_{L_2},g^H_V\)\\
	=-\int_{M/B}\cho\(\nabla^F,g^F\)
			\,\(\nabla\TX h\)^*\psi\(\nabla\TX,g\TX\)
		+\hat p_*\((-1)^{\ind_h}\,{}^0\!J(T^sX-T^uX)\)\,\rk F\;.
\endmultline\tag\FormelNummer\MainTheoremFormel$$
modulo exact forms on~$B$.
\MainTheorem\ will be proved at the end of this section.

\Remark\MainBGzweiRemark
The formulation of \MainTheorem\ is the same
as that of Theorem~0.1 in~\BGzwei.
The only difference is in the definition of the ``combinatorial
torsion form'' $T $.
\endremark


We discuss an important special case of \MainTheorem.
Assume therefore that there are parallel metrics~$g^F$ on~$F\to M$
and~$g^H$ on~$H\to B$,
and recall the definition of the higher torsion classes~$T (M/B;F,h)$
and~$\Cal T (M/B;F)$ in~\HigherChernTorsionDef,
cf.\ \MorseTorsionDef\ and~\AnalyticTorsionClassDef.
\MainTheorem\ immediately implies

\Corollary\MainCorollary.
Assume that the vector bundles~$F\to M$ and~$H\to B$ carry parallel metrics.
Then
	$$\Cal T (M/B;F)-T (M/B;F,h)
	=\hat p_*\((-1)^{\ind_h}\,{}^0\!J(T^sX-T^uX)\)\,\rk F\;.$$

We will give an application of \MainCorollary\ in \BeispielKapitel.


\Abschnitt\MainCompatibilityAbschnitt Compatibility
with known results on analytic torsion forms

We check that \MainTheorem\ is compatible with the transgression
and variation formulas for higher analytic torsion forms
of~\BL\ and~\BGzwei,
and with Poincar\'e duality,
cf.~\BGzwei, sections~7.4--7.6.
On the other hand,
we remark that most of the results of sections~7.8--7.10 in~\BGzwei\
fail if~$h$ does not admit a fibre-wise Morse-Smale gradient field.

\Remark\MainCompatibilityRemarkOne
From equations~\MainTanTransFormel--\MainDhPsiTransFormel\
it is clear that applying the exterior differential~$d^B$ on~$B$
to both sides of~\MainTheoremFormel\
gives the trivial identity
$$%\multline
	\int_{M/B}e\(\nabla\TX\)\,\cho\(\nabla^F,g^F\)
		-\cho\(\nabla^V,g^V\)%\\
	=\int_{M/B}\cho\(\nabla^F,g^F\)\,e\(\nabla\TX\)
		-\hat p_*\cho\(\nabla^F,g^F\)\;.
%\endmultline
$$
Thus,
\MainTheorem\ is compatible with the transgression formulas
satisfied by~$\Cal T $, $T $, $\chslo$ and~$\psi$.
This automatically implies that \MainTheorem\
is also compatible with the variation formulas
for~$\Cal T $, $T $, $\cho$, $\chslo$ and~$\psi$
under smooth variations of~$T^HM$, $g\TX$, and~$g^F$.
In particular,
we may choose these data such that the simplifying assumptions
of \SimplifyingAssumptions\ are satisfied
when we prove \MainTheorem.
\endremark

Let~$\nabla^{F,s}$ be a smooth one-parameter family
of flat connections on~$F$,
and let~$g^{F,s}$ be a smooth one-parameter family of Hermitian metrics,
with~$s\in[0,1]$.
Let~$\(H_s,\nabla^{H,s},g^{H,s}_{L_2}\)$
be the fibre-wise cohomology with values in~$\(F,\nabla^{F,s}\)$
with its induced $L_2$-metric.
We assume that~$\dim H_s$ is constant,
so that~$\(H_s,\nabla^{H,s}\)$ becomes
a smooth vector bundle on~$B\times[0,1]$.

For~$k\ge0$,
let~$(\punkt)\dg{\ge k}$ denote the sum of the homogeneous parts
of degree~$\ge k$ of a differential form.
In~\BGzwei, Definition~2.4,
we defined the so-called
Bott-Chern classes~$L \(\nabla^{F,s},g^{F,s}\)
\in\Omega^{\ge2}(M)/d\Omega^*(M)$
and~$L \(\nabla^{H,s},g^{H,s}_{L_2}\)
\in\Omega^{\ge2}(B)/d\Omega^*(B)$,
see Definitions~\NummerVon\LDef\ and~\NummerVon\ChernNormDef,
such that
$$\split
	d^ML \(\nabla^{F,s},g^{F,s}\)
	&=\cho\(\nabla^{F,1},g^{F,1}\)\dg{\ge3}
		-\cho\(\nabla^{F,0},g^{F,0}\)\dg{\ge3}\\
		\Land
	d^BL \(\nabla^{H,s},g^{H,s}\)
	&=\cho\(\nabla^{H,1},g^{H,1}\)\dg{\ge3}
		-\cho\(\nabla^{H,0},g^{H,0}\)\dg{\ge3}\;,
\endsplit\Formel\MainChoVarFormel$$
see \LOddClassThm\ and \ChernNormCor~(2).
In~\BGzwei, Theorem~3.45,
it was shown that
$$\multline
	\Cal T \(T^HM,g\TX,\nabla^{F,1},g^{F,1}\)\dg{\ge2}
		-\Cal T \(T^HM,g\TX,\nabla^{F,0},g^{F,0}\)\dg{\ge2}\\
	=\int_{M/B}e\(\nabla\TX\)\,L \(\nabla^{F,s},g^{F,s}\)
		-L \(\nabla^{H,s},g^{H,s}\)\;,
\endmultline\tag\FormelNummer\MainTanVarFormel$$
see \TanThm~(3) and \ChernNormCor~(8).
Similarly, we have shown in \MVBTorsionRigidityThm\ and \ChernNormCor~(7)
that
$$%\multline
	T \(A^{\prime1},g^{V,1},h^V\)\dg{\ge2}
		-T \(A^{\prime0},g^{V,0},h^V\)\dg{\ge2}%\\
	=L \(\nabla^{V,s},g^{V,s}\)-L \(\nabla^{H,s},g^{H,s}_V\)\;.
%\endmultline
\tag\FormelNummer\MainThVarFormel$$

\Remark\MainCompatibilityRemarkTwo
By equations~\MainChoVarFormel--\MainThVarFormel,
subtracting~\MainTheoremFormel\ for~$\Fb_0$
from~\MainTheoremFormel\ for~$\Fb_1$ in degree~$\ge2$
after a straightforward partial integration
gives the trivial identity
$$\multline
	\int_{M/B}e\(\nabla\TX\)\,L \(\nabla^{F,s},g^{F,s}\)
		-L \(\nabla^{V,s},g^{V,s}\)\\
	=\int_{M/B}L \(\nabla^{F,s},g^{F,s}\)\,e\(\nabla\TX\)
		-\hat p_*L \(\nabla^{F,s},g^{F,s}\)\;.
\endmultline$$
In particular,
\MainTheorem\ is compatible with the known variation formulas
for analytic torsion forms in dependence of the flat connection on~$F$.
\endremark


Let~$o(TX)\to M$ denote the orientation line bundle of~$TX$,
and let~$n$ denote the dimension of the fibres of~$p\colon M\to B$.
Fibre-wise Poincar\'e duality implies that there exists a canonical
isomorphism
	$$\overline{H^k(M/B;F)}^*
	\cong H^{n-k}\(M/B;\overline F^*\otimes o(TX)\)\;.$$
As in~\BGzwei, one verifies that the analytic torsion form satisfies
$$\multline
	\Cal T \Bigl(T^HM,g\TX,\nabla^{\overline F^*\otimes o(TX)},
		g^{\overline F^*\otimes o(TX)}\Bigr)\\
	=(-1)^{n+1}\,\Cal T \(T^HM,g\TX,\nabla^F,g^F\)
	\quad\in\quad\Omega^*(B)/d\Omega^*(B)\;.
\endmultline\tag\FormelNummer\PDTanFormel$$

If we replace the fibrewise Morse function~$h$ by~$-h$
and~$F$ by~$\overline F^*\otimes o(TX)$,
then the bundles~$V^k\to B$ get replaced by~$\overline{V^{n-k}}^*\to B$.
The superconnections~$A'$ and~$A''$ on~$V$ get replaced by~$\overline{A'}^*$
and~$\overline{A''}^*$,
which are conjugates of~$A''$ and~$A'$ by the metric~$g^V$.
As in~\BGzwei, \MVBTorsionDef\ implies that
	$$T \Bigl(\overline{A'}^*,g^{\overline V^*}
			,-h^{\overline V^*}\Bigr)
%	=T \(A'',g^V,-h^V\)
	=(-1)^{n+1}\,T \(A',g^V,h^V\)
	\quad\in\quad\Omega^*(B)/d\Omega^*(B)\;.
	\Formel\PDMVBFormel$$

\Remark\PDRem
It has been checked in~\BGzwei\ that the remaining terms in \MainTheoremFormel\
satisfy equations analogous to~\PDTanFormel\ and~\PDMVBFormel\
if one replaces~$h$ by~$-h$ and~$F$ by~$\overline F^*\otimes o(TX)$.
Thus,
\MainTheorem\ is compatible with Poincar\'e duality.
\endremark

\Remark\NoBGzweiCorRem
In~\BGzwei, sections~7.8 and~7.9,
we drew some conclusions in the special case that the metric~$g^F|_C$
is parallel.
We found that then the combinatorial torsion vanishes in higher degrees on~$B$,
and that the metric~$g^H_V$ induced by~$g^V$ on~$H\to B$ is parallel;
both due to the fact
that~$A'$ takes the special form~$a_0+\nabla^V$
with~$a_0$ of degree~$0$ on~$B$ and~$\nabla^V$ unitary with respect to~$g^V$.

If~$\nabla\TX h$ does not satisfy Smale's transversality condition,
then~$A'$ is more complicated,
and we will give an example in~\TeilZwei\
where~$T (A',g^V,h^V)$ carries cohomological information
even though~$\nabla^V$ is unitarily flat and~$H=0$.

Moreover in our situation,
$\nabla^H$ is induced by~$\nabla^V+a_1$,
which is not necessarily unitary,
so we cannot conclude any longer that~$g^H$ is a parallel metric.
Thus,
the arguments in~\BGzwei, sections~7.8 and~7.9 fail in our context.

Note also that in~\BGzwei, Thm~7.11,
we deduced that if the fibres are even-dimensional,
then the bundle
	$$\hat p_*\((-1)^{\ind_h}\,TX|_C\)$$
becomes trivial in~$KR_0(B)\otimes\Q$.
By~\TeilZweiTheorem\ that will be proved in~\TeilZwei,
the above bundle gives the difference between the higher analytic torsion
and the higher Franz-Reidemeister torsion
	$$\int_{M/B}e(TX)\,\^0J(TX)\,\rk F
	=\hat p_*\((-1)^{\ind_h}\,\^0J(TX|_C)\)\,\rk F\;,$$
which may be nonzero.
\endremark

\Remark\NoBzweiCorRemZwei
In~\BGzwei, section~7.10,
we used Smale's transversality condition to show that
for every subcovering~$C'\subset C\to B$ that leads
to an acyclic direct summand~$V'\subset V$,
we have
	$$\(\hat p|_{C'}\)_*\((-1)^{\ind_h}\^0J(T^sX-T^uX)\)=0\;.$$
We will see in \HatcherNonSmaleRem\
that this is not the case in Hatcher's example,
which shows that the Morse function in Hatcher's example
does not admit a global Smale gradient field,
and that Theorem~7.12 in~\BGzwei\ becomes false
without the transversality condition.
\endremark


\Abschnitt\MainLottSubsection Morse functions and Lott's pushdown

In this subsection,
we give a short review of Lott's $\Kqo_R$-groups
and describe the pushdown map~$p_!\colon\Kqo_R(M)\to\Kqo_R(B)$
associated to a proper submersion~$p\colon M\to B$
in the special case that~$M$ carries a fibre-wise Morse function.
In \ExampleLottSubsection,
we will apply the $\Kqo_R$-functor to Hatcher's example
and discuss a conjectural relation of~$p_!$ to a Becker-Gottlieb transfer.

Let~$R$ be a right-regular, right-Noetherian ring,
e.g., $R=\Z$, $R=\R$ or~$R=\C$.
Let~$\rho\colon R\to\End(\C^n)$ be a complex representation of~$R$
such that~$\C^n$ becomes a flat $R$-module.
If~$M$ is a manifold and~$F\to M$ is a local system of right-$R$-modules,
let~$F_\C=F\otimes_\rho\C^n$
denote the complex flat vector bundle associated to~$F$
with flat connection~$\nabla^F$.

Lott first defines a Grothendieck group~$\Kdo_R(M)$
with generators~$(F,g^F,\eta)$,
where~$F$ is a local system of right-$R$-modules,
$g^F$ is a Hermitian metric on~$F_\C$,
and~$\eta\in\Omega^\even(M)/d\Omega^\odd(M)$
is a class of even differential forms modulo exact forms on~$M$.

To define the relations in~$\Kdo_R(M)$,
let
	$$\CD 0@>>>F_1@>>>F_2@>>>F_3@>>>0\endCD
	\Formel\LottShortSeqEins$$
be a short exact sequence,
choose metrics~$g^{F_i}$ on~$F_{i,\C}$
and forms~$\eta_i\in\Omega^\even(M)/d\Omega^\odd(M)$,
and let~$T$ denote the Bismut-Lott analytic torsion
of the finite-dimensional flat family of acyclic complexes
	$$\CD 0@>>>\(F_{1,\C},g^{F_1}\)
	@>>>\(F_{2,\C},g^{F_2}\)
	@>>>\(F_{3,\C},g^{F_3}\)@>>>0\endCD
	\Formel\LottShortSeqZwei$$
over~$M$.
Then in~$\Kdo_R(M)$,
	$$\bigl[F_2,g^{F_2},\eta_2\bigr]
	=\bigl[F_1,g^{F_1},\eta_1\bigr]\oplus\bigl[F_3,g^{F_3},\eta_3\bigr]
	\Mtext{if}
	\eta_2
	=\eta_1+\eta_3+T\;.
	\Formel\KdachRelations$$

There is a well-defined map~$\cho\colon\Kdo_R(M)\to\Omega^\odd(M)$
given on generators by
	$$\cho\bigl[F,g^F,\eta\bigr]
	=\cho\(\nabla^F,g^F\)-d\eta\;,$$
and~$\Kqo_R(M)\subset \Kdo_R(M)$ is defined
as the kernel of~$\cho$.
Then~$\Kqo_R(M)$ fits into an exact sequence
	$$H^\even(M,\R)@>>>\Kqo_R(M)
		@>>>K^0_R(M)@>>> H^\odd(M,\R)\;,
	\Formel\LottExactSeq$$
where~$K^0_R(M)$ is simply the Grothendieck group
of local systems of right-$R$-modules on~$M$,
and the rightmost arrow is the characteristic class~$\cho$.
For any smooth map~$f\colon M\to N$ there is a natural
pullback~$f^*\colon\Kdo_R(N)\to\Kdo_R(M)$
that restricts to
a pullback~$f^*\colon\Kqo_R(N)\to\Kqo_R(M)$
and is compatible with~\LottExactSeq.

Now let~$p\colon M\to B$ be a smooth proper submersion,
and fix a horizontal subbundle~$T^HM$ and a vertical metric~$g\TX$.
Let~$H=H^*(M/B;F)$ denote the fibre-wise cohomology of~$F$,
which is again a local system of right-$R$-modules over~$B$,
and let~$g^H_{L_2}$ denote the $L_2$-metric on~$H_\C\to B$.
Then a pushdown in~$\Kdo_R(M)$ is given on generators by
	$$p_!\bigl[F,g^F,\eta\bigr]
	=\biggl[H,g^H_{L_2},\int_{M/B}e\(TX,\nabla\TX\)\,\eta
		-\Cal T\(T^HM,g\TX,\nabla^F,g^F\)\biggr]\;.
	\Formel\LottPShriekFormel$$
It is proved in~\LottSec\ that~$p_!$ is compatible with~\KdachRelations.
Moreover,
the pushdown restricts
to a map~$p_!\colon\Kqo_R(M)\to\Kqo_R(B)$
that is independent of~$T^HM$ and~$g\TX$.

If~$h\colon M\to B$ is a fibre-wise Morse-function,
we define~$C$, $(V,\nabla^V,A',g^V)\to B$ and~$g^H_V$ as above.
We define another pushdown on~$\Kdo_R(M)$ by
	$$p_{!h}\bigl[F,g^F,\eta\bigr]
	=\Bigl[H,g^H_V,
		\hat p_*\Bigl((-1)^{\ind_h}\,\(\^0J(T^sX-T^uX)\,\rk F_\C
			+\eta\)\Bigr)-T\(A',g^V,h^V\)\Bigr]\;.
	\Formel\OurPshriekFormel$$
As in~\LottSec, one can show that this pushdown is compatible
with~\KdachRelations.

\Theorem\MainLottThm.
Suppose that~$h\colon M\to\R$ is a fibre-wise Morse function.
Then~$p_{!h}$ restricts
to the pushdown~$p_!\colon\Kqo_R(M)\to\Kqo_R(B)$.

\Proof
The relations~\KdachRelations imply that
	$$\bigl[F,g^{\prime F},\eta'\bigr]
	=\bigl[F,g^F,\eta'-\chslo\(\nabla^F,g^F,g^{\prime F}\)\bigr]
	\quad\in\quad \Kdo_R(M)\;.
	\Formel\LottVariationFormel$$
Also,
a straightforward partial integration shows that
	$$\int_{M/B}e\(TX,\nabla\TX\)\,\eta-\hat p_*\eta
	=\int_{M/B}\(\nabla\TX h\)^*\psi\(\nabla\TX,g\TX\)\,d\eta$$
modulo exact forms on~$B$.
Using \MainTheoremFormel\ and the two observations above,
we may rewrite~\LottPShriekFormel\ as
$$\align
	p_!\bigl[F,g^F,\eta\bigr]
	&=\biggl[H,g^H_V,\hat p_*\eta
		+\int_{M/B}\(\nabla\TX h\)^*\psi\(\nabla\TX,g\TX\)\,d\eta\\
	&\kern6em
		-\Cal T\(T^HM,g\TX,\nabla^F,g^F\)
		-\chslo\(\nabla^H,g^H_{L_2},g^H_V\)\biggr]\\
\noalign{\smallskip}\allowdisplaybreak
	&=p_{!h}\bigl[F,g^F,\eta\bigr]
		+\biggl[0,0,\int_{M/B}\(\nabla\TX h\)^*\psi\(\nabla\TX,g\TX\)\,
			\Bigl(d\eta-\cho\(\nabla^F,g^F\)\Bigr)\biggr]\;.
\endalign$$
The theorem follows because in~$\Kqo_R(M)$,
we have
	$$d\eta-\cho\(\nabla^F,g^F\)=-\cho\bigl[F,g^F,\eta\bigr]=0\;.
	\quad\qed$$
\enddemo


\Abschnitt\MainProofSubsection A proof of Theorem 0.1

In order to prove \MainTheorem,
we refer to the proof of Theorem~0.1 of~\BGzwei.
However,
we have to replace all arguments
that rely on the Smale transversality condition
for the negative gradient flow,
or on its consequences.
We start by summarizing the places in~\BGzwei\ that depend on
the Smale condition,
then we indicate how to extend~\BGzwei\ to the more general case
of fibre-wise Morse functions that do not necessarily admit
a fibre-wise Morse-Smale gradient field,
and finally,
we show how to fit the contents of this paper
into the general framework of~\BGzwei,
thereby proving \MainTheorem.

In Chapter~5 of~\BGzwei,
we used the transversality condition
to prove a families' version of the result of Laudenbach
in the appendix of~\BZeins.
In particular,
we constructed a vector bundle~$V=C^*\(W^u;F\)\to B$
with a flat superconnection
	$$A'=\nabla^{V}+v
		\Mtext{with}
	v\in\Omega^0\(B;\End^1V\)
	\Formel\BGzweiSupConn$$
of total degree one,
and an $\Omega^*(B)$-linear quasi-isomorphism
	$$\bold P^\infty\colon\Omega^*(M;F)\to\Omega^*\(B;V\)\;,
	\Formel\BGzweiIntMap$$
given as integration over the fibres of the unstable cells.

In Chapters~1 and~2 of~\BGzwei,
we recalled and extended the construction of torsion forms
from Section~2\thinspace(f) in~\BL.
The torsion forms
	$$T_f\(A',g^{V}\)
	\in\Omega^*(B)/d^B\Omega^*(B)
	\Formel\BGzweiTorsionForm$$
are only defined for flat superconnections
of the type~\BGzweiSupConn,
in particular,
they converge only if there is no component
in~$\Omega^{\ge 2}\(B;\End V\)$.
Moreover,
the identity
	$$dT_f\(A',g^{V}\)
	=f\(\nabla^{V},g^{V}\)
	-f\(\nabla^{H^*(V,v)},g^{H^*(V,v)}\)
	\Formel\BGzweiTorsionProperty$$
only holds if the connection part of~$A'$
equals~$\nabla^{V}$.

In Theorem~9.8 of~\BGzwei,
we proved that up to a few divergences and correction terms,
the analytic torsion form of~\BL\ converges
to the form~$T_f\(A',g^{V}\)$ of~\BGzweiTorsionForm.
We gave two proofs of this theorem
in Chapters~10 and~11 of~\BGzwei.
While the proof in Chapter~11 needs the transversality condition
in its full strength,
the proof in Chapter~10 uses only that~$A'$ is of the form~\BGzweiSupConn,
and this assumption is needed only in very few places.

We now indicate the changes that lead to \MainTheorem.
In \ComplexKapitel\ of this paper,
we constructed a flat superconnection
	$$A'=\nabla^V+\sum_ja_j
		\Mtext{with}
	a_j\in\Omega^j\(B;\End^{1-j}V\)$$
of total degree one on~$V$,
which replaces the one of~\BGzweiSupConn.
If we view~$h|_C$ as the ``height'' of a critical point,
then the coefficients~$a_j$ ``move'' elements of~$V$ ``up''
by at least~$\eps^2>0$.
We also constructed an $\Omega^*(B)$-linear
quasi-isomorphism
	$$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$$
that takes the role of~$\bold P^\infty$ of~\BGzweiIntMap.
The main properties of~$I$ and~$A'$ are summarised in \PatchThm.
%Let us note that our constructions are intimately related
%to the notion of a ``differential Whitehead space'',
%analogous and homotopy equivalent
%to the Whitehead spaces developed in~\IBuch.

In \TorsionKapitel,
we generalised the torsion forms of~\BL, \BGzwei\
to general flat superconnections of total degree one
that are compatible with an endomorphism~$h^V$
given by the Morse function~$h|_C$.
The necessary condition is that~$\(V,A',h^V\)$
is a family Thom-Smale complex in the sense of \MVBDef\
with an adapted metric~$g^V$,
and the torsion forms~$T_f\(A',g^V,h^V\)$
are constructed in \MVBTorsionDef.

In \WittenKapitel,
we summarise Chapter~10 of~\BGzwei\ and prove our \SmallEvTorsionThm,
which replaces Theorem~9.8 of~\BGzwei.
We still use the Witten deformation as in~\BZzwei\ and~\BGzwei.
In fact,
our proof is a bit more direct
because our definition of~$T_f\(A',g^V,h^V\)$
is modelled itself on a ``Witten deformation'' of the complex~$(V,A')$
with metric~$g^V$.

Let us recall Theorem~9.8 of~\BGzwei.
The definition of~$T_f\(A',g^V\)$ in~\BL, Chapter~2
and~\BGzwei, Definition~1.29 was recalled in~\BGzweiTorsionForm.
It implies in particular that
$$\multline
	T_f\(A',g^V\)
		-\(\chi'(V)-\chi'(H)\)
			\,\int_1^\infty f'\(\sqrt{-t}/2\)\,{dt\over2t}\\
\split
	&\qquad
		-\(\chi'(V)-\chi'(H)\)
			\,\int_0^1\Bigl(f'\(\sqrt{-t}/2\)
				-f'(0)\Bigr)\,{dt\over2t}\\
	&=-(2\pi i)^{-\textstyle{N^B\over2}}\,\int_1^\infty\Bigl(
		\strace_V\(N^V\gamma\,f'(X_{t,1})\)
		-\chi'(H)f'(0)\Bigr)
		\,{dt\over2t}\\
	&\qquad
	-(2\pi i)^{-\textstyle{N^B\over2}}\,\int_0^1\Bigl(
		\strace_V\(N^V\gamma\,f'(X_{t,1})\)
		-\chi'(V)f'(0)\Bigr)
		\,{dt\over2t}\;,
\endsplit\endmultline$$
where~$f(z)=z\,e^{z^2}$,
so in particular, $f'(0)=1$.

We must also recall the definition of~$\tilde\chi^{\prime\pm}(F)$
in~\BGzwei.
Starting from~(9.6) in~\BGzwei,
an elementary calculation shows that
$$\align
	\tilde\chi^{\prime+}(F)
	&=\sum_{c\in C/B}(-1)^{\ind_h(c)}
		\,\(n-\ind_h(c)\)\,\rk F\\
	&=\strace_V\((n-N^V)\)
	=n\,\chi(V)-\chi'(V)\;,\\
\allowdisplaybreak
	\Land
	\tilde\chi^{\prime-}(F)
	&=\sum_{c\in C/B}(-1)^{\ind_h(c)}
		\,\ind_h(c)\,\rk F\\
	&=\strace_V\(N^V\)
	=\chi'(V)\;.
\endalign$$
With this preliminaries,
we can now reformulate Theorem~9.8 of~\BGzwei\ in our notation.

\Theorem\BGzweiNeunAchtThm.
In the limit~$T\to\infty$,
we have
$$\multline
	\tilde f\(\nabla^H,g^H,g^H_T\)
	+S_f\(\Aa',g^{TX,F}_T\)
	-\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,\log T
	+T\,\strace_V\(h^V\gamma\)\\
\split
	&\to\tilde f\(\nabla^H,g^H,g^H_V\)
		-\biggl({\chi'(V)\over2}
			-{n\,\chi(V)\over4}\biggr)\,\log\pi
		+T_f\(A',g^V\)\\
	&\kern4em
		-\(\chi'(V)-\chi'(H)\)
			\,\int_1^\infty f'\(\sqrt{-t}/2\)\,{dt\over2t}\\
	&\kern4em
		-\(\chi'(V)-\chi'(H)\)
			\,\int_0^1\Bigl(f'\(\sqrt{-t}/2\)
				-1\Bigr)\,{dt\over2t}
\endsplit\endmultline$$
modulo exact forms on~$B$.
\qed

In \WittenKapitel,
we prove \SmallEvTorsionThm,
which equals the theorem above,
except that we do no longer require the existence of a Smale gradient field
(which is used implicitly in \BGzwei, Theorem~9.8),
and that~$T_f\(A',g^V\)$ gets replaced by~$T_f\(A',g^V,h^V\)$.

\demo{Proof of \MainTheorem}
By \MainCompatibilityRemarkOne,
we may assume that the simplifying assumptions of \SimplifyingAssumptions\
are satisfied.
By the definitions and arguments of \ChernSubsection,
we may replace the Chern normalization in \MainTheorem\
by the normalisation~$f(z)=z\,e^{z^2}$ of~\GenFnChoice.
If we define
	$${}^0\!I_f(E)
	={1\over2}\,\sum_{k=1}^\infty{(4k+1)!\over2^{4k}\,(2k)!^2}
		\,\zeta'(-2k)\,\ch(E)\dg{4k}
	\Formel\IfDef$$
in analogy with~\JDef,
then an elementary calculation shows that
	$${}^0\!J(E)
	=\int_0^1\(4s(1-s)\)^{\textstyle{N^B\over2}}\;{}^0\!I_f(E)\,ds\;.$$
It remains to establish
$$\multline
	\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)
		-T_f\(A',g^V,h^V\)
		-\tilde f\(\nabla^H,g^H_{L_2},g^H_V\)\\
	=-\int_{M/B}f\(\nabla^F,g^F\)
			\,\(\nabla\TX h\)^*\psi\(\nabla\TX,g\TX\)
		+\hat p_*\((-1)^{\ind_h}\,{}^0\!I_f(T^sX-T^uX)\)\,\rk F\;.
\endmultline\tag\FormelNummer\MainTheoremFNormFormel$$
As indicated above,
this equation now follows from the proof given in Chapter~9 of~\BGzwei,
where \SmallEvTorsionThm\ replaces Theorem~9.8.
\qed\enddemo




\enddocument



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refs.tex
%auto-ignore
\input morse.def

\document

\Refs
\widestnumber\key{DWW}

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J.-M.\ Bismut, J.\ Lott:
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J.-M. Bismut, W. Zhang:
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\bysame: %J.-M. Bismut, W. Zhang:
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M. B\"okstedt:
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U.~Bunke:
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K-Theory~25 (2002), 51--58

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\finalinfo preprint (1998)


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\endRefs

\enddocument





torsion.tex
%auto-ignore
\input morse.def

\document

\Kapitel\TorsionKapitel=2 Torsion forms for family Thom-Smale complexes

Inspired by the previous section,
we define the notion of a family Thom-Smale complex
and construct torsion forms for it.
These torsion forms generalise the torsion forms
for families of finite-dimensional complexes of~\BL,
Section~2.f.

Before we recall
the definitions of odd Chern classes
and transgression forms for flat superconnections from~\BL\
in Sections~\NummerVon\OddClassSubsection,
and~\NummerVon\PreTorsionSubsection,
we introduce some sign conventions in \TorSignSubsection.
In~\MVBSubsection,
we define family Thom-Smale complexes~$(V,A',h^V)$ in \MVBDef.
Their torsion forms~$T(A',g^V,h^V)$ are constructed in \MVBTorsionDef\
using a finite-dimensional version of the Witten deformation.
\MVBTorsionThm\ states the most important properties of~$T(A',g^V,h^V)$,
which are similar to those of~$T(A',g^V)$ in~\BL.
We show that these forms are rigid under deformation of the underlying flat
superconnection in~\MVBRigiditySubsection,
cf.~\BGzwei.
In~\MorseTorsionSubsection,
we use~$T(A',g^V,h^V)$ to define the higher torsion classes~$T(M/B;F,h)$
when~$F$ and the fibrewise cohomology~$H$ carry parallel metrics,
see \MorseTorsionDef.

In~\GenMetSubsection,
we collect some facts about generalised metrics (cf.~\BGzwei)
that will be needed in \WittenKapitel.
We recall the construction of analytic torsion forms from~\BL\
and the Chern normalisation of the odd classes and torsion forms from~\BGzwei\
in Sections~\NummerVon\TanSubsection\ and~\NummerVon\ChernSubsection.
We also define a higher analytic torsion class~$\Cal T(M/B;F)$
if~$F$ and~$H$ are unitarily flat
in \AnalyticTorsionClassDef.


\Abschnitt\TorSignSubsection Notes on Sign Conventions

We recall the sign conventions of~\BL\ and~\BGzwei\
regarding superconnections and their adjoints.
In particular,
we hope to disarm a few well-concealed traps here
and in \GenMetSubsection.

For most of this paper,
we strictly follow the Koszul sign convention,
by which swapping to terms of degrees~$p$ and~$q$ modulo~2
introduces a sign factor of~$(-1)^{pq}$.
However,
there are a few exceptions in connection with metrics
and their generalisations.
This is best motivated by the fact
that a Hermitian sesquilinear form in the $\Z_2$-graded sense
would have to be Hermitian on the even part
and skew-Hermitian on the odd degree part of a vector space,
whereas for analytic reasons,
we need positive definite symmetric forms both in even and in odd degree.

Let~$V=V^+\oplus V^-\to M$ be $\Z_2$-graded complex vector bundle,
and let~$\overline V^*$ be its antidual.
We define~$\Omega^*(M;V)=\Gamma\bigl(\Lambda^*M\tend V\bigr)$
to be the space of sections of a graded tensor product.
The natural $\C$-sesquilinear pairing~$\overline V^*\otimes V\to\C$
extends to a pairing
$$\split
	\Omega^*\(M;\overline V^*\)\otimes\Omega^*(M;V)
	&\to\Omega^*(M)\\
	\Ltext{with}
	\(\alpha\tend f\)\otimes\(\beta\tend v\)
	&\mapsto\(\alpha\tend f,\beta\tend v\)
	=(-1)^{\textstyle\bigl[{\deg\beta\over2}\bigr]}
			\,\alpha\overline\beta\,f(v)
\endsplit\Formel\TorSignPairing$$
for all~$\alpha$, $\beta\in\Omega^*(M)$,
$f\in\Gamma(\overline V^*)$ and~$v\in\Gamma(V)$.
The factor~$(-1)^{[{\deg\beta\over2}]}$ comes from reversing
the one-form components of~$\overline\beta$.
Note the absence of a sign factor~$(-1)^{\deg V\cdot\deg\beta}$;
in particular,
the pairing~$(\punkt,\punkt)$ violates the sign convention.

A metric~$g^V=g^{V^+}\oplus g^{V^-}$ on~$V$
which respect the $\Z_2$-grading can be regarded
as an even complex linear isomorphism~$g^V\colon V\to\overline V^*$.
Together with~\TorSignPairing,
this induces a pairing
$$\split
	g^V\colon\Omega^*(M,V)\otimes\Omega^*(M;V)
	&\to\Omega^*(M)\\
	\Ltext{with}
	\(\alpha\tend v\)\otimes\(\beta\tend w\)
	&\mapsto \<\alpha\tend v,\beta\tend w\>
	=(-1)^{\textstyle\bigl[{\deg\beta\over2}\bigr]}
			\,\alpha\overline\beta\,\<v,w\>\;.
\endsplit\Formel\TorSignMetric$$
The fact that~$g^V$ is Hermitian is equivalent to the relation
	$$\<\beta\tend w,\alpha\tend v\>
	=(-1)^{\textstyle\bigl[{\deg(\alpha\beta)\over2}\bigr]}
		\,\overline{\<\alpha\tend v,\beta\tend w\>}\;,
	\Formel\TorSignSymmetry$$
where the sign factor is once more equivalent to a reversal of the order
of the one-form factors in each decomposable summand.

As in~\BL\ and~\BGzwei,
we extend complex conjugation to~$\Omega^*(M;\C)$ and~$\Omega^*(M;\End V)$.

\Definition\BarStarDef
Define a $\C$-anti-linear operator on~$\Omega^*(M;\C)$
by
	$$\overline\alpha^*
	=(-1)^{\textstyle\left[\topsmash{N^B}+1\over2\right]}\overline\alpha\;.
		\Formel\BarStarDef$$
\enddefinition

\Remark\BarStarRem
This operator has the following properties:
$$\alignat2
	\overline f^*
	&=\overline f
		&&\Rtext{for all
			$f\in\Omega^0(B;\C)$,}\\
	\overline\alpha^*
	&=-\overline\alpha
		&&\Rtext{for all
			$\alpha\in\Omega^1(B;\C)$,}\\
		\Ltext{and}
	\overline{(\alpha\wedge\beta)}^*
	&=\overline\beta^*\wedge\overline\alpha^*\;.
		&&\Rtext{for all
			$\alpha$, $\beta\in\Omega^*(B;\C)$.}
\endalignat$$
Note that this operator differs
from the operator~$(-1)^{\left[{\deg(\punkt)\over2}\right]}\overline\punkt$
encountered in~\TorSignPairing--\TorSignSymmetry\
by a factor of~$(-1)^{\deg(\punkt)}$.
\endremark

We let~$E^*$ denote the classical (ungraded) adjoint
of an endomorphism~$E\in\End V$ with respect to~$g^V$,
and we let~$\nabla^{V,*}$ denote the adjoint of a connection~$\nabla^V$.

\Definition\AdjointDef
The adjoint of a differential form~$\alpha\tend E\in\Omega^*(B;\End V)$
is defined as
	$$(\alpha\tend E)^*
	=(-1)^{\deg\alpha\cdot\deg E}\,\overline\alpha^*\,E^*\;.$$
The adjoint of a superconnection~$A=\nabla^V+a$ with~$a\in\Omega^*(M;\End V)$
is given by
	$$A^*
	=\nabla^{V,*}+a^*\;.$$
We call~$E$ or~$A$ {\em self-adjoint\/} ({\em skew-adjoint\/})
iff~$E^*=E$ or~$A^*=A$ ($E^*=-E$ or~$A^*=-A$) respectively.
\enddefinition

\Remark\AdjointRemEins
The definition of~$A^*$ is independent of the splitting~$A=\nabla^V+a$.
For odd elements~$E\in\Omega^*(M;\End V)^\odd$
and for superconnections~$A$ on~$V$,
\AdjointDef\ is compatible with the sign convention of~\TorSignMetric\
in the sense that
$$\split
	0
	&=\<E^*v,w\>+(-1)^{\deg E\,(\deg v+\deg w)}\,\<v,Ew\>\\
		\Land
	d\<v,w\>
	&=\<A^*v,w\>+(-1)^{\deg E\,(\deg v+\deg w)}\,\<v,Aw\>
\endsplit$$
for all~$v$, $w\in\Omega^*(M;V)$.

Symmetry of~$g^V$ together with \AdjointDef\ implies
that~$(E^*)^*=E$
and~$(A^*)^*=A$ for all superconnections~$A$
and all~$E\in\Omega^*(M;\End V)$.
\endremark

\Remark\AdjointRem
Sometimes it is useful to regard~$g^V$
as an element of~$\Gamma\bigl(\Hom(V,\overline V^*)\bigr)$.
In this setting,
the adjoint of a superconnection~$A$ is given as
	$$A^*=\(g^V\)^{-1}\,\overline{A}^*\,g^V\;,
	\Formel\BLAdjointDef$$
where~$\overline{A}^*$ is the superconnection that is
naturally induced on~$\overline V^*$ by~$A$,
cf.~\BL, Sections~1(c), 1(d).
Note that once more~$\overline A^*$ is compatible with~\TorSignPairing\
in the sense that
	$$d(f,v)
	=(\overline A^*f,v)
		+(-1)^{\deg f+\deg v}
			\,(f,Av)$$
for all~$f\in\Omega^*(M;\overline V^*)$ and all~$v\in\Omega^*(M,V)$.
\endremark


\Abschnitt\OddClassSubsection
Odd characteristic classes for flat superconnections

We recall the construction of Kamber-Tondeur classes in de Rham theory,
following~\BL\ and~\BGzwei\ (cf.~\KT, \Dupont).
Let~$B$ be a smooth manifold,
and let~$N^B$ denote the number operator on~$\Omega^*(B)$.

We will fix for the rest of the paper a generating function
	$$f(z)=z\,e^{z^2}\;,
		\Formel\GenFnChoice$$
although in a few explicitly marked steps,
we need a different choice of~$f$.
Then~$f$ is an odd holomorphic function,
which is real, i.e., $f(\R)\subset\R$,
and there exists~$c>0$ such that
	$$\abs{\partial^kf(z)\over\partial z^k}=O\((1+\abs{\Re z})^{-l}\)
		\Rtext{uniformly on~$\bigl\{\,z\in\C
			\bigm|\abs{\Re(z)}<c\,\bigr\}$}
		\Formel\GenFnProperty$$
for all pairs of integers~$k$, $l\ge 0$.

Now let~$A'$ be a flat superconnection on a $\Z_2$-graded
vector bundle~$V\to B$.
Fix a Hermitian metric~$g^V$ on~$V$,
and let~$A''=(A')^*$ be the adjoint superconnection
in the sense of \AdjointDef.
Define a self-adjoint superconnection~$A$ and a skew-adjoint operator~$X$ by
	$$A={1\over2}\,\(A''+A'\)\Mand
	X={1\over2}\,\(A''-A'\)\quad\in\quad\Omega^*\(B;\End V\)\;,
		\Formel\XDef$$
then clearly
	$$[A,X]=0\quad\in\quad\Omega^*\(B;\End V\)$$
because~$A'$ and~$A''$ are flat.

\Definition\FDef
Let~$f\(A',g^V\)\in\Omega^*(B)$ be defined by
	$$f\(A',g^V\)
	=(2\pi i)^{\textstyle{1-N^B\over 2}}
		\,\strace_V\(f(X)\)\;.$$
\enddefinition

Because supertraces vanish on supercommutators,
we have~$\strace_V\(X^{2k}\)={1\over2}\,\strace_V\([X,X^{2k-1}]\)=0$
for~$k\ge 1$,
so we have lost no information by specifying~$f$ to be an odd function.
Since~$f$ and~$X$ are odd and~$\strace_V$ vanishes on~$\End^\odd V$,
only the part of~$f(X)$ in~$\Omega^\odd\(B;\End^\even V\)$ contributes,
so~$f\(A',g^V\)$ is an odd form.

Because~$[A,X]=0$,
we have
	$$d^B\strace_V\(f(X)\)
	=\strace_V\([A,f(X)]\)
	=0\;,$$
which means that~$f\(A',g^V\)$ is closed.

Finally, since~$f$ is odd and real,
it follows that~$f(X)$ is skew-adjoint in the sense of \AdjointDef.
Using also that~$\strace_V\(f(X)\)$ is odd,
we find
	$$\strace_V\(f(X)\)
	=-\strace_V\(\overline{f(X)}^*\)
%	=-(-1)^{N^B+1\over 2}\,\overline{\strace_V\(f(X)\)}
	=(-1)^{\textstyle{N^B-1\over2}}
		\,\overline{\strace_V\(f(X)\)}$$
by \BarStarDef,
so the form~$f\(A',g^V\)$ is real.

\Theorem\FVThm(\BL, {Theorem~1.8}).
The form~$f\(A',g^V\)$ is odd, closed and real.
Its cohomology class does not depend on~$g^V$.

Let~$g^V_0$ and~$g^V_1$ be two metrics on~$V$.
Then there exists a path~$\(g^V_s\)_{s\in[0,1]}$ of metrics
connecting~$g^V_0$ and~$g^V_1$.
Let~$A''_s$ denote the adjoint of~$A'$ with respect to~$g^V_s$,
and define~$A_s$ and~$X_s$ as in \XDef.
Let~$\bar g^V$ be the metric on the pullback of~$V$ to~$B\times[0,1]$
that restricts to~$g^V_s$ on~$B\times\{s\}$.
We write~$\overline A'$ for the pull-back of~$A'$ to~$B\times[0,1]$.
Then we have
	$$f\(\overline A',\bar g^V\)_{p,s}
	=f\(A',g^V_s\)_p
	+(2\pi i)^{-\textstyle{N^B\over2}}
		\,\strace_V\biggl(\(g_s^V\)^{-1}\,{\del g_s^V\over\del s}
			\,f'\(X_s\)\biggr)_p\,ds\;.$$

\Definition\FslDef
Let~$\tilde f\(A',g^V_0,g^V_1\)$ be defined by
	$$\tilde f\(A',g^V_0,g^V_1\)
	=(2\pi i)^{-\textstyle{N^B\over2}}\int_0^1
		\strace_V\biggl({1\over2}\,\(g_s^V\)^{-1}
			\,{\del g_s^V\over\del s}\,f'\(X_s\)\biggr)\,ds
		\quad\in\Omega^*(B)/d^B\Omega^*(B)\;.$$
\enddefinition

Then by a standard argument,
\FVThm\ immediately implies~\BL, Theorem~1.11.

\Corollary\FVCor.
The class~$\tilde f\(A',g^V_0,g^V_1\)\in\Omega^*(B)/d^B\Omega^*(B)$
is independent of the path~$g^V_s$ chosen in its definition.
It is even and real,
and satisfies
	$$d^B\tilde f\(A',g^V_0,g^V_1\)
	=f\(A',g^V_1\)-f\(A',g^V_0\)\;.$$
%	\Formel\FVariationFormel

We recall the behaviour of~$f\(A',g^V\)$ under smooth changes of~$A'$,
following~\BGzwei, Chapter~2.
Let~$(A'_s)_{s\in[0,1]}$ be a family of flat superconnections on~$V\to B$,
let~$(g^V_s)_{s\in[0,1]}$
be a family of metrics that respect the $\Z_2$-grading,
and define~$A_s$, $X_s$ as in~\XDef.
Define an odd holomorphic function~$k$ by
	$$k(z)={f'(z)-f'(0)\over2z}\;.$$
On~$\overline B=B\times[0,1]$,
we calculate
$$\split
	d^{\overline B}\Biggl(\hochmhalb{N^B}
		\,\strace_V\biggl(k(X_s)\,{\del A_s\over\del s}\biggr)\,ds
		\Biggr)
	&=(2\pi i)\hochhalb{1-N^B}\,\strace_V\biggl(\biggl[A_s,
			k(X_s)\,{\del A_s\over\del s}\biggr]\biggr)\,ds\\
	&=-(2\pi i)\hochhalb{1-N^B}
		\,\strace_V\biggl(k(X_s)\,{\del A^2_s\over\del s}\biggr)\,ds\\
	&={\del\over\del s}\,\biggl(f\(A'_s,g^V_s\)
		-\strace_V\(X_s\)\,f'(0)\biggr)\,ds\;.
\endsplit\Formel\OddClassDLFormel$$
This motivates the following definition.

\Definition\LDef
Set
	$$L_k\(A'_s,g^V_s\)
	=(2\pi i)\hochhalb{1-N^B}\int_0^1
		\strace_V\biggl(k(X_s)\,{\del A_s\over\del s}\biggr)\,ds
	\quad\in\Omega^*(B;\C)\;.$$

These classes resemble the Bott-Chern classes in complex geometry.
We now state Theorem~2.5 of~\BGzwei\ without proof.
\enddefinition

\Theorem\LOddClassThm.
The form~$L_k\(A'_s,g^V_0,g^V_1\)$ is even and real.
If~$\deg k\ge 3$,
then~$L_k\(A'_s,g^V_s\)$
depends modulo~$d^B\Omega^*(B)$ only on~$(A'_s,g^V_s)$ at~$s=0$, $1$
and on the homotopy class of the path~$(A'_s)_{s\in[0,1]}$
in the space of flat superconnections on~$V$.
For~$k(z)={f'(z)-f'(0)\over 2z}$,
	$$d^BL_k\(A'_s,g^V_0,g^V_1\)
	=f\(A'_1,g^V_1\)\dg{\ge3}-f\(A'_0,g^V_0\)\dg{\ge3}\;.$$






\Abschnitt\PreTorsionSubsection Superconnections of total degree 1
and the form~$S_f\(A',g^V\)$

We recall the definition of a superconnection of total degree~$1$.
A flat superconnection of degree~$1$ on a Hermitian vector bundle~$V\to B$
gives rise to a fibre-wise cochain complex
with a metric~$g^H$ and a connection~$\nabla^H$
on its cohomology bundle~$H\to B$.
Then we define a natural form~$S_f\(A',g^V\)\in\Omega^*(B)$
relating the form~$f\(\nabla^H,g^H\)$ to~$f\(A',g^V\)$.



\Definition\TotDegOneDef
A superconnection~$A$ on a $\Z$-graded vector bundle~$V=\bigoplus_jV^j$
is {\em of total degree~$1$\/} iff
	$$A\(\Omega^j(V^k)\)
	\subset\bigoplus_{l\ge 0}\Omega^{j+l}\(V^{k+1-l}\)\;,$$
and~$A$ is {\em of total degree~$-1$\/} iff
	$$A\(\Omega^j(V^k)\)
	\subset\bigoplus_{l\ge 0}\Omega^{j+l}\(V^{k-1+l}\)\;.$$
\enddefinition

\Remark\TotDegAdjRem
Let~$V$ be a $\Z$-graded vector bundle with a Hermition metric~$g^V$.
We say that~$V$ is a {\em $\Z$-graded Hermitian vector bundle\/}
iff~$g^V$ respects the grading,
i.e., iff~$V^j\perp V^k$ for all~$j\ne k$.
In this cases,
a superconnection is of total degree~$1$
iff its adjoint is of total degree~$-1$.
\endremark

Now suppose that~$V\to B$ is a $\Z$-graded Hermition vector bundle,
and let~$\nabla^V$ be any connection on~$V$ with respect to which
all subbundles~$V^j$ are parallel.
Let~$A'$ be a flat superconnection of total degree~$1$,
then
	$$A'=\nabla^V+\sum_ja'_j
		\Mtext{with}
	a'_j\in\bigoplus_k\Omega^j\(B;\Hom(V^k,V^{k+1-j})\)\;,
		\Formel\TotDegOneSuperconn$$
and if~$\nabla^{V,*}$ denotes the adjoint of~$\nabla^V$
and~$A''$ the adjoint of~$A'$ with respect to~$g^V$,
then
	$$A''=\nabla^{V,*}+\sum_ja''_j
		\Mtext{with}
	a''_j=(a'_j)^*\in\bigoplus_k\Omega^j\(B;\Hom(V^k,V^{k+1-j})\)\;.
		\Formel\TotDegMinusOneSuperconn$$

Flatness of~$A'$ implies in particular
	$$(a'_0)^2=0\;,\qquad
	\bigl[\nabla^V+a'_1,a'_0\bigr]=0\;,\Mand
	\(\nabla^V+a'_1\)^2+[a'_0,a'_2]=0\;.
		\Formel\TotDegOneFlatConseq$$
This says first of all that~$(V^*,a'_0)$ is a bundle of cochain complexes.
Next, the coboundary operator~$a'_0$ is parallel
with respect to~$\nabla^V+a'_1$,
so the fibre-wise cohomology~$H^*(V^*,a'_0)$ forms a bundle over~$B$
denoted~$H\to B$,
and~$\nabla^V+a'_1$ induces a connection~$\nabla^H$ on~$H$.
Finally,
the term~$a'_2$ provides a cochain homotopy
between the curvature of~$\nabla^V+a'_1$ and~$0$,
so the induced connection~$\nabla^H$ is flat.
The connection~$\nabla^H$ is called the {\em Gau\ss-Manin connection\/} on~$H$.

If~$\(V,g^V\)\to B$ is a $\Z$-graded Hermitian vector bundle as in
\TotDegAdjRem,
let~$a''_0$ denote the adjoint of~$a'_0$.
By finite dimensional Hodge theory,
we may identify~$H_b$ with~$\ker\(a''_0+a'_0\)_b=\ker\(a''_0-a'_0\)_b$
for~$b\in B$,
so we get an identification of vector bundles
	$$H=\ker\(a''_0+a'_0\)=\ker\(a''_0-a'_0\)\subset V\to B\;.$$
Let~$P^0\in\Gamma\(\Hom(V,H)\)$ denote the orthogonal projection
onto~$H$,
and let~$g^H$ denote the restriction of~$g^V$ to~$H$.
If we take adjoints with respect to~$g^V$ and~$g^H$,
then by~\BL, Proposition~2.6,
we have
	$$\nabla^H=P^0\,\(\nabla^V+a'_1\)\,P^0
		\Mand
	\nabla^{H,*}=P^0\,\(\nabla^{V,*}+a''_1\)\,P^0\;.
		\Formel\NablaVCompressionFormel$$

We replace~$B$ by~$\overline B=B\times\R$,
and let~$\overline V$ be the pullback of~$V$ to~$\overline B$,
with induced superconnection
	$$\overline A'=A'+{\del\over\del t}\,dt$$
as in \TotDegOneSuperconn.
Let~$N^V\in\Gamma(\End V)$ be the number operator
that acts by multiplication with~$i$ on~$V^i$,
and consider the metric~$g^{\overline V}$ on~$\overline V$
with
	$$g^{\overline V}(\punkt,\punkt)_{(b,t)}
	=g^V\(t^{N^V}\punkt,\punkt\)_b\;.$$
If~$A''$ is the adjoint of~$A'$ with respect to~$g^V$
as in~\TotDegMinusOneSuperconn,
then the adjoint of~$\overline A'$ with respect to~$g^{\overline V}$
is given by
	$$\overline A''
	=t^{-N^V}\,\biggl(A''+{\del\over\del t}\,dt\biggr)
	=\nabla^{V,*}+\sum_jt^{1-j}a_j
		+\biggl({\del\over\del t}+{N^V\over t}\biggr)\,dt\;.$$
We have an operator
	$$\overline X={1\over2}\,\(\overline A''-\overline A'\)
	\quad\in\Omega^*\(\overline B;\End\overline V\)\;,$$
and we denote its restriction to~$B\times\{t\}$ by~$X_t$.


If we conjugate by~$t^{N^V\over2}$,
we obtain superconnections
$$\align
	\widetilde A'
	&=t^{\textstyle{N^V\over2}}\,\overline A'\,t^{-\textstyle{N^V\over2}}
	=\nabla^V+\sum_jt^{\textstyle{1-j\over2}}a'_j
		+\biggl({\del\over\del t}-{N^V\over2t}\biggr)\,dt\;,\\
		\Land
	\widetilde A''
	&=t^{\textstyle{N^V\over2}}\,\overline A''\,t^{-\textstyle{N^V\over2}}
	=\nabla^{V,*}+\sum_jt^{\textstyle{1-j\over2}}a''_j
		+\biggl({\del\over\del t}+{N^V\over2t}\biggr)\,dt\;,
\endalign$$
and~$\widetilde A''$ and~$\widetilde A'$ are adjoint to each other
with respect to the pull-back of~$g^V$ with to~$\overline V\to\overline B$
that we still denote by~$g^V$.
Let
	$$\omega^V=\nabla^{V,*}-\nabla^V=\(g^V\)^{-1}\,\bigl[\nabla^V,g^V\bigr]
		\quad\in\Omega^1\(B;\End V\)\;,
	\Formel\OmegaVDef$$
and as in~\XDef, set
	$$\widetilde X={1\over2}\,\(\widetilde A''-\widetilde A'\)
	={\omega^V\over2}
		+\sum_jt^{\textstyle{1-j\over2}}\,{a''_j-a'_j\over2}
		+N^V\,{dt\over2t}\;,$$
and we write~$\widetilde X_t$ for the restriction of~$\widetilde X$
to the submanifold~$B\times\{t\}$.

We define~$\hat f$ to be twice the form with the same name in~\BL,
	$$\hat f\(A',g^V\)
	=(2\pi i)^{-\textstyle{N^B\over2}}
		\,\strace_{\overline V}\biggl(N^V
			\,f'\(\widetilde X\)\biggr)\;.
	\Formel\FHatDef$$
If~$g^V_t=g^V\(t^{N^V}\punkt,\punkt\)$ denotes the restriction
of~$g^{\overline V}$ to~$B\times\{t\}$,
then clearly
	$$f\(\overline A',g^{\overline V}\)
	=f\(\widetilde A',g^V\)
	=f\(\widetilde A'_t,g^V\)
		+\hat f\(\widetilde A'_t,g^V\)\,{dt\over 2t}
	=f\(A',g^V_t\)
		+\hat f\(A',g^V_t\)\,{dt\over 2t}\;,
	\Formel\FVTransgressionFormel$$
and this form is closed on~$B\times\R$.

Again differing from~\BL,
we define
	$$\chi(V)=\strace_V(1)=\sum_j(-1)^j\,\rk(V^i)
		\Mand
	\chi'(V)=\strace_V\(N^V\)
		=\sum_j(-1)^j\,j\,\rk(V^i)\;,$$
so~$\chi(V)=\chi(H)$.
Then by~\BL, Theorem~2.9,
as~$t\to\infty$,
$$\split
	f\(A',g^V_t\)
	&=f\(\nabla^H,g^H\)+O\Bigl(t^{-\textstyle{1\over2}}\Bigr)\;,\\
		\Land
	\hat f\(A',g^V_t\)
	&=f'(0)\,\chi'(H)+O\Bigl(t^{-\textstyle{1\over2}}\Bigr)\;.
\endsplit\Formel\FVInftyFormel$$

We define~$S_f\(A',g^V\)$ as in~\BL, \BGzwei,
where convergence is granted by~\FVInftyFormel.

\Definition\SVDef
Set
	$$S_f\(A',g^V\)
	=-\int_1^\infty\Bigl(\hat f\(A',g^V_t\)
		-f'(0)\,\chi'(H)\Bigr)\,{dt\over2t}
	\quad\in\Omega^*(B)\;.$$
\enddefinition

Then we have the following important result of~\BL, Theorems~2.16,
and~\BGzwei, Theorem~1.25.

\Theorem\SVThm.
The form~$S_f\(A',g^V\)$ is even and real,
and satisfies
	$$d^B\,S_f\(A',g^V\)
	=f\(A',g^V\)-f\(\nabla^H,g^H\)\;.$$

\Proof
This follows from~\FVTransgressionFormel\ and~\FVInftyFormel.
\qed\enddemo

Let now~$g^V$, $g^{\prime V}$ be two metrics on~$V\to B$,
and let~$g^H$, $g^{\prime H}$ be the corresponding metrics on~$H\to B$.
From~\SVThm\ and~\FslDef,
we easily deduce that
	$$S_f\(A',g^{\prime V}\)-S_f\(A',g^V\)
	=\tilde f\(A',g^V,g^{\prime V}\)
		-\tilde f\(A',g^H,g^{\prime H}\)\;.
	\Formel\SVVariationFormel$$

In order to control the behaviour of~$S_f\(A',g^{\prime V}\)$
under variation of~$A'$,
we assume for a moment that~$f$ is an odd homolomorphic function
of degree~$\ge 3$ that satisfies~\GenFnProperty.
Then~$k(z)={f'(z)\over2z}$ is also holomorphic.
Let~$(A'_s,g^V_s)_{s\in[0,1]}$ be a family of flat superconnections
and $\Z$-graded metrics on~$V$.
We assume that the dimension of the cohomology bundles~$H_s$
is independent of~$s$,
then these bundle combine to form a family over~$B\times[0,1]$,
and we get a family~$(\nabla^{H,s},g^H_s)_{s\in[0,1]}$ 
of flat connections and metrics.
We recall Theorem~2.15 of~\BGzwei,
again without proof.

\Theorem\LSVThm.
Modulo exact forms on~$B$,
we have
	$$S_f\(A'_1,g^V_1\)-S_f\(A'_0,g^V_0\)
	=L_k\(A'_s,g^V_0,g^V_1\)-L_k\(\nabla^{H,s},g^H_0,g^H_1\)\;.$$





\Abschnitt\MVBSubsection Family Thom-Smale complexes
and their torsion forms

We define a family Thom-Smale complex
to be a $\Z$-graded complex vector bundle~$V\to B$
with a flat superconnection~$A'=\nabla^V+a'$ of total degree~$1$
such that the connection~$\nabla^V$ is diagonal
and the coefficients~$a'$ are upper triangular
with respect to the eigenspace decomposition of an endomorphism~$h^V$,
in a sense to be made precise.
For such a vector bundle, together with an adapted metric~$g^V$,
we define a torsion form~$T_f\(A',g^V,h^V\)$
relating~$f\(\nabla^H,g^H\)$ to~$f\(\nabla^V,g^V\)$.
In the special case that~$a'_j=0$ for~$j\ne 0$,
we may take~$h^V=N^V$,
and our torsion forms equal those of~\BL, Chapter~2,
that appeared in Theorem~0.1 of~\BGzwei.

\Definition\MVBDef
Let~$V$ be a $\Z$-graded complex vector bundle
as in \TotDegAdjRem,
let~$A'=\nabla^V+a'$ be a flat superconnection of total degree~$1$,
and let~$h^V\in\Gamma\(\End^0V\)$ be an endomorphism of~$V$ of degree~$0$.
Then~$\(V,A',h^V\)$ is called a {\em family Thom-Smale complex\/}
iff for each compact subset~$K\subset B$ there exists~$\eps>0$,
and for each point~$b\in K$ there exists a neighbourhood~$U$ of~$b$,
such that
\roster
\item for each~$j$,
there is a splitting~$V^j|_U=\bigoplus_\alpha V^j_\alpha$,
which is preserved by~$\nabla^V$;
\item
the endomorphism~$h^V$ acts on~$V^j_\alpha$
by a function~$h^j_\alpha\colon U\to\R$;
\item
for each~$j$ and each~$\alpha$,
	$$a'\(V^j_\alpha\)
	\subset\bigoplus_{\textstyle{k,\beta
			\atop h^j_\alpha+\eps\le h^k_\beta}}
		\Omega^*\(B;V^k_\beta\)\;.$$
\endroster
If~$F\colon M\to B$ is a smooth map,
then the pullback~$F^*\(V,A',h^V\)=\(F^*V,F^*A',F^*h^V\)$
is defined in the obvious way.
\enddefinition

It seems that we should include~$\nabla^V$ in the notation
for a family Thom-Smale complex.
However,
the following statement shows that this is not necessary.

\Proposition\MVBFlatConnProp.
The connection~$\nabla^V$ in \MVBDef\ is uniquely determined
by~$A'$ and~$h^V$.
Moreover, it is flat.

\Proof
Fix a compact subset~$K\subset B$, a point~$b\in K$
and a small open neighbourhood~$U$ of~$b$ in~$B$ as above.
Let~$V|_U=\bigoplus_j\bigoplus_\alpha V^j_\alpha$ be the local splitting
of~(1) above,
and let~$P^j_\alpha\colon V|_U\to V^j_\alpha$
be the corresponding product projections.
By~(3),
we clearly have
	$$P^j_\alpha\,A'\,P^j_\alpha
	=P^j_\alpha\,\nabla^V\,P^j_\alpha
	=\nabla^{V^j_\alpha}$$
over~$U$,
and since~$\nabla^V$ preserves~$V^j_\alpha$,
the connection~$\nabla^{V^j_\alpha}$ is simply
the restriction of~$\nabla^V$ over~$U$ to~$V^j_\alpha$.
In particular, $\nabla^V|_U=\bigoplus_j\bigoplus_\alpha\nabla^{V^j_\alpha}$
is uniquely determined by~$A'$.

Similarly, we can calculate the curvature of~$\nabla^{V^j_\alpha}$ by
	$$\(\nabla^{V^j_\alpha}\)^2
	=P^j_\alpha\,\(\nabla^V\)^2\,P^j_\alpha
	=P^j_\alpha\,\(A'\)^2\,P^j_\alpha
	=0\;,$$
again using~(3) to show that the derivatives of~$a'$
do not contribute to~$\(\nabla^{V^j_\alpha}\)^2$.
\qed\enddemo

For example,
if~$A'=\nabla^V+a'_0$ is a flat family of complexes as in~\BL, section~2.f,
that is, $a'_0\in\Omega^0\(B;\End^1V\)$,
we may take~$h^V=N^V$ in order to obtain a family Thom-Smale complex.
If we constructed~$V$, $\nabla^V$ and~$A'$ as in \ComplexKapitel,
then we may choose~$h^V$ to act on~$V_\alpha\subset V$ by~$h_\alpha$.

\Definition\MorseMetricDef
Let~$\(V,A',h^V\)$ be a family Thom-Smale complex.
An {\em adapted metric\/} on~$V$ is a metric
	$$g^V=\bigoplus_ig^{V^i}\Rtext{on~$V$}$$
such that the splitting of \MVBDef~(1) is $g^V$-orthogonal.
\enddefinition

\Remark\MVBMetricRem
If~$(V,A',h^V)$ is a family Thom-Smale complex and~$g^V$ is an adapted metric,
then the endomorphisms~$h^V$ and~$N^V$ are self-adjoint with respect to~$g^V$.
Because~$\nabla^V$ and~$g^V$ respect the splitting of~$V$ in \MVBDef~(1),
so do~$\nabla^{V,*}$ and~$\omega^V$.
If we set
	$$\nabla^{V,\roman u}={1\over2}\,\(\nabla^{V,*}+\nabla^V\)\;,$$
then~$\nabla^{V,\roman u}$ also preserves that splitting.
If we write
	$$dh^V=\Bigl[\nabla^V,h^V\Bigr]
	=\Bigl[\nabla^{V,*},h^V\Bigr]=\Bigl[\nabla^{V,\roman u},h^V\Bigr]\;,$$
then~$dh^V$ acts on~$V^j_\alpha$ by~$dh^j_\alpha$.
We have the obvious relations
	$$\Bigl[\nabla^{V,\roman u},dh^V\Bigr]
	=\Bigl[\nabla^{V,\roman u},\omega^V\Bigr]
	=\Bigl[h^V,\omega^V\Bigr]
	=\Bigl[dh^V,\omega^V\Bigr]
	=0\;.$$
\endremark

Let~$\overline A'=A'+{\del\over\del t}\,dt+{\del\over\del T}\,dT$
denote the trivial extension of~$A'$ to the pullback~$\overline V$
of~$V$ to~$B\times\R_>\times\R$.
Define a metric~$g^{\overline V}$ on~$\overline V$ by
	$$g^{\overline V}=g^V\(t^{N^V}\,T^{h^V}\punkt,\punkt\)\;.
	\Formel\GtTDef$$
If we take the adjoint of~$A'$ with respect to~$g^{\overline V}$,
we obtain a flat superconnection
	$$\overline A''
	=\nabla^{V,*}+\sum_jt^{1-j}\,T^{-h^V}\,a''_j\,T^{h^V}
		+\log T\,dh^V
		+\biggl({\del\over\del t}+{N^V\over t}\biggr)\,dt
		+\biggr({\del\over\del T}+{h^V\over T}\biggr)\,dT$$
and an operator
	$$\overline X
	={1\over2}\,\(\overline A''-\overline A'\)\;.$$
We denote the restrictions of~$g^{\overline V}$, $\overline A'$,
$\overline A''$, $\overline X$ to~$B\times\{(t,T)\}$
by~$g^V\tT$, $A'\tT$, $A''\tT$ and~$X\tT$.

If we conjugate everything with~$t\hochhalb{N^V}\,T\hochhalb{h^V}$,
we obtain two superconnections
$$\split
	\widetilde A'
	&=\nabla^V+\sum_jt^{\textstyle{1-j\over2}}\,T^{\textstyle{h^V\over2}}
			\,a'_j\,T^{-\textstyle{h^V\over2}}
		-{\log T\over2}\,dh^V
% dh^V=\bigl[\nabla^V,h^V\bigr]
		+\biggl({\del\over\del t}-{N^V\over2t}\biggr)\,dt
		+\biggr({\del\over\del T}-{h^V\over2T}\biggr)\,dT\\
			\Land
	\widetilde A''
	&=\nabla^{V,*}+\sum_jt^{\textstyle{1-j\over2}}
			\,T^{-\textstyle{h^V\over2}}
			\,a''_j\,T^{\textstyle{h^V\over2}}
		+{\log T\over2}\,dh^V
% dh^V=\bigl[\nabla^{V,*},h^V\bigr]
		+\biggl({\del\over\del t}+{N^V\over2t}\biggr)\,dt
		+\biggr({\del\over\del T}+{h^V\over 2T}\biggr)\,dT
\endsplit\Formel\MVBAslFormel$$
that are adjoints of each other with respect to~$g^V$,
and a $g^V$-skew-adjoint operator
	$$\widetilde X
	={1\over2}\,\(\widetilde A''-\widetilde A'\)\;.
	\Formel\MVBXslFormel$$

Fix a compact subset~$K$ of~$B$ and let~$\eps$ be the constant of \MVBDef.
From now on, we keep~$t$ fixed.
Then
$$\split
	\widetilde X\tT
	&={\omega^V\over2}+{1\over2}\sum_jt^{\textstyle{1-j\over2}}\Bigl(
		T^{-\textstyle{h^V\over2}}\,a''_j\,T^{\textstyle{h^V\over2}}
		-T^{\textstyle{h^V\over2}}\,a'_j\,T^{-\textstyle{h^V\over2}}
		\Bigr)+{\log T\over2}\,dh^V
% dh^V=\bigl[\nabla^{V,\roman u},h^V\bigr]
\\
	&={\omega^V\over2}+{\log T\over2}\,dh^V
		+O\Bigl(T^{\textstyle{\eps\over2}}\Bigr)
\endsplit\Formel\XtildeZeroLimit$$
uniformly on~$K$ as~$T\to 0$.

Define~$\nabla^{V,\roman u}$ as in \MVBMetricRem,
then since~$f$ and~$\omega^V+\log T\,dh^V$ are odd
with~$[\omega^V,dh^V]=\(dh^V\)^2=0$,
and~$\omega^V+\log T\,dh^V$ commutes with~$h^V$,
we have
$$\alignat2
	&&\strace_V\biggl(h^V
		\,f'\Bigl({\omega^V+\log T\,dh^V\over2}\Bigr)\biggr)
	&=\strace_V\(h^V\)\,f'(0)\;,\\
		\text{and}
	&&\strace_V\biggl(f\Bigl({\omega^V+\log T\,dh^V\over2}\Bigr)\biggr)
	&=\strace_V\biggl(f\Bigl({\omega^V\over2}\Bigr)\biggr)
		+{\log T\over2}\,\strace_V\biggl(dh^V
			\,f'\Bigl({\omega^V\over2}\Bigr)\biggr)\;,\\
		\text{with}
	&&\strace_V\biggl(dh^V
			\,f'\Bigl({\omega^V\over2}\Bigr)\biggr)
	&=\strace_V\biggl(\biggl[\nabla^{V,\roman u}
		,h^Vf'\Bigl({\omega^V\over2}\Bigr)\biggr]\biggr)\\
	&&&=d^B\strace_V\biggl(h^Vf'\Bigl({\omega^V\over2}\Bigr)\biggr)
	=\strace_V\(dh^V\)\,f'(0)
\endalignat$$
by \MVBMetricRem,
so
$$\align
	f\biggl(\nabla^V+{\del\over\del T}\,dT
				,g^{\overline V}\biggr)
	&=f\Biggl(\nabla^V-{\log T\over2}\,dh^V
			+\biggl({\del\over\del T}-{h^V\over2T}\biggr)\,dT
			,g^V\Biggr)\\
	&=(2\pi i)^{\textstyle{1-N^B\over2}}
		\,\strace_V\Biggl(
			\,f\biggl({\omega^V+\log T\,dh^V\over2}\biggr)\Biggr)\\
	&\qquad+(2\pi i)^{-\textstyle{N^B\over2}}
		\,\strace_V\Biggl(h^Vf'\biggl({\omega^V
			+\log T\,dh^V\over2}\biggr)\Biggr)\,{dT\over2T}\\
	&=f\(\nabla^V,g^V\)
		+d^{\overline B}\biggl({\log T\over2}
			\,\strace_V\(h^V\)\biggr)\,f'(0)\;.
\endalign$$

Define an odd closed and real form
	$$\alpha
	=f\Bigl(\overline A',g^{\overline V}\Bigr)
		-f\biggl(\nabla^V+{\del\over\del t}\,dt+{\del\over\del T}\,dT
				,g^{\overline V}\biggr)
	\quad\in\Omega^*\(B\times(0,\infty)\times\R\)\;,
	\Formel\MVBAlphaDef$$
then for fixed~$t$,
$$\split
	\alpha_t
%	&=f\(\widetilde A',g^V\)|_{B\times\{t\}\times\R}
%		-f\Biggl(\nabla^V-{\log T\over2}\,dh^V
%			+\biggl({\del\over\del T}-{h^V\over2T}\biggr)\,dT
%			,g^V\Biggr)\\
	&=f\(\widetilde A'\tT,g^V\)-f\(\nabla^V,g^V\)\\
	&\qquad
		+(2\pi i)^{-\textstyle{N^B\over2}}\,\strace_V\Bigl(h^V
			\,f'\(\widetilde X_{t,T}\)\Bigr)\,{dT\over2T}
		-d^{\overline B}\biggl({\log T\over2}
			\,\strace_V\(h^V\)\biggr)\,f'(0)\;.
\endsplit\Formel\FVBarTransgression$$

For a fixed compact~$K\subset B$,
let~$\eps$ be as in \MVBDef,
and choose~$0<\eps'<{\eps\over2}$.
Using~\XtildeZeroLimit,
we compute
$$\split
	\alpha_t
	&=(2\pi i)^{\textstyle{1-N^B\over2}}
		\,\strace_V\Biggl(f\biggl({\omega^V+\log T\,dh^V\over2}
			+O\Bigl(T^{\textstyle{\eps\over2}}\Bigr)\biggr)\Biggr)
	-f\(\nabla^V,g^V\)\\
	&\qquad
		+(2\pi i)^{-\textstyle{N^B\over2}}\,\strace_V\Biggl(h^V
			\,f'\biggl({\omega^V+\log T\,dh^V\over2}
				+O\Bigl(T^{\textstyle{\eps\over2}}\Bigr)\biggr)
			\Biggr)\,{dT\over2T}\\
	&\qquad
		-d^{\overline B}\biggl({\log T\over2}
			\,\strace_V\(h^V\)\biggr)\,f'(0)\\
	&=O\(T^{\eps'}\)+O\(T^{\eps'}\)\,{dT\over2T}
\endsplit\Formel\AlphaTZeroLimit$$
as~$T\to 0$,
where the terms~$O\(T^{\eps'}\)$ do not involve the exterior variable~$dT$.
Using this formula and~\FVInftyFormel,
we find that the integrals in the following definition converge.

\Definition\MVBTorsionDef
Let~$(V,A',h^V)$ be a family Thom-Smale complex
with an adapted metric~$g^V$,
then we define its {\em torsion form\/}
in~$\Omega^*(B)/d^B\Omega^*(B)$ by
$$\align
	T_f\(A',g^V,h^V\)
	&=-(2\pi i)^{-\textstyle{N^B\over2}}\,\int_1^\infty\Bigl(
		\strace_V\(N^Vf'(X_{t,1})\)-\chi'(H)f'(0)\\
	&\kern10em
		-\(\chi'(V)-\chi'(H)\)
			\,f'\(\sqrt{-t}/2\)\Bigr)
		\,{dt\over2t}\\
\noalign{\smallskip}
	&\qquad
		-(2\pi i)^{-\textstyle{N^B\over2}}\,\int_0^1\Bigl(
		\strace_V\(h^Vf'(X_{1,T})\)
			-\strace_V\(h^V\)\,f'(0)\\
	&\kern10em
		-\(\chi'(V)-\chi'(H)\)
			\,\(f'(\sqrt{-T}/2)-f'(0)\)\Bigr)
		\,{dT\over2T}\;.
\endalign$$
\enddefinition

\Remark\MVBTorsionRem
Suppose that~$A'=\nabla^V+a_0$ is a flat superconnection
on a $\Z$-graded Hermitian vector bundle~$(V,g^V)$,
then the form~$T_f\(A',g^V,N^V\)$
clearly equals the form
$$\multline
	T_f\(A',g^V\)
	=-(2\pi i)^{-\textstyle{N^B\over2}}\,\int_0^\infty\Bigl(
		\strace_V\(N^Vf'(X_{t,1})\)\\
		-\chi'(H)f'(0)
		-\(\chi'(V)-\chi'(H)\)\,f'\(\sqrt{-t}/2\)\Bigr)
		\,{dt\over2t}
\endmultline\tag\BGzweiTorsionForm$$
of~\BL, \BGzwei.
This motivates some of the correction terms in \MVBTorsionDef.
In particular,
the component of degree~$0$
can be written as
	$$T_f\(A',g^V,N^V\)\dg0
	={1\over2}\,\sum_j\strace_{H^\perp}\(N^V\,\log(a'_0+a''_0)^2\)\;.$$

\endremark


\Theorem\MVBTorsionThm.
The form~$T_f\(A',g^V,h^V\)\in\Omega^*(B)$ is even and real,
and if~$F\colon M\to B$ is a smooth map,
then
	$$T_f\(F^*A',F^*g^V,F^*h^V\)
	=F^*T_f\(A',g^V,h^V\)\;.$$
We have
	$$d^BT_f\(A',g^V,h^V\)
	=f\(\nabla,g^V\)-f\(\nabla^H,g^H\)\;.$$
The component of~$T_f\(A',g^V,h^V\)$ in degree zero is given by
	$$T_f\(A',g^V,h^V\)\dg0
	={1\over2}\,\strace\(N^V\,\log(a'_0+a''_0)^2|_{\ker(a'_0+a''_0)^\perp}\)\;.$$

\Proof
The form~$T_f\(A',g^V,h^V\)$ is even and real
because~$f\bigl(\overline A',g^{\overline V}\bigr)$
is odd on~$B\times\R_>\times\R$ and real by \FVThm.
The first equation of \MVBTorsionThm\ is clear from \MVBTorsionDef.

It follows from~\SVDef\ and \SVThm\ that
$$\multline
	-d^B\Biggl((2\pi i)^{-\textstyle{N^B\over2}}\,\int_1^\infty\Bigl(
		\strace_V\(N^Vf'(X_{t,1})\)-\chi'(H)f'(0)\\
		-\(\chi'(V)-\chi'(H)\)
			\,f'\(\sqrt{-t}/2\)\Bigr)
		\,{dt\over2t}\Biggr)\\
	=d^B\,S_f\(A',g^V\)
	=f\(A',g^V\)-f\(\nabla^H,g^H\)\;.
\endmultline\tag\FormelNummer\MVBProofOne$$
Fix~$K\subset B$ compact and~$\eps'>0$ as above,
then equations~\FVBarTransgression\ and~\AlphaTZeroLimit\
imply that similarly,
$$\multline
	-d^B\Biggl((2\pi i)^{-\textstyle{N^B\over2}}\,\int_0^1\Bigl(
		\strace_V\(h^Vf'(X_{1,T})\)
			-\strace_V\(h^V\)\,f'(0)\\
		-\(\chi'(V)-\chi'(H)\)
			\,\(f'(\sqrt{-T}/2)-f'(0)\)\Bigr)
		\,{dT\over2T}\Biggr)\\
	=-d^B\int_{B\times[0,1]/B}\alpha_1
	=-\alpha_{1,1}
	=f\(\nabla^V,g^V\)-f\(A',g^V\)\;.
\endmultline\tag\FormelNummer\MVBProofTwo$$
The second equation in \MVBTorsionThm\ follows from~\MVBProofOne\
and~\MVBProofTwo.

Now fix~$b\in B$ and consider the $\Z$-graded Hermitian
vector bundle~$\(V_b\times[0,1],g^{V_b}\)\to[0,1]$
with the trivial flat connection~$\nabla^0$.
This bundle admits a flat superconnection~$A'_b=a'_0+\nabla^0$.
Let~$u$ denote the coordinate of~$[0,1]$,
then~$\(V_b\times[0,1],A'_b,u\,h^V+(1-u)\,N^V\)$
is a family Thom-Smale complex with adapted metric~$g^{V_b}$.
We have
	$$dT_f\(A'_b,g^{V_b},u\,h^V+(1-u)\,N^V\)
	=f\(\nabla^0,g^{V_b}\)-f\(\nabla^H,g^H\)
	=0\quad\in\quad\Omega^*([0,1])\;.$$
On the other hand, Theorem~2.25 in~\BL\
shows that at~$u=0$,
we obtain the classical Reidemeister torsion of the complex~$(V,a_0')$,
which proves the last assertion.
\qed\enddemo

\Remark\MVBEquivRem
Suppose that~$\gamma\in\Gamma\(\End V\)$
acts by fibre-wise isometries on~$V$,
leaving~$A'$ and~$h^V$ invariant.
Then there is an obvious equivariant analogue~$T_{f,\gamma}\(A',g^V,h^V\)$
satisfying an equivariant version of \MVBTorsionThm.
\endremark


\Abschnitt\MVBRigiditySubsection Rigidity of the forms~$T_f\(A',g^V,h^V\)$

We analyse the behaviour of~$T_f\(A',g^V,h^V\)$ under smooth variation
of~$A'$, $g^V$ and~$h^V$.

\Definition\MVBVariationDef
Let~$V\to B$ be a $\Z$-graded complex vector bundle,
and let~$(A'_s,g^V_s,h^V_s)_{s\in[0,1]}$ be family of flat superconnections
of total degree~$1$ with~$A'_s=\nabla^{V,s}+a'_s$,
$\Z$-graded Hermitian metrics, and $\Z$-graded endomorphisms on~$V$.
We call~$(A'_s,g^V_s,h^V_s)_{s\in[0,1]}$ a
{\em family of family Thom-Smale complexes\/}
iff for each compactum~$K\subset B\times[0,1]$ there exists~$\eps>0$,
and for each point~$b\in K$ there exists a neighbourhood~$U$ of~$b$
in~$B\times[0,1]$,
such that \MVBDef~(1)--(3) hold over~$U$.
\enddefinition

Note that both in \MVBDef\ and here,
the splittings of~$V^j|_U=\bigoplus_\alpha V^j_\alpha$
are allowed to depend on~$U$.
In particular,
it may happen that several eigenvalues of~$h^V$ run together
over a certain subset of~$B\times[0,1]$.

Let us assume for the moment that~$f$ is an odd holomorphic function
of degree~$\ge 3$ that satisfies~\GenFnProperty, cf.\ \LSVThm,
and set~$k(z)={f'(z)\over2z}$ as before.
We also assume that the dimension of the cohomology bundles~$H_s$
is independent of~$s\in[0,1]$,
so that they combine to form a bundle on~$B\times[0,1]$.
Let~$(\nabla^{H,s},g^H_s)_{s\in[0,1]}$ be the induced family
of flat connections and Hermitian metrics.

\Proposition\MVBRigidityProp.
Modulo exact forms on~$B$,
we have
	$$T_f\(A'_1,g^V_1,h^V_1\)-T_f\(A'_0,g^V_0,h^V_0\)
	=L_k\(\nabla^{V,s},g^V_0,g^V_1\)-L_k\(\nabla^{H,s},g^H_0,g^H_1\)\;.$$


\Proof
Because we already know \LSVThm,
we only have to care about the integral over~$T$ in \MVBTorsionDef.
For fixed~$t$,
define a form
$$%\multline
	\ell_t
	=(2\pi i)\hochmhalb{N^B}\,\Biggl(\strace_V\biggl(k\(\overline X_s\)
			\,{\del\overline A_s\over\del s}\biggr)%\\
		-\strace_V\biggl(k\biggl({\omega^V+\log T\,dh^V_s\over2}
				+{h^V_s\over2T}\,dT\biggr)
			\,{\del\nabla^{V,s,\roman u}\over\del s}\biggr)
		\Biggr)\,ds
%\endmultline
$$
on~$B\times[0,1]\times\{t\}\times\R$.
As in~\OddClassDLFormel,
we have
	$$d^{B\times[0,1]\times\{t\}\times\R}\ell_t
	=f\Bigl(\overline A'_s,g^{\overline V}_s\Bigr)
		-f\biggl(\nabla^{V,s}+{\del\over\del T}\,dT
			,g^{\overline V}_s\biggr)
	=\alpha_{s,t}\,ds\;,$$
where~$\alpha_{s,t}$ is the form defined in~\MVBAlphaDef.
In particular,
the form~$\ell_t+\alpha_t$ is closed on~$B\times[0,1]\times\{t\}\times\R$.

As~$T\to 0$,
we have
$$\align
	{\del\widetilde A_{s,t}\over\del s}
	&=\nabla^{V,s,\roman u}+O\Bigl(T\hochhalb\eps\Bigr)\\
		\Land
	\widetilde X_{s,t}
	&={\omega^{V,s}+\log T\,dh^V_s\over2}+{h^V_s\over2T}\,dT
		+O\Bigl(T\hochhalb\eps\Bigr)\;.
\endalign$$
This implies that
	$$\ell_t=O\Bigl(T\hochhalb\eps\Bigr)
		+O\Bigl(T\hochhalb\eps\Bigr)\,{dT\over2T}
	\Formel\EllTLimitFormel$$
uniformly in~$s$ as~$T\to 0$,
where the forms~$O\(T\hochhalb\eps\)$
do not contain the exterior variable~$dT$.

By~\AlphaTZeroLimit, \EllTLimitFormel\ and Stokes' theorem,
	$$d^B\int_{B\times[0,1]\times[0,1]/B}(\ell_1+\alpha_1)
	=\int_{B\times[0,1]/B}(\alpha_{1,1}-\alpha_{0,1})
		-\int_{B\times[0,1]/B}\ell_{1,1}$$
for~$t=1$ fixed,
where the first integration on the right hand side is in the direction of~$T$,
and the second in the direction of~$s$.
Together with \LSVThm,
this proves our proposition.
\qed\enddemo

The next step towards the general case is an analogue of Theorem~2.18
in~\BGzwei.
Let~$f$ be an odd holomorphic function satisfying~\GenFnProperty.
Let~$a>0$, let~$R$ be a polynomial,
and set
	$$f_a(z)
	=R\biggl({\del\over\del a}\biggr)\Bigr|_{a=1}
		\,{h\bigl(\sqrt a\,z\bigr)\over\sqrt a}\;.
	\Formel\RescaledGenFn$$
Then~$f_a$ also satisfies~\GenFnProperty.

\Proposition\MVBTorsionRescalingProp.
The torsion with respect to the generating function~$f_a$ is
up to exact forms on~$B$ given by
	$$T_{f_a}\(A',g^V,h^V\)
	=R\biggl({\del\over\del a}\biggr)\Bigr|_{a=1}
		\Bigl(a\hochhalb{N^B}\,T_f\(A',g^V,h^V\)\Bigr)\;.$$

\Proof
Let us regard the form~$\alpha$ of~\MVBAlphaDef\ for variable~$t$,
but~$T=1$.
Because
	$$f\biggl(\nabla^V+{\del\over\del t}\,dt+{\del\over\del T}\,dT
				,g^{\overline V}\biggr)\Bigr|_{T=1}
	=f\(\nabla^V,g^V_t\)+\chi'(V)\,{dt\over2t}\;,$$
we have
	$$\alpha|_{T=1}
	=f\(\overline A',g^{\overline V}\)
		-f\(\nabla^V,g^V_t\)-\chi'(V)\,f'(0)\,{dt\over2t}\;.$$
Setting
	$$\beta
	=\(\chi'(V)-\chi'(H)\)\,\Bigl(f'\(\sqrt{-t}/2\)-f'(0)\Bigr)
			\,\biggl({dt\over2t}+{dT\over2T}\biggr)\;,$$
we may therefore rewrite \MVBTorsionDef\ as
	$$T_f\(A',g^V,h^V\)
	=-\int_{B\times{\textstyle(}\{1\}\times[0,1]
			\;\cup\;(1,\infty)\times\{1\}{\textstyle)/}B}
		(\alpha-\beta)\;.$$

Rescaling~$t$ to~$at$ gives
$$\split
	\widetilde X_{at}
	&={\omega^V\over2}
		+\sum_j(at)^{1-j\over2}\Bigl(T\hochmhalb{h^V}
				\,a''_j\,T\hochhalb{h^V}
			-T\hochhalb{h^V}\,a'_j\,T\hochmhalb{h^V}\Bigr)
		+{\log T\over2}\,dh^V+{h^V\over2T}\,dT\\
	&=a\hochhalb{1-N^B}\,\widetilde X_t\,a\hochhalb{N^B}\;.
\endsplit$$
This implies that
	$$\biggl({f\bigl(\sqrt a\punkt\bigr)\over\sqrt a}\biggr)'(X\tT)
	=f'\(\sqrt a\,X\tT\)
%	=f'\Bigl(a\hochhalb{N^B}\,X_{at,T}\,a\hochmhalb{N^B}\Bigr)
	=a\hochhalb{N^B}\,f'(X_{at,T})\,a\hochmhalb{N^B}\;.$$

We may thus express the torsion form with respect
to~${f(\sqrt a\punkt)\over\sqrt a}$ as
$$\align
	\kern3em&\kern-3em
	T_{\textstyle{f(\sqrt a\punkt)\over\sqrt a}}\(A',g^V,h^V\)\\
	&=-a\hochhalb{N^B}\,\int_{B\times(1,\infty)\times\{1\}/B}
		\Bigl(\strace_V\(f'(X_{at,T})\)-\chi'(H)\,f'(0)\\
	&\kern13em
			-\(\chi'(V)-\chi'(H)\)\,f'\(\sqrt{-at}/2\)
			\Bigr)\,{dt\over2t}\\
	&\qquad
	-a\hochhalb{N^B}\,\int_{B\times\{a\}\times[0,1]/B}
		\biggl(\alpha-\(\chi'(V)-\chi'(H)\)
			\,\Bigl(f'\(\sqrt{-aT}/2\)-f'(0)\Bigr)
			\,{dT\over2T}\biggr)
\allowdisplaybreak
	&=-a\hochhalb{N^B}\int_{B\times{\textstyle(}[1,a]\times\{0\}
			\;\cup\;\{a\}\times[0,1]
			\;\cup\;(a,\infty)\times\{1\}{\textstyle)/}B}
		(\alpha-\beta)\;.
\endalign$$
Here we have used that~$\alpha$ vanishes on~$B\times[1,a]\times\{0\}$
by~\AlphaTZeroLimit,
and that~$\beta$ is symmetric in~$t$ and~$T$.
Since the integrand above is closed,
we may integrate along any path from~$(\infty,1)$ to~$(1,0)$,
changing the result only by an exact form,
so
	$$T_{\textstyle{f(\sqrt a\punkt)\over\sqrt a}}\(A',g^V,h^V\)
	=a\hochhalb{N^B}\,T_f\(A',g^V,h^V\)
		+a\hochhalb{N^B}\,d^B\int_{B\times[1,a]\times[0,1]/B}
			(\alpha-\beta)\;.$$

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\centerline{\NeueFigur\RescalingPathFig
\RescalingPathFig. Paths in the construction of~$T_f$ and~$T_{f_\alpha}$.}
\endinsert


Our theorem now follows,
since
$$\align
	T_{f_a}\(A',g^V,h^V\)
	&=R\biggl({\del\over\del a}\biggr)\Bigr|_{a=1}
		\,T_{\textstyle{f(\sqrt a\punkt)\over\sqrt a}}\(A',g^V,h^V\)\\
	&=R\biggl({\del\over\del a}\biggr)\Bigr|_{a=1}
		\Bigl(a\hochhalb{N^B}\,T_f\(A',g^V,h^V\)\Bigr)\;.\quad\qed
\endalign$$
\enddemo

We can now finally prove the analogue of~\BGzwei, Theorem~2.20.
We return to the convention~\GenFnChoice\ that~$f(z)=z\,e^{z^2}$,
and we set~$k(z)={f'(z)-f'(0)\over2z}$.

\Theorem\MVBTorsionRigidityThm.
Modulo exact forms on~$B$,
we have
	$$\Bigl(T_f\(A'_1,g^V_1,h^V_1\)-T_f\(A'_0,g^V_0,h^V_0\)\Bigr)\dg{\ge 2}
	=L_k\(\nabla^{V,s},g^V_0,g^V_1\)
		-L_k\(\nabla^{H,s},g^H_0,g^H_1\)\;.$$


\Remark\MVBLRem
Note that the right hand side is independend modulo~$d\Omega^*(B)$
of the path connecting~$\nabla^{V,0}$ with~$\nabla^{V,1}$
and~$\nabla^{H,0}$ with~$\nabla^{H,1}$,
even though both $L_k$-terms may depend on the homotopy class of these paths
in degree~$\ge 4$,
and on the paths themselves in degree~$2$.
\endremark


\Proof
Choose~$f(z)=z\,e^{z^2}$ as in~\GenFnChoice\
and set~$R(X)=X$ in~\RescaledGenFn,
so that~$f_a(z)=z^3\,e^{z^2}$ is now of degree~$3$,
and let~$k_a(z)={f'_a(z)\over2z}$.
By~\MVBRigidityProp\ and~\MVBTorsionRescalingProp,
$$\align
	\kern6em&\kern-6em
	L_{k_a}\(\nabla^{V,s},g^V_0,g^V_1\)
	-L_{k_a}\(\nabla^{H,s},g^H_0,g^H_1\)\\
%	&=T_{f_a}\(A'_1,g^V_1,h^V_1\)-T_{f_a}\(A'_0,g^V_0,h^V_0\)\\
	&={\del\over\del a}\Bigr|_{a=1}
		\biggl(a\hochhalb{N^B}\,\Bigl(T_f\(A'_1,g^V_1,h^V_1\)
			-T_f\(A'_0,g^V_0,h^V_0\)\Bigr)\biggr)\\
	&={N^B\over2}\,\Bigl(T_f\(A'_1,g^V_1,h^V_1\)
		-T_f\(A'_0,g^V_0,h^V_0\)\Bigr)\;.
\endalign$$

On the other hand,
we have
	$$k_a(z)
	={3z+z^3\over2}\,e^{z^2}
	={\del\over\del a}\Bigr|_{a=1}
		\biggl({(1+2az^2)\,e^{az^2}-1\over2z}\biggr)
	={\del\over\del a}\Bigr|_{a=1}\Bigl(\sqrt a\,k\(\sqrt a\,z\)\Bigr)\;.$$
It is now easy to check that
$$\align
	L_{k_a}\(\nabla^{V,s},g^V_0,g^V_1\)
	&={\del\over\del a}\Bigr|_{a=1}\int_0^1
		\strace_V\Biggl(k\biggl(\sqrt a\,{\omega^{V,s}\over2}\biggr)
			\,\sqrt a\,{\del\nabla^{V,s,\roman u}\over\del s}
			\Biggr)\,ds\\
	&={\del\over\del a}\Bigr|_{a=1}\int_0^1a\hochhalb{N^B}
		\,\strace_V\Biggl(k\biggl({\omega^{V,s}\over2}\biggr)
			\,{\del\nabla^{V,s,\roman u}\over\del s}\Biggr)\,ds\\
	&={N^B\over2}\,L_k\(\nabla^{V,s},g^V_0,g^V_1\)\;.
\endalign$$

Together, these equations give
	$${N^B\over2}\,\Bigl(T_f\(A'_1,g^V_1,h^V_1\)
		-T_f\(A'_0,g^V_0,h^V_0\)\Bigr)
	={N^B\over2}\,\Bigl(L_k\(\nabla^{V,s},g^V_0,g^V_1\)
		-L_k\(\nabla^{H,s},g^H_0,g^H_1\)\Bigr)\;,$$
from which our theorem finally follows.
\qed\enddemo


\Abschnitt\MorseTorsionSubsection Fibre-wise Morse functions
and torsion classes

We show how to associate torsion classes
to families of compact manifolds with fibre-wise Morse functions.
If we start with a flat bundle~$\(F,\nabla^F\)\to M$
that admits a parallel metric~$g^F$,
such that the fibre-wise cohomology~$\(H,\nabla^H\)\to B$
also admits a parallel metric,
then we obtain a well-defined
cohomology class~$T_f(\nabla^F,h,g^F,g^H)\in H^{\even,\ge2}(B)$
that is invariant under smooth deformations of~$h$
in the space of all fibre-wise Morse-functions on~$M$.


Let~$p\colon M\to B$ be a bundle with compact fibres,
and assume that~$\(F,\nabla^F\)\to M$ is a flat bundle
and~$h\colon M\to\R$ is a Morse function on each fibre of~$p$.
If~$\hat p\colon C\to B$ denotes the covering of~$B$
by the fibre-wise critical points of~$h$,
we define the bundle~$V=\hat p_*(\Lambda^{\max}T^uX\otimes F|_C)$,
with the $\Z$-grading by the fibre-wise index of~$h$ as in \MorseBundleDef.
Then~$h$ acts by multiplication on~$\Lambda^{\max}T^uX\otimes F|_C$,
and we let~$h^V\in\Gamma\(\End^0V\)$ be the pushdown of this action.
Let~$A'=\nabla^V+a$ be the superconnection of \PatchSuperconnectionDefinition,
then~$a\in\Omega^*(B;\End_+^*V)$ is of total degree~$1$ by \PatchThm.

\Remark\MorseMVBRem
The bundle~$\(V,A',h^V\)$ is a family Thom-Smale complex
in the sense of \MVBDef,
and any Hermitian metric~$g^F$ on~$F\to M$ induces by restriction to~$F_C$
and push-down to~$V\to B$ an adapted metric~$g^V$ on~$V$.
In fact, if we denote by~$C^j_\alpha$ the various local leaves of~$\hat p^j$
where~$h$ has index~$j$,
we may take the $\nabla^V$-parallel splitting
	$$V^j=\bigoplus_\alpha F|_{C^j_\alpha}\;,$$
such that~$h^V$ acts on~$C^j_\alpha$
by multiplication with~$h^j_\alpha=h|_{C^j_\alpha}$,
so points~(1) and~(2) of \MVBDef\ are satisfied.
We have~$A'=\nabla^V+a'$ with~$a\in\Omega^*\(B;\End_+^*V\)$
by \HIncDef\ and~\PatchThm,
so~(3) also holds.
\endremark

Suppose that~$A'_0$ and~$A'_1$ are two flat superconnections
of total degree one on~$V\to B$,
both constructed as in \ComplexKapitel.
In particular, both~$\(V,A'_0,h^V\)$ and~$\(V,A'_1,h^V\)$
are family Thom-Smale complexes.
Then we know by \RelativityProp\
that there exists a flat superconnection
	$$\overline A'=\nabla^{\overline V}+\sum_j\bar a'_j$$
of total degree one
on the pullback~$\overline V$ of~$V$ to~$\overline B=B\times[0,1]$,
also constructed as in \ComplexKapitel,
such that~$\overline A'|_{B\times\{s\}}=A'_s$ for~$s=0$, $1$.

We fix a metric~$g^F$ on~$F\to M$,
let~$g^V$ be the induced metric o~$V$,
and let~$g^{\overline V}$ be its pull-back to~$\overline V\to\overline B$.
Similarly, $h^V$ pulls back to an endomorphism~$h^{\overline V}$
of~$\overline V$,
and~$\bigl(\overline V,\overline A',h^{\overline V}\bigr)$
is a family Thom-Smale complex with adapted metric~$g^{\overline V}$.

Let~$\overline H=H^*\bigl(\overline V,\bar a'_0\bigr)\to\overline B$
denote the fibre-wise cohomology
with its Gau\ss-Manin connection~$\nabla^{\overline H}$,
and let~$g^{\overline H}$ denote the metric induced on~$\overline H$
by~$g^{\overline V}$ using finite-dimensional Hodge theory.
Then for~$s=0$, $1$,
the restriction
of~$\bigl(\overline H,\nabla^{\overline H},g^{\overline H}\bigr)$
to~$B\times\{s\}$ is naturally isomorphic to the cohomology bundle~$H\to B$
with Gau\ss-Manin connection~$\nabla^H$
and metric~$g^H_s$ induced by~$g^V$.
Thus,
we have
	$$\tilde f\(\nabla^H,g^H_0,g^H_1\)
	=\int_0^1 f\(\nabla^{\overline H},g^{\overline H}\)$$
as in~\FslDef.


\Proposition\MorseTorsionWellDefProp.
Let~$A'_0$ and~$A'_1$ be two superconnections on~$V$
that are constructed as in \PatchSuperconnectionDefinition,
and let~$g^V$ be induced by a metric~$g^F$ on~$F\to M$.
Then
	$$T_f\(A'_1,g^V,h^V\)-T_f\(A'_0,g^V,h^V\)
	=-\tilde f\(\nabla^H,g^H_0,g^H_1\)
		\Rtext{modulo~$d^B\Omega^*(B)$.}$$

\Proof
By \MVBTorsionThm\ and Stokes' Theorem,
$$\align
	T_f\(A'_1,g^V,h^V\)-T_f\(A'_0,g^V,h^V\)
	&=d\int_{\overline B/B}T_f\bigl(\overline A'
			,g^{\overline V},h^{\overline V}\bigr)\\
	&\qquad
	+\int_{\overline B/B}\Bigl(f\(\nabla^{\overline V}
			,g^{\overline V}\)
		-f\(\nabla^{\overline H},g^{\overline H}\)\Bigr)\\
	&=-\tilde f\(\nabla^H,g^H_0,g^H_1\)
		\Rtext{modulo~$d^B\Omega^*(B)$,}
\endalign$$
because~$f\bigl(\nabla^{\overline V},g^{\overline V}\bigr)$
is just the pullback of~$f\(\nabla^V,g^V\)$ to~$\overline B$,
so~$\int_{\overline B/B}f\(\nabla^{\overline V},g^{\overline V}\)$
vanishes.
\qed\enddemo

In particular,
if we fix metrics~$g^F$ on~$F$ and~$g^H$ on~$H$,
we can define a torsion form for the bundle~$M\to B$,
equipped with a flat vector bundle~$\(F,\nabla^F\)\to M$
and a fibre-wise Morse function~$h$
that is independent of all choices made during the constructions
in \ComplexKapitel,
by setting
	$$T_f\(\nabla^F,h,g^F,g^H\)
	=T_f\(A',g^V,h^V\)+\tilde f\(\nabla^H,g^H,g^H_V\)
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;.
	\Formel\MorsePreTorsionDef$$
It follows from \MorseTorsionWellDefProp\
that~$T_f\(\nabla^F,h,g^F,g^H\)$ is indeed well-defined.

\Corollary\MorseTorsionCor.
Assume that the metrics~$g^F|_C$ and~$g^H$ %on~$F|_C$ and~$H$
are parallel,
then~$T_f\(\nabla^F,h,g^F,g^H\)$ defines a cohomology class
in~$H^\even\dR(B)$.
Moreover,
only the homogeneous part of degree~$0$ depends on the particular
choice of~$g^F|_C$ and~$g^H$.

\Proof
The first assertion is immediate from \MVBTorsionThm\
and \MorseTorsionDef.
The second claim follows because any two parallel metrics~$g^F_0$, $g^F_1$
on a flat bundle are related
by family of flat metrics~$g^F_s=(1-s)\,g^F_0+s\,g^F_1$.
Let~$g^{\overline F}$ be the metric induced by this family
on the pullback~$\overline F\to\overline B=B\times [0,1]$.
Let~$\nabla^{\overline F,*}$ be the adjoint
of the pullback~$\nabla^{\overline F}$ of~$\nabla^F$ to~$\overline F$
with respect to~$g^{\overline F}$.
Then
	$$\omega^{\overline F}
	=\nabla^{\overline F,*}-\nabla^{\overline F,*}
	\quad\in\Omega^1\(\overline B,\End\overline F\)$$
vanishes on all vectors in~$T\overline B$ that are tangent to the factor~$B$.
In particular,
$\bigl(\omega^{\overline F}\bigr)^k=0$ for~$k>1$,
so
	$$\tilde f\(\nabla^F,g^F_0,g^F_1\)
	=\tilde f\(\nabla^F,g^F_0,g^F_1\)\dg0\;.
	\quad\qed$$
\enddemo


\Definition\MorseTorsionDef
Assuming that the metrics~$g^F|_C$ on~$F|_C\to C$ and~$g^H$ on~$H\to B$
are parallel,
we define the {\em higher Morse torsion\/} 
	$$T_f(M/B;F,h)
	=T_f\(A',g^V,h^V\)\dg{\ge2}+\tilde f\(\nabla^H,g^H,g^H_V\)\dg{\ge2}
	\quad\in H^{\even,\ge2}\dR(B)\;.$$
\enddefinition



\Abschnitt\GenMetSubsection Generalised metrics
and the classes~$S_f\(A',\bold g^V,g^H\)$

We recall the notion of a generalised metric
following~\BGzwei, Section~2.9.
Our presentation will follow the lines of \TorSignSubsection\ above.
We also recall the definition of~$S_f$ for generalised metrics.
The content of this subsection will be needed
only in \WittenKapitel,
where generalised metrics appear naturally as restrictions of ``classical''
metrics to certain subbundles of~$\Lambda^*TB\tend\Omega(M/B;F)$
formed by ``generalised eigenspaces'' of the Witten-deformed operator~$\Xx\tT$.

Recall that in \TorSignSubsection,
we extended a metric~$g^V$ on a vector bundle~$V\to B$
to an $\R$-bilinear map~$g^V\colon\(\Lambda^*TB\tend V\)^2\to\Lambda^*TB$
with the properties~\TorSignMetric\ and~\TorSignSymmetry.
A natural generalisation of metrics is the following.


\Definition\GenMetDef
A {\em generalised Hermitian metric\/}~$\bold g^V$
on a $\Z_2$-graded complex vector bundle~$V\to B$ is an $\R$-bilinear
map~$\bold g^V\colon\(\Lambda^*TB\tend V\)^2\to\Lambda^*TB$,
such that
\roster
\item
the component~$\(\bold g^V\)\dg0\colon\(g^V\)^2\to\C$
is a Hermitian metric;
\item
$\bold g^V$ is even with respect to the total grading
of~$\Lambda^*TB\tend V$;
\item
for~$v$, $w\in V$, we have
	$$\bold g^V(w,v)
	=(-1)^{\textstyle\bigl[{\deg(\bold g^V(w,v))\over2}\bigr]}
		\,\overline{\bold g^V(v,w)}\;;$$
\item
for~$\alpha$, $\beta\in\Lambda^*TB$ and~$v$, $w\in V$,
we have
	$$\bold g^V\(\alpha\tend v,\beta\tend w\)
	=(-1)^{\textstyle\bigl[{\deg(\beta)\over2}\bigr]}
		\,\alpha\wedge\bold g^V\(v,w)\wedge\overline\beta\;.$$
\endroster
\enddefinition

Properties~(3) and~(4) are the natural generalisations of~\TorSignMetric\
and \TorSignSymmetry.
Together,
they imply that~(3) still holds for~$v$, $w\in\Lambda^*TB\tend V$.
Note that as in \AdjointRem,
we may regard~$\bold g^V$ as an even element
of~$\Omega^*\bigl(B;\Hom(V,\overline V^*)\bigr)$.
Clearly, there is an analogous notion of a generalised Euclidean metric.
Note that by~(2),
the part~$\(\bold g^V\)\dg0$ of horizontal degree~$0$
respects the $\Z^2$-grading of~$V$.

\Remark\GenMetContractRem
Since any convex combination of generalised metrics is again a generalised
metric,
the space of generalised metrics on a given vector bundle is contractible.
In particular,
any generalised metric can be deformed onto its component~$(\bold g^V)\dg0$.
\endremark

When we define adjoints of endomorphisms and superconnections
with respect to general metrics,
we have to take care of the fact
that~$\bold g^V(v,w)$ may be an odd form for~$v$, $w\in V$.

\Definition\GenAdjointDef
For an endomorphism~$E\in\Omega^*(B;\End V)$,
we define the adjoint endomorphism by
	$$E^*=(\bold g^V)^{-1}\,\overline E^*\,\bold g^V\;,$$
and for a superconnection~$A$ on~$V$,
we set
	$$A^*=(\bold g^V)^{-1}\,\overline A^*\,\bold g^V\;.$$
\enddefinition

\Remark\GenAdjointRem
For an endomorphism~$E\in\End V$,
\GenAdjointDef\ is equivalent to
	$$\bold g^V(E^*w,v)
	=(-1)^{\abs E\,(\abs E+\abs w+\abs v)}\,\bold g^V(w,Ev)$$
for all~$v$, $w\in V$.
Note that~$\abs E+\abs w+\abs v$ has the same parity as the exterior degree
of the form~$\bold g^V\(w,Ev)$.

For a connection~$\nabla^V$,
we get
	$$\nabla^{V,*}=\nabla^V+\omega^V
		\Mtext{with}
	\omega^V=\(\bold g^V\)^{-1}\,\bigl[\nabla^V,\bold g^V\bigr]
	\quad\in\quad\Omega^*(B;\End V)\;.$$

\GenMetDef~(3), (4) and \GenAdjointDef\ imply
that~$(E^*)^*=E$
and~$(A^*)^*=A$ for all superconnections~$A$
and all~$E\in\Omega^*(M;\End V)$.

For odd elements~$\alpha\tend E\in\Omega^*(M;\End V)^\odd$
and for superconnections~$A$ on~$V$,
\GenMetDef~(4) and \GenAdjointDef\ imply that
$$\split
	0
	&=\<E^*v,w\>+(-1)^{\deg E\,(\deg v+\deg w)}\,\<v,Ew\>\\
		\Land
	d\<v,w\>
	&=\<A^*v,w\>+(-1)^{\deg E\,(\deg v+\deg w)}\,\<v,Aw\>
\endsplit$$
for all~$v$, $w\in\Omega^*(M;V)$.
Thus,
most of the properties of metrics, superconnections and their adjoints
generalise to the present setting.
\endremark

Let~$f\colon\C\to\C$ be a holomorphic function satisfying~\GenFnProperty,
e.g., $f(z)=z\,e^{z^2}$ as in~\GenFnChoice.
If~$\bold g^V$ is a generalised Hermitian metric
and~$A'$ is a superconnection,
let~$A''$ be the adjoint of~$A'$ with respect to~$\bold g^V$,
and set~$X={1\over2}\,\(A''-A'\)$ as in~\XDef.
We have a natural generalisation of \FDef.

\Definition\GenFDef
Let~$f\(A',\bold g^V\)\in\Omega^*(B)$ be defined by
	$$f\(A',\bold g^V\)
	=(2\pi i)^{\textstyle{1-N^B\over 2}}
		\,\strace_V\(f(X)\)\;.$$
\enddefinition

\FVThm\ extends literally to this generalised setting,
so in particular,
$f\(A',\bold g^V\)$ is odd, closed,
and real.
Similarly,
we define~$\tilde f\(A',\bold g^V_0,\bold g^V_1\)
\in\Omega^*(B)/d^B\Omega^*(B)$ as in \FslDef.
Again, \FVCor\ extends literally to this situation and gives in particular
	$$d^B\tilde f\(A',\bold g^V_0,\bold g^V_1\)
	=f\(A',\bold g^V_1\)-f\(A',\bold g^V_0\)\;.$$

We also need a generalisation of \SVDef\ to this setting.
Therefore,
let~$\(V,g^V\)\to B$ be a $\Z$-graded Hermitian vector bundle,
let~$\(\bold g^V_t\)_{t\in(0,\infty)}$ be a smooth family
of generalised metrics on~$V$,
and let~$g'\in\Omega^*\(B;\End V\)$ be even,
such that as~$t\to\infty$,
$$\split
	t^{\textstyle{n-N^V\over2}}\,\bold g^V_t\,t^{-\textstyle{N^V\over2}}
	&=g^V\Biggl(\biggl(\id_V+{g'\over\sqrt t}+O\biggl({1\over t}\biggr)
			\biggr)\punkt,\punkt\Biggr)\;,\\
		\Land
	t^{\textstyle{N^V\over2}}\,\Bigl(\bold g^V_t\Bigr)^{-1}
		\,{\del\bold g^V_t\over\del t}\,t^{-\textstyle{N^V\over2}}
	&=\biggl(N^V-{n\over2}\biggr)\,{1\over t}
		+O\Bigl(t^{-\textstyle{3\over2}}\Bigr)\;,
\endsplit\Formel\UMetricInftyCondition$$
where both~$O(\punkt)$ are even elements of~$\Omega^*\(B;\End V\)$.
Let~$\bar{\bold g}^V$ be the generalised metric
on~$\overline V=V\times(0,\infty)\to\overline B=B\times(0,\infty)$
induced by~$\bold g^V_t$,
in particular,
$\bar{\bold g}^V$ does not contain the exterior variable~$dt$.

Let~$A'$ be a flat superconnection of total  degree~$1$ as in \TotDegOneDef,
so
	$$A'=\nabla^V+\sum_ia_i
		\Mtext{with}
	a_i\in\Omega^i\(B;\End^{1-i}V\)\;,$$
and let~$\overline A'=A'+{\del\over\del t}\,dt$
be the trivial extension of~$A'$ to~$\overline V$.
Let~$\overline A''$ be the adjoint of~$\overline A'$
with respect to~$\bar{\bold g}^V$.
Note that~$\overline A''$ is not necessarily of total degree~$-1$,
since~$\bar{\bold g}^V$ can shuffle degrees.
Nevertheless,
using that~$\overline A'$ is of total degree~$1$
together with~\UMetricInftyCondition,
we establish the following properties of~$A'$ and~$A''_t$.

\Proposition\USupConnProp{(\BGzwei, Proposition~2.46)}.
As~$t\to\infty$,
$$\align
	t^{\textstyle{N^V\over2}}\,A'\,t^{-\textstyle{N^V\over2}}
	&=\sqrt t\,a_0+\nabla^V+a_1
		+O\Bigl(t^{-\textstyle{1\over2}}\Bigr)\;,\\
		\Land
	t^{\textstyle{N^V\over2}}\,A''_t\,t^{-\textstyle{N^V\over2}}
	&=\sqrt t\,a^*_0+\nabla^{V,*}+a^*_1+[a_0^*,g']
		+O\Bigl(t^{-\textstyle{1\over2}}\Bigr)\;,\\
\endalign$$
where adjoints are taken with respect to~$g^V$.
\qed

As in~\XDef, we define
	$$\overline X
	={1\over 2}\,\(\overline A''-\overline A'\)
	={1\over 2}\,\(A''_t-A'\)
		+{1\over2}\,\Bigl(\bold g^V_t\Bigr)^{-1}
			\,{\del\bold g^V_t\over\del t}\,dt\;.$$
We define an odd closed differential form on~$\overline B$ by
$$\multline\split
	\beta
	&=f\(\overline A',\bar{\bold g}^V\)-\biggl(\chi'(H)
		-{n\over2}\,\chi(V)\biggr)\,f'(0)\,{dt\over2t}\\
	&=f\(A',\bold g^V_t\)
		+(2\pi i)^{-\textstyle{N^B\over2}}\Biggl(
			\,\strace_V\biggl({1\over2}
				\,\Bigl(\bold g^V_t\Bigr)^{-1}
				\,{\del\bold g^V_t\over\del t}
				\,f'\(X_t\)\biggr)
\endsplit\\
			-\biggl(\chi'(H)
				-{n\over2}\,\chi(V)\biggr)
			\,{f'(0)\over2t}\Biggr)\,dt\;.
\endmultline\tag\FormelNummer\UBetaDef$$


Let~$g^H$ be the metric induced by~$g^V$ on~$H=H^*(V,a_1)\to B$,
and let~$\nabla^H$ be the Gau\ss-Manin connection on~$H$ induced by~$A'$.
If~$P^H\colon V\to H$ denotes the $g^V$-orthogonal projection
onto the $(a_0+a^*_0)$-harmonic elements of~$V$,
then clearly~$P^H\,[a_0^*,g']\,P^H=0$.
Now the proof of~\BL, Theorem~2.9,
translates to this situation
and gives an estimate for~$\beta$ as~$t\to\infty$,
cf.~\BGzwei, Proposition~2.48.

\Proposition\UInftyLimitProp.
As~$t\to\infty$,
	$$\beta
	=f\(\nabla^H,g^H\)+O\Bigl(t^{-\textstyle{1\over2}}\Bigr)
		+O\Bigl(t^{-\textstyle{3\over2}}\Bigr)\,dt\;,$$
where the~$O(\punkt)\in\Omega^*\bigl(\overline B;\End\overline V\bigr)$
do not contain~$dt$.
\qed

Let~$\bold g^V_{s,t}$ be two generalised metrics on~$V$,
together with~$g^V_s$ and~$g'_s$,
such that~\UMetricInftyCondition\ holds for~$s=0$, $1$.
Then there exist families~$\bold g^V_{s,t}$, $g^V_s$ and~$g'_s$
for~$s\in[0,1]$ that restrict to the given metrics for~$s=0$, $1$,
such that~\UMetricInftyCondition\ holds uniformly in~$s$.
We define a form~$\overline\beta$ on~$B\times[0,1]\times(0,\infty)$,
and we let~$\beta_s$ denote its restriction to~$B\times\{s\}\times(0,\infty)$.
Let~$g^H_s$ denote the metric induced on~$H$ by~$g^V_s$,
then the obvious generalisation of~\SVVariationFormel\ is
	$$\int_{(B\times(1,\infty))/B}\(\beta_1-\beta_0\)
	=\tilde f\(A',\bold g^V_{0,1},\bold g^V_{1,1}\)
		-\tilde f\(\nabla^H,g^H_0,g^H_1\)
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;.
	\Formel\UVVariationFormel$$
Writing~$\bold g^V=\bold g^V_1$,
me can now define a generalisation of the class~$S_f\(A',g^V\)$.

\Definition\UVDef
Set
	$$S_f\(A',\bold g^V,g^H\)
	=\int_{(B\times(1,\infty))/B}\beta
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;.$$
\enddefinition

Note that the class~$S_f\(A',\bold g^V,g^H\)$
was called~$U_f\(A',\bold g^V_t\)$ in~\BGzwei.

\Theorem\UVThm{(\BGzwei, Theorem~2.50)}.
The form~$S_f\(A',\bold g^V,g^H\)$ is even
%and not real in general !
and satisfies
	$$d^BS_f\(A',\bold g^V,g^H\)
	=f\(A',\bold g^V\)-f\(\nabla^H,g^H\)\;.$$
If~$\bold g^{\prime V}$ is another generalised metric on~$V$
and~$g^{\prime H}$ is another metric on~$H$,
them
	$$S_f\(A',\bold g^{\prime V},g^{\prime H}\)
		-S_f\(A',\bold g^V,g^H\)
	=\tilde f\(A',\bold g^V,\bold g^{\prime V}\)
		-\tilde f\(\nabla^H,g^H,g^{\prime H}\)\;.$$

Let~$g^V$ be a fixed metric on~$V$ that respects the $\Z$-grading.
Note that the family~$g^V\,t^{N^V}$
does not quite satisfy~\UMetricInftyCondition.
Me may however set
	$$g^V_t=g^V\,t^{N^V-\textstyle{n\over2}}\;,$$
and since~$t^{-{n\over2}}$ commutes with all operators involved,
we have
	$$A''_t
	=t^{-N^V+\textstyle{n\over2}}\,A''\,t^{N^V-\textstyle{n\over2}}
	=t^{-N^V}\,A''\,t^{N^V}\;.$$
Because of this,
our form~$\beta$ of~\UBetaDef\ becomes
$$\split
	\beta
	&=f\(A',g^V\,t^{N^V}\)
		+(2\pi i)^{-\textstyle{N^B\over2}}\,\Biggl(
			\strace_V\biggl(\Bigl(N^V-{n\over2}\Bigr)
				\,f'(X_t)\biggr)\\
	&\kern15em
			-\biggl(\chi'(H)
					-{n\over2}\,\chi(V)\biggr)
				\,f'(0)\Biggr)\,{dt\over2t}\\
	&=f\(A',g^V\,t^{N^V}\)
		+(2\pi i)^{-\textstyle{N^B\over2}}\,\Bigl(
			\strace_V\(N^V\,f'(X_t)\)
			-\chi'(H)\,f'(0)\Bigr)\,{dt\over2t}\;.
\endsplit$$
Let again~$g^H$ be the metric induced on~$H$ by~$g^V$.
A comparison with our definition of~$S_f\(A',g^V\)$
in \SVDef\
shows that
	$$S_f\(A',g^V,g^H\)=S_f\(A',g^V\)
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;,
	\Formel\SUGammmaIdentity$$
which justifies our notation.



\Abschnitt\TanSubsection Analytic torsion forms

We recall the construction of analytic torsion forms in~\BL.

Let~$p\colon M\to B$ be a smooth fibre bundle,
and let~$\(F,\nabla^F\)\to M$ be a flat complex vector bundle.
Let~$\Omega^*(M;F)$ denote the space of $F$-valued differential forms on~$M$.
If we fix a horizontal subbundle~$T^HM\subset TM$ with~$T^HM\oplus TX=TM$,
then~$T^HM\cong p^*TB$,
and we may split~$\Lambda^*TM=p^*\Lambda^*TB\tend\Lambda^*TX$.
We denote the space of vertical forms
by~$\Omega^*(M/B;F)=p_*\(\Lambda^*TX\otimes F\)\to B$,
then we have
	$$\Omega^*(M;F)=\Omega^*\(B;\Omega^*(M/B;F)\)\;.
	\Formel\OmegaSplittingFormel$$

Let~$\Omega\in\Gamma(p^*\Lambda^2TB\otimes TX)$ denote the fibre-bundle
curvature of~$T^HM$,
given by
	$$\Omega(v,w)=-\bigl[\bar v,\bar w\bigr]^\perp\in TX\;,$$
where~$\bar v$, $\bar w\in\Gamma(T^HM)$ are horizontal lifts
of vectorfields~$v$, $w$ on~$B$
and~$(\punkt)^\perp$ denote the projection~$TM\to TX$ along~$T^HM$.
With respect to the splitting~\OmegaSplittingFormel,
we may interpret the exterior differential~$d^M$ induced by~$\nabla^F$
on~$\Omega^*(M;F)$ as a flat superconnection~$\Aa'$ of total degree~$1$
on the infinite-dimensional vector bundle~$\Omega^*(M/B;F)\to B$.
Classically,
this superconnection decomposes as
	$$\Aa'=d^M=d^X+\nabla^{\Omega^*(M/B;F)}+\iota_\Omega
	\Formel\AaDef$$
with respect to horizontal degree, cf.~\BGV.

Now let us fix a vertical metric~$g\TX$,
then~$g\TX$ induces a metric on~$\Lambda^*TX$
that we still denote by~$g\TX$.
We obtain a natural $L_2$-metric on~$\Omega^*(M/B;F)$,
which we write as
	$$g^{TX,F}(\omega,\omega')_b
	=\int_{p^{-1}(b)}\(g\TX\otimes g^F\)(\omega,\omega')_x\,d\Vol(x)\;,
	\Formel\LzweiMetricDef$$
where~$d\Vol$ is the fibre-wise volume element induced by~$g\TX$.
Let~$\Aa''$ be the adjoint of~$\Aa'$ with respect to~$g^{TX,F}$
in the sense of \AdjointDef.
Then
$$\multline
	\Xx={1\over2}\,\(\Aa''-\Aa'\)
	={1\over2}\,\Bigl(\(d^{X,*}-d^X\)
		+\omega^{\Omega^*(M/B;F)}
		-\eps_\Omega-\iota_\Omega\Bigr)\\
	\quad\in\Omega^*\(B;\End\Omega^*(M/B;F)\)
\endmultline\tag\FormelNummer\XxDef$$
is a fibre-wise elliptic differential operator on~$\Omega^*(M;F)$
with coefficients in~$\Omega^*(B)$.
Moreover, $\Xx$ is skew-adjoint in the sense of \AdjointDef.

By Hodge theory,
we can identify the fibre-wise cohomology~$H=H^*\dR(M/B;F)$
with the bundle of $(d^{X,*}-d^X)$-harmonic forms in~$\Omega^*(M/B;F)$.
Then~$\nabla^{\Omega^*(M/B;F)}$ induces the Gau\ss-Manin connection~$\nabla^H$
on~$H$.
We let~$g^H_{L_2}$ denote the restriction of~$g^{TX,F}$ to~$H$.

Because~$-\Xx^2$ has the same spectrum
as the Hodge Laplacian~$-\(\Xx\dg0\)^2={1\over4}\,\(d^{X,*}+d^X\)^2$,
the odd heat kernel~$\Xx\,e^{\Xx^2}$ is of trace-class.
In particular,
as in \FDef,
we obtain an odd, closed and real form
	$$f\(\Aa',g^{TX,F}\)
	=\strace\(\Xx\,e^{\Xx^2}\)\;.
	\Formel\FanDef$$

Let~$N^X$ denote the fibre-wise number operator on~$\Omega^*(M;F)$
and~$\Omega^*(M/B;F)$.
We set~$g\TX_t=g\TX\(t^{-N^X}\punkt,\punkt\)$.
Let~$g^{TX,F}_t$ denote the induced $L_2$-metric on~$\Omega^*(M/B;F)$,
then
	$$g^{TX,F}_t
	=g^{TX,F}\(t^{N^X-\textstyle{n\over2}}\punkt,\punkt\)\;,
	\Formel\GtDef$$
where the additional~$t^{n\over2}$ is due to the change of the fibre-wise
volume element.
Let~$\bar g^{TX,F}$ denote the metric on~$\Omega^*(M/B;F)\times(0,\infty)$
such that~$\bar g^{TX,F}|_{B\times\{t\}}=g^{TX,F}_t$.
Let~$\overline\Aa''$ be the adjoint of the obvious extension~$\overline\Aa'$
of the superconnection~$\Aa'$
to the bundle~$\Omega^*(M/B;F)\times(0,\infty)\to B\times(0,\infty)$,
and set~$\overline\Xx={1\over2}\,\bigl(\overline\Aa''-\overline\Aa'\bigr)$.
As before, restriction to~$B\times\{t\}$ will be written~$(\punkt)_t$.

The form
$$\multline
	f\(\overline\Aa',\bar g^{TX,F}\)-\chi'(H)\,f'(0)\,{dt\over2t}\\
	=f\(\Aa'_t,g^{TX,F}\)
		+(2\pi i)^{-\textstyle{N^B\over2}}
		\,\Bigl(\strace\(N^X\,f'(\Xx_t)\)-\chi'(H)\,f'(0)\Bigr)\,{dt\over2t}
\endmultline$$
is closed and satisfies
$$%\multline	
	f\(\overline\Aa',\bar g^{TX,F}\)-\chi'(H)\,f'(0)\,{dt\over2t}%\\
	=f\(\nabla^H,g^H_{L_2}\)+O\Bigl(t^{-\textstyle{1\over2}}\Bigr)
		++O\Bigl(t^{-\textstyle{3\over2}}\Bigr)\,dt\;,
%\endmultline
$$
as~$t\to\infty$ by \BL, Theorem~3.21,
where the $O(\punkt)$-terms do not involve~$dt$.

\Definition\SanDef
Set
	$$S_f\(T^HM,g\TX,\nabla^F,g^F\)
	=-(2\pi i)^{-\textstyle{N^B\over2}}\int_1^\infty
		\Bigl(\strace\(N^X\,f'(\Xx_t)\)
			-\chi'(H)\,f'(0)\Bigr)\,{dt\over2t}\;.$$
\enddefinition

Then clearly,
$S_f\(T^HM,g\TX,\nabla^F,g^F\)$ is even and real,
and satisfies
	$$d^BS_f\(T^HM,g\TX,\nabla^F,g^F\)
	=\cho\(\Aa',g^{TX,F}\)-\cho\(\nabla^H,g^H\)\;.$$

Let~$\nabla\TX$ be the natural connection on~$TX\to M$
that is induced by~$T^HM$ and~$g\TX$,
and let~$e\(\nabla\TX\)$ be its Euler form.
As~$t\to 0$, again by~\BL, Theorem~3.21,
$$\multline
	f\(\overline\Aa',\bar g^{TX,F}\)-\chi'(H)\,{dt\over2t}\\
	=e\(\nabla\TX\)\,f\(\nabla^F,g^F\)
		+\biggl({n\over 2}\,\chi(H)-\chi'(H)\biggr)
			\,f'(0)\,{dt\over2t}
		+O\(t^{\textstyle{1\over2}}\)
		+O\(t^{-\textstyle{1\over2}}\)\,dt\;,
\endmultline$$
where the $O(\punkt)$-terms do not contain~$dt$.

\Definition\TanDef
Set
$$\multline
	\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)
	=-(2\pi i)^{-\textstyle{N^B\over2}}\int_0^\infty
		\Biggl(\strace\(N^X\,f'(\Xx_t)\)-\chi'(H)\,f'(0)\\
			-\biggl({n\over2}\,\chi(H)-\chi'(H)\biggr)
				\,f'\biggl({\sqrt{-t}\over2}\biggr)\Biggr)
			\,{dt\over2t}\;.
\endmultline$$
\enddefinition

Let~$T^{H,s}M$, $g\TX_s$, $\nabla^{F,s}$ and~$g^F_s$ denote families
of horizontal subbundles and vertical metrics on~$TM$,
and of flat connections and Hermitian metrics on~$F$ for~$s\in[0,1]$,
where we assume that the rank of~$H_s=H^*\dR\(M/B;(F,\nabla^{F,s})\)$
does not depend on~$s$.
Let~$\nabla^{H,s}$ and~$g^{H,s}_{L_2}$
denote the families of induced flat connections and $L_2$-metrics.
Let~$\nabla^{TX,s}$ denote the induced connections on the vertical tangent
bundle,
and let~$\tilde e(\nabla^{TX,0},\nabla^{TX,1})$
be the Chern-Simons class associated to the Euler class.
Let~$k(z)={f'(z)-f'(0)\over2z}$.
We now summarise~\BL, Theorems~3.23, 3.24, 3.29 and~\BGzwei, Theorem~3.45.

\Theorem\TanThm.
The form~$\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)$ is even and real,
and satisfies
	$$d^B\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)
	=\int_{M/B}e\(\nabla\TX\)\,f\(\nabla^F,g^F\)
		-f\(\nabla^H,g^H_{L_2}\)\;.\tag1$$
The component~$\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)\dg0$ of degree~$0$
is equal to the Ray-Singer
analytic torsion~$\Cal T_{\roman{RS}}\(g\TX,\nabla^F,g^F\)$ of the fibre.
Moreover,
modulo exact forms on~$B$,
$$\align
\split
	\Cal T_f\(T^{H,1}M,g\TX_1,\nabla^F,g^F_1\)
	\kern-7em&\kern7em
		-\Cal T_f\(T^{H,0}M,g\TX_0,\nabla^F,g^F_0\)\\
	&=\int_{M/B}\tilde e(\nabla^{TX,0},\nabla^{TX,1})
		\,f\(\nabla^F,g^V_0\)\\
	&\qquad+\int_{M/B}e(\nabla^{TX,1})\,\tilde f\(\nabla^F,g^V_0,g^V_1\)
		-\tilde f\(\nabla^H,g^{H,0}_{L_2},g^{H,1}_{L_2}\)\;,
\endsplit\tag2\\
\noalign{\smallskip}\allowdisplaybreak
\split
	\Land
	\Cal T_f\(T^HM,g\TX,\nabla^{F,1},g^F_1\)\dg{\ge 2}
	\kern-8em&\kern8em
		-\Cal T_f\(T^HM,g\TX,\nabla^{F,0},g^F_0\)\dg{\ge 2}\\
	&=\int_{M/B}e(\nabla\TX_0)\,L_k\(\nabla^{F,s},g^V_0,g^V_1\)
		-L_k\(\nabla^{H,s},g^{H,0}_{L_2},g^{H,1}_{L_2}\)\;.
\endsplit\tag3
\endalign$$

\Remark\AnalyticLRem
Note again that the right hand side of~(3) is independend modulo~$d\Omega^*(B)$
of the paths connecting~$\nabla^{F,0}$ with~$\nabla^{F,1}$
and~$\nabla^{H,0}$ with~$\nabla^{H,1}$,
even though both $L_k$-terms may depend on the homotopy class of these paths
in degree~$\ge 4$,
and on the paths themselves in degree~$2$.
\endremark

Suppose now that both the bundles~$F\to M$ and $H\to B$
carry parallel metrics~$g^F$ and~$g^H$.
Then as in~\MorseTorsionCor,
the form
	$$\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)
		+\tilde f\(\nabla^H,g^H,g^H_{L_2}\)$$
is closed,
and its cohomology class is independend of the choice
of the parallel metrics~$g^F$ and~$g^H$ in degree~$\ge2$.

\Definition\AnalyticTorsionClassDef
If the metrics~$g^F$ and~$g^H$ are parallel,
define the {\em higher analytic torsion\/}
	$$\Cal T_f\(M/B;F\)
	=\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)\dg{\ge2}
		+\tilde f\(\nabla^H,g^H,g^H_{L_2}\)\dg{\ge2}
	\quad\in H\dR^{\even,\ge2}(B)\;.$$
\enddefinition



\Abschnitt\ChernSubsection The Chern normalisation

We recall the Chern normalisation of the odd characteristic classes
of \FDef\ and the torsion forms of \MVBTorsionDef\ and \TanDef\
following~\BGzwei.
Note that ``normalisation'' means the choice of a particular generating
function~$f$.
\MainTheorem\ does not depend on~$f$,
and the construction of torsion forms using the function~$f$ of~\GenFnChoice\
is very convenient for proving it.
Nevertheless,
results like Theorem~0.1 of~\BGdrei\
depend on the choice of normalisation.
This suggests that the most natural normalisation
of torsion forms is the one we are introducing now.

Let~$\(F,\nabla^F\)\to M$ be a flat vector bundle.
A metric~$g^F$ on~$F$ determines an adjoint connection~$\nabla^{F,*}$
which is also flat,
so
	$$\ch\(\nabla^F\)
	=\trace\Bigl(e^{-\textstyle{(\nabla^F)^2\over2\pi i}}\Bigr)
	=\ch\(\nabla^{F,*}\)
	=\rk F$$
as differential forms.
Let~$\chsl\(\nabla^F,\nabla^{F,*}\)$ be the natural Chern-Simons class
in~$\Omega^*(M)/d^M\Omega^*(M)$, then
	$$d^M\chsl\(\nabla^F,\nabla^{F,*}\)
	=\ch\(\nabla^{F,*}\)-\ch\(\nabla^F\)
	=0\;,$$
so~$\chsl\(\nabla^F,\nabla^{F,*}\)$ is in fact a cohomology class.

In order to construct a natural representative for this class,
we choose a connection
	$$\nabla^{\overline F}
	=(1-s)\,\nabla^F+s\,\nabla^{F,*}+{\del\over\del s}\,ds$$
on~$\overline F=F\times[0,1]\to\overline B=B\times[0,1]$.
Set~$\omega^F=\nabla^{F,*}-\nabla^F$ as before.
Because~$\nabla^F$ and~$\nabla^{F,*}$ are flat,
the curvature of~$\nabla^{\overline F}$ is given by
	$$\(\nabla^{\overline F}\)^2
	=s(1-s)\,\bigl[\nabla^F,\nabla^{F,*}\bigr]
		-\(\nabla^{F^*}-\nabla^F\)\,ds
	=-s(1-s)\,(\omega^F)^2-\omega^F\,ds\;.$$

\Definition\ChoDef
Set
	$$\cho\(\nabla^F,g^F\)
	=\int_0^1\trace\biggl({\omega^V\over2}
		\,e^{\textstyle{s(1-s)\,(\omega^V)^2\over2\pi i}}
		\biggr)\,ds\;.$$
\enddefinition

\Theorem\ChoThm.
The form~$\cho\(\nabla^F,g^F\)$ is real, odd and closed
and represents the Chern-Simons class~$\pi i\,\chsl\(\nabla^F,\nabla^{F,*}\)$
in~$H^*\dR(M)$.
Moreover, for~$f(z)=z\,e^{z^2}$,
%the following relation holds.
	$$\cho\(\nabla^F,g^F\)
	=\int_0^1\(4s(1-s)\)^{\textstyle{N^B-1\over2}}
		\,f\(\nabla^F,g^F\)\,ds\;.$$

\Proof
It is clear that~$\cho\(\nabla^F,g^F\)$
represents~$\pi i\,\chsl\(\nabla^F,\nabla^{F,*}\)$
by the definition of Chern-Simons classes.
Moreover,
$$\align
	\cho\(\nabla^F,g^F\)
	&=\int_0^1\trace\biggl({\omega^V\over2}
		\,e^{\textstyle{s(1-s)\,(\omega^V)^2\over2\pi i}}
		\biggr)\,ds\\
	&=\int_0^1\(4s(1-s)\)^{\textstyle{N^B-1\over2}}
		\,\trace\biggl({\omega^V\over2}
		\,e^{\textstyle{(\omega^V/2)^2\over2\pi i}}\biggr)\,ds\\
	&=\int_0^1\(4s(1-s)\)^{\textstyle{N^B-1\over2}}
		\,f\(\nabla^F,g^F\)\,ds\;.
\endalign$$
Finally, $\cho\(\nabla^F,g^F\)$ is real, odd and closed
because~$f\(\nabla^F,g^F\)$ is.
\qed\enddemo

This motivates the following definitions,
where we make the same assumptions as before.
In particular,
let~$(V,A',h^V)\to B$ be a family Thom-Smale complex
with an adapted metric~$g^V$.
Let~$p\colon M\to B$ be a smooth bundle with compact fibres,
with horizontal subbundle~$T^HM$ and vertical metric~$g^V$.
Let~$(F,\nabla^F)\to M$ be a flat complex vector bundle,
and let~$g^F$, $g^F_0$ and~$g^F_1$ be metrics on~$F$.
Let~$(V,A'_s,h^V_s)$ be a family of family Thom-Smale complexes
with adapted metrics~$g^V_s$ for~$s\in[0,1]$,
and let~$\nabla^{F,s}$ be a family of flat connections on~$F\to M$,
such that the fibre-wise cohomology~$H_s\to B$ has constant rank,
with~$\nabla^{H,s}$, $g^{H,s}_V$ and~$g^{H,s}_{L_2}$ in both cases
denoting the respective induced family
of metrics and connections on~$H_s$, respectively.

\Definition\ChernNormDef
Let~$\tilde f\(\nabla^F,g^F_0,g^F_1\)$, $L_k\(A'_s,g^V_0,g^V_1\)$,
$T_f\(A',\penalty-100g^V,h^V\)$,
and~$\Cal T_f\(T^HM,g\TX,\penalty-100\nabla^F,g^F\)$
be defined as in \FslDef, \LDef, \MVBTorsionDef\ and \TanDef,
and set
$$\alignat2
	\chslo\(\nabla^F,g^F_0,g^F_1\)
	&=\int_0^1\(4s(1-s)\)^{\textstyle{N^B\over2}}
		\,\tilde f\(\nabla^F,g^F_0,g^F_1\)\,ds
	&&\quad\in\Omega^*(M)\;,\\
	L \(A'_s,g^V_0,g^V_1\)
	&=\int_0^1\(4s(1-s)\)^{\textstyle{N^B\over2}}
		\,L_k\(A'_s,g^V_0,g^V_1\)\,ds\;,\\
	T \(A',g^V,h^V\)
	&=\int_0^1\(4s(1-s)\)^{\textstyle{N^B\over2}}
		\,T_f\(A',g^V,h^V\)\,ds\;,\\
	\Land
	\Cal T \(T^HM,g\TX,\nabla^F,g^F\)
	&=\int_0^1\(4s(1-s)\)^{\textstyle{N^B\over2}}
		\,\Cal T_f\(T^HM,g\TX,\nabla^F,g^F\)\,ds
	&&\quad\in\Omega^*(B)\;.
\endalignat$$
\enddefinition

Let us assume that~$\(H,\nabla^H\)$ is the fibre-wise cohomology
of~$(V,A')$ (or~$\(\Omega^*(M/B;F),d^M\)$ respectively),
with induced metric~$g^H_V$ (or~$g^H_{L_2}$).
Then we have the obvious analogues of \FVCor, \MVBTorsionThm,
\MVBTorsionRigidityThm\ and \TanThm.

\Corollary\ChernNormCor.
The forms~$\chslo\(\nabla^F,g^F,g^{F\prime}\)$,
$L \(A'_s,g^V_0,g^V_1\)$, $T \(A',\penalty-100g^V,h^V\)$
and~$\Cal T \(T^HM,\penalty-100g\TX,\penalty-100\nabla^F,g^F\)$ are even and real,
and satisfy
$$\align
	d^M\chslo\(\nabla^F,g^F_0,g^F_1\)
	&=\cho\(\nabla^F,g^F_1\)-\cho\(\nabla^F,g^F_0\)\;,\tag1\\
	d^BL \(A'_s,g^V_0,g^V_1\)
	&=\cho\(A'_1,g^V_1\)\dg{\ge3}-\cho\(A'_0,g^V_0\)\dg{\ge3}\;,\tag2\\
	d^BT \(A',g^V,h^V\)
	&=\cho\(\nabla^V,g^V\)-\cho\(\nabla^H,g^H_V\)\;,\tag3\\
	\Land
	d^B\Cal T \(T^HM,g\TX,\nabla^F,g^F\)
	&=\int_{M/B}e\(\nabla\TX\)\,\cho\(\nabla^F,g^F\)
		-\cho\(\nabla^H,g^H_{L_2}\)\;.\tag4
\endalign$$
Modulo exact forms on~$B$,
$$\align
\split
	T \(A',g^V_1,h^V\)\kern-3em&\kern3em-T \(A',g^V_0,h^V\)\\
	&=\chslo\(\nabla^V,g^V_0,g^V_1\)
		-\chslo\(\nabla^H,g^{H,0}_V,g^{H,1}_V\)\;,
\endsplit\tag5\\
\noalign{\smallskip}
\split
	\Cal T \(T^H_1M,g\TX_1,\nabla^F,g^F_1\)
	\kern-6em&\kern6em
		-\Cal T \(T^H_0M,g\TX_0,\nabla^F,g^F_0\)\\
	&=\int_{M/B}\tilde e\(\nabla^{TX,0},\nabla^{TX,1}\)
			\,\cho\(\nabla^F,g^F_1\)\\
	&\qquad
		+\int_{M/B}e\(\nabla^{TX,0}\)\,\chslo\(\nabla^F,g^F_0,g^F_1\)
		-\chslo\(\nabla^H,g^{H,0}_{L_2},g^{H,1}_{L_2}\)\;,
\endsplit\tag6\\
\noalign{\smallskip}\allowdisplaybreak
\split
	T \(A'_1,g^V_1,h^V_1\)\dg{\ge2}
	\kern-5em&\kern5em-T \(A'_0,g^V_0,h^V_0\)\dg{\ge2}\\
	&=L \(\nabla^{V,s},g^V_0,g^V_1\)
		-L \(\nabla^{H,s},g^{H,0}_V,g^{H,1}_V\)\;,
\endsplit\tag7\\
\noalign{\smallskip}
\split
	\Land
	\Cal T \(T^HM,g\TX,\nabla^{F,1},g^F_1\)\dg{\ge2}
	\kern-8em&\kern8em
		-\Cal T \(T^HM,g\TX,\nabla^{F,0},g^F_0\)\dg{\ge2}\\
	&=\int_{M/B}e\(\nabla\TX\)L \(\nabla^{F,s},g^F_0,g^F_1\)
		-L \(\nabla^{H,s},g^{H,0}_{L_2},g^{H,1}_{L_2}\)\;.
\endsplit\tag8
\endalign$$


\MVBTorsionThm\ and \ChernNormCor~(7)
imply that~$T (A',g^V,h^V)$ is independent of~$h^V$
modulo exact forms in all degrees.
In~\TeilZwei,
we will give a slightly more complicated definition
of~$T (A',g^V,h^V)$
that does not depend on~$h^V$ explicitly,
but uses only the splitting~$A'=\nabla^V+a'$ induced by~$h^V$.

Assume that the metrics~$g^F$ on~$F\to B$ and~$g^H$ on~$H\to B$ are parallel.
Then \ChernNormCor\ implies in particular that we get higher torsion classes
$$\split
	T (M/B;F,h)
	&=T \(A',g^V,h^V\)\dg{\ge2}
		+\chslo\(\nabla^H,g^H,g^H_V\)\dg{\ge2}\\
	\Cal T (M/B;F)
	&=\Cal T \(T^HM,g\TX,\nabla^F,g^F\)\dg{\ge2}
		+\chslo\(\nabla^H,g^H,g^H_{L_2}\)\dg{\ge2}
\endsplit\Formel\HigherChernTorsionDef$$
in~$H^{\even,\ge2}\dR(B)$
as in \MorseTorsionDef\ and \AnalyticTorsionClassDef.
Moreover, the classes~$T (M/B;F,h)$ and~$\Cal T (M/B;F)$
are independent of the particular parallel metrics on~$F$ and~$H$
and of all other secondary data needed to define the respective analytic
torsion forms.

\enddocument

witten.tex
%auto-ignore
\input morse.def

\document

\Kapitel\WittenKapitel=4 The Witten deformation

We consider the Witten deformation of the fibre-wise de Rham complex
with parameter~$T>0$.
Proceeding as in \BGzwei,
we show that for~$T$ sufficiently large,
the de Rham complex splits naturally into two sub-complexes.
One of these sub-complexes represents all eigenvalues that grow
at least linearly in~$T$ (this will be made precise),
so that it does not contribute to the torsion integral in the limit.
The other one can be identified with the sections of a finite-dimensional
vector bundle~$V\to B$,
generalising the families' Thom-Smale complex of~\BGzwei.
While in~\BGzwei,
the identification was provided by the integration over the unstable bundles
of the fibre-wise gradient flow,
we now use the integration map~$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$
constructed in \ComplexKapitel.

In Sections~\NummerVon\ModelSubsection--\NummerVon\SmallTSubsection,
we recall several results of~\BGzwei\
related to the model operator near the fibre-wise critical points,
the space spanned by the generalised eigenforms with small eigenvalues,
and the behaviour of the torsion integrand for large values of~$t$,
always using~$I$ instead of the de~Rham map employed in~\BGzwei.
In \LargeTSubsection\ we show how the definition of~$T(A',g^V,h^V)$
via finite-dimensional Witten deformation on~$\Omega^*(B;V)$
fits into the picture
of the infinite-dimensional Witten deformation on~$\Omega^*(M;F)$,
thereby establishing \SmallEvTorsionThm,
which was used in \MainProofSubsection\ to prove \MainTheorem.



\Abschnitt\ModelSubsection The model operator near the critical points

We recall constructions from~\BGzwei,
sections~4 and~10 related to a model operator~$\widehat\Xx_T$
acting on forms on the total space of a $\Z_2$-graded Euclidean
vector bundle~$E=E^+\oplus E^-$ over a smooth manifold~$C$,
with values in a flat vector bundle~$F\to E$ that is pulled back from~$C$.
This will be applied as follows.
Assume that~$p\colon M\to B$
carries a fibre-wise Morse function~$h\colon M\to\R$
with fibre-wise critical set~$C$,
and there exists a vertical metric~$g\TX$
on the vertical tangent bundle~$TX\to M$,
such that~$h$ and~$g\TX$ satisfy the simplifying assumptions
of \SimplifyingAssumptions.
Then the restriction of the vertical tangent bundle to~$C$
splits orthogonally as~$TX|_C=T^sX\oplus T^uX$.
The Witten deformed operator~$\widetilde\Xx_T$ acting on~$\Omega^*(M;F)$
coincides with~$\widehat\Xx_T$ on~$E=TX|_C$ in a tubular neighbourhood of~$C$.
It has been shown in~\BGzwei\
that for~$T\to\infty$,
the operator~$e^{\widehat\Xx_T^2}$ provides us with a good approximation
for the heat kernel~$e^{\widetilde\Xx_T^2}$,
in a sense to be made precise later.

Let~$\pi\colon(E,g^E)=\(E^+,g^{E^+}\)\oplus\(E^-,g^{E^-}\)\to C$
be a $\Z_2$-graded Euclidean vector bundle
of dimension~$n=n^++n^-$.
Let~$\nabla^E=\nabla^{E^+}\oplus\nabla^{E^-}$
be a Euclidean connection that respects the grading,
with curvature~$F^E$.
If we identify~$\pi^*E\to E$ with the vertical tangent bundle of~$E\to C$,
we can define a vertical radial vector field on the total space~$E$ by
	$$\Cal R=\Cal R^++\Cal R^-
		\Mtext{with}
	\Cal R^\pm(p,e^+,e^-)=e^\pm\in\pi^*E\subset TE\;,$$
for all~$p\in C$ and all~$(e^+,e^-)\in E_p$.
%If no confusion can arise,
%we will sometimes write~``$\punkt$'' instead of~$\Cal R$.

The connection~$\nabla^E$ induces a bundle connection on~$TE$,
i.e., a projection~$TE\to\pi^*E$ with kernel~$T^HE$,
and~$g^E$ becomes a vertical metric.
These data induce a connection on the vertical tangent bundle
as in~\Bzehn, Theorem~1.1,
which coincides with the pull back~$\nabla^{\pi^*E}$
of the connection~$\nabla^E$.
The fibre bundle curvature of~$T^HE\subset TE$
is given by~$\Omega^E=F^E\Cal R$,
see~\VectorBundleCurvFormel.

Let~$\Omega^*(E/C)\to C$ denote the bundle of differential forms
on the fibres of~$E$,
equipped with the $L_2$-metric
	$$g^{\Omega^*(E/C)}(s,s')_p
	=\int_{E_p}g^E\(s(e),s'(e)\)\,d\Vol_E(e)\;,$$
where~$d\Vol_E$ is the volume element induced by the vertical metric~$g^E$.
Then~$\nabla^E$ induces a natural connection~$\nabla^{\Omega^*(E/C)}$
on~$\Omega^*(E/C)$,
and~$\nabla^{\Omega^*(E/C)}$ is compatible
with the $L_2$-metric~$g^{\Omega^*(E/C)}$.

If we identify~$E$ with~$E^*$,
then~$E$ acts on~$\Omega^*(E/C)$ by exterior multiplication~$\eps_e$,
interior multiplication~$\iota_e$, and two types of Clifford multiplication
	$$c_e=\eps_e-\iota_e\Mand\hat c_e=\eps_e+\iota_e\;,$$
satisfying
	$$[c_e,c_{e'}]=-g^E(e,e')\;,\qquad
	[c_e,\hat c_{e'}]=0\;,\Mand
	[\hat c_e,\hat c_{e'}]=g^E(e,e')\;,$$
where we use~$[A,B]=AB-(-1)^{\deg A\cdot\deg B}BA$
to denote the supercommutator of~$A$ and~$B$.
Fixing a local orthonormal basis~$e_1$, \dots, $e_n$ of~$E$,
we abbreviate~$\eps_j=\eps_{e_j}$, $\iota_j=\iota_{e_j}$ etc.

Finally,
let~$\(F,\nabla^F\)\to C$ be a flat complex vector bundle
with a Hermitian metric~$g^F$
that is not necessarily parallel.
Let~$\nabla^{F,*}$ be the adjoint of~$\nabla^F$ with respect to~$g^F$,
and set
	$$\omega^F
	=\nabla^{F,*}-\nabla^F
	=\(g^F\)^{-1}\bigl[\nabla^F,g^F\bigr]\;,
		\Mand
	\nabla^{F,\roman u}
	={1\over2}\,\(\nabla^{F,*}+\nabla^F\)
	=\nabla^F+{1\over2}\,\omega^F\;.$$
Then~$\nabla^{F,\roman u}$ is unitary with respect to~$g^F$.
By an abuse of notation,
we write~$\(F,\nabla^F\)\to E$ and~$g^F$
for the pull-backs~$\(\pi^*F,\pi^*\nabla^F\)$ and~$\pi^*g^F$.
Then~$g^E$ and~$g^F$ induce an $L_2$-metric
	$$g^{\Omega^*(E/C;F)}(s,s')_p
	=\int_{E_p}\(g^E\otimes g^F\)\(s(e),s'(e)\)\,d\Vol_E(e)$$
on the bundle~$\Omega^*(E/C;F)=\Omega^*(E/C)\otimes F\to C$.
The connections~$\nabla^F$, $\nabla^{F,*}$ and~$\nabla^{F,\roman u}$
induce connections on~$\Omega^*(E/C;F)$ denoted by
$$\gather
	\nabla^{\Omega^*(E/C;F)}\;,\qquad
	\nabla^{\Omega^*(E/C;F),*}
	=\nabla^{\Omega^*(E/C;F)}+\omega^F\;,\\
		\Land
	\nabla^{\Omega^*(E/C;F),\roman u}
	={1\over2}\,\(\nabla^{\Omega^*(E/C;F),*}+\nabla^{\Omega^*(E/C;F)}\)\;.
\endgather$$

We are now ready to construct two superconnections on
the bundle~$\Omega^*(E/C;F)\to C$ by
$$\split
	\widehat\Aa'_0
	&=d^E
	=\eps_i\,\nabla^F_{e_i}+\nabla^{\Omega^*(E/C;F)}+\iota_{F^E\Cal R}\;,\\
		\Land
	\widehat\Aa''_0
	&=-\iota_i\,\nabla^F_{e_i}+\nabla^{\Omega^*(E/C;F),*}
		-\eps_{F^E\Cal R}\;.
\endsplit\Formel\HatANullDef$$
Then~$\widehat\Aa''$ is the adjoint of~$\widehat\Aa'$
with respect to~$g^{\Omega^*(E/C;F)}$.

Now suppose that~$h^C\colon C\to\R$ is any smooth function,
then the function~$h^E\colon E\to\R$ given by
	$$h^E=h^C\circ\pi+{1\over2}\,\(\abs{\Cal R^+}^2-\abs{\Cal R^-}^2\)$$
is a fibre-wise Morse function.
The critical set is the zero section~$C\subset E$ of~$E\to C$,
and its index is~$n^-=\dim E^-$.
We extend the superconnections~$\widehat\Aa'_0$ and~$\widehat\Aa''_0$
to superconnection on the pullback~$\Omega^*\(\widehat E/\widehat C;F\)$
of~$\Omega^*(E/C;F)$ to~$\widehat C=C\times(0,\infty)$
by setting
	$$\widehat\Aa'
	=e^{-Th^E}\,\biggl(\widehat\Aa'_0+{\del\over\del T}\,dT\biggr)
		\,e^{Th^E}\;,
	\Mand
	\widehat\Aa''
	=e^{Th^E}\,\biggl(\widehat\Aa''_0+{\del\over\del T}\,dT\biggr)
		\,e^{-Th^E}\;,$$
where~$T$ is the standard coordinate on~$(0,\infty)$.
Then~$\widehat\Aa''$ is the adjoint of~$\widehat\Aa''$
with respect to the pullback of the $L_2$-metric~$g^{\Omega^*(E/C;F)}$.
We denote the restriction of any of these connections
to the submanifold~$C\times\{T\}$
by adding a subscript~$(\punkt)_T$.
%; this notation is of course consistent with~\HatANullDef.

Define an operator~$\widehat\Xx$ on~$\Omega^*\(\widehat E/\widehat C;F\)$
by
$$\multline
	\widehat\Xx
	={1\over2}\,\(\widehat\Aa''-\widehat\Aa'\)
	=-{1\over2}\(\hat c_j\,\del_j+T\,c_{\Cal R^+-\Cal R^-}\)
	+{1\over2}\,\omega^F
		-{1\over2}\,\hat c_{F^E\Cal R}\\
		-T\,dh^E
		-{\abs{\Cal R^s}^2-\abs{\Cal R^u}^2\over2}\,dT
		-h^C\,dT
	=\widehat\Xx_T-h^E\,dT\;.
\endmultline$$
The square of this operator is given by
$$\split
	\widehat\Xx^2
	&=-{1\over4}\Bigl(\Delta^E+T^2\abs{\Cal R}^2
			-\bigl\<F^E\Cal R,F^E\Cal R\bigr\>\Bigr)
		+{1\over2}\nabla_{F^E\Cal R}\\
	&\qquad
		-{T\over2}\biggl(N^+-N^-+{n^--n^+\over2}\biggr)
		+{1\over4}\<F^Ee_j,e_k\>\hat c_j\hat c_k\\
	&\qquad
		+{1\over4}\bigl(\omega^F\bigr)^2
		+{1\over2}\,\hat c_{\Cal R^+-\Cal R^-}\,dT\\
	&=\widehat\Xx_T^2+{1\over2}\,\hat c_{\Cal R^+-\Cal R^-}\,dT\;,
\endsplit\Formel\XhatSquareFormel$$
where~$\Delta^E$ denotes the fibre-wise Laplacian on~$E$,
and~$N^+$, $N^-$ are number operators
that act on~$\Lambda^pE^+\otimes\Lambda^qE^-\subset\Lambda^*E$ by~$p$
and~$q$ respectively.


The model operator~$\widehat\Xx$ is invariant under an $\R^*_+$-action
that we now define.
Let~$o(E^-)=\lambda^{\max}\widehat E^-\to\widehat C$
be the orientation bundle of~$\widehat E^-\to\widehat C$.
Let~$N^{dT}$ denote the number operator
on~$\Lambda^*T(0,\infty)=\R\oplus\R\,dT$.

\Definition\RlambdaDef
For~$\lambda>0$,
set
$$\alignat3
	r_\lambda(\hat p)
	&=\(p,T/\lambda\)&\quad&
		&&\Rtext{for~$\hat p=(p,T)\in\widehat C$,}\tag1\\
	r_\lambda^{o(E^-)}(o)
	&=o
		&&\in o(E^-)_{r_\lambda\hat p}
		&&\Rtext{for~$o\in o(E^-)_{\hat p}$,}\tag2\\
	r_\lambda^F(f)
	&=f
		&&\in F_{r_\lambda\hat p}
		&&\Rtext{for~$f\in F_{\hat p}$,}\tag3\\
	r_\lambda^{\Lambda^*T\widehat C}(\alpha)
	&=\lambda^{\textstyle{N^{\widehat C}\over2}}\,\alpha
		&\quad&\in\Lambda^*T_{r_\lambda\hat p}\widehat C
		&&\Rtext{for~$\alpha\in\Lambda^*T_{\hat p}\widehat C$,}\tag4\\
	\(r_\lambda^{\Omega^*\(\widehat E/\widehat C\)}(\omega)\)
		\(r_\lambda\hat p,e\)
	&=\lambda^{-\textstyle{n\over4}}\,\omega\(\hat p,e/\sqrt\lambda\)
		\kern-5em&&
		&&\Rtext{for~$\omega
			\in\Omega^*\(\widehat E/\widehat C\)_{\hat p}$,}\tag5
\endalignat$$
where we agree that the rescaling of the coordinates~$T$ and~$e$
affects~$dT$, but not the vertical forms.
For~$\omega\in\Omega^*\bigl(\widehat E;F\bigr)
=\Omega^*\bigl(\widehat C;\Omega^*(\widehat E/\widehat C;F)\bigr)$,
set
	$$(r_\lambda^{\Omega^*\(\widehat E;F\)}\omega)(\hat p,e)
	=\Bigl(r_\lambda^{\Lambda^*T\widehat C
			\otimes\Omega^*(\widehat E/\widehat C;F)}
		\,\omega_{r_\lambda^{-1}\hat p}\Bigr)(e)
	=\lambda^{\textstyle{N^{\widehat C}\over2}-{n\over4}}
		\,\omega\Bigl(p,\lambda T,e/\sqrt\lambda\Bigr)\;.\tag6$$
\enddefinition

Note that~$r_\lambda^{o(E^-)}$, $r_\lambda^F$,
and~$r_\lambda^{\Omega^*\(\widehat E/\widehat C\)}$
are isometries with respect to the pull-backs to~$\widehat C$
of the bundle metrics~$g^{o(E^-)}$, $g^F$ and~$g^{\Omega^*(E/C;F)}$.
When no confusion can arise,
we will drop the superscript and write simply~$r_\lambda$.

\Proposition\XhatScalProp.
The operator~$\widehat\Xx^2$ satisfies
	$$\widehat\Xx^2
	=\lambda\,r_\lambda
		\,\widehat\Xx^2
		\,r_\lambda^{-1}\;.$$

\Proof
This is clear from~\XhatSquareFormel\ and~\RlambdaDef.
Note that~$dT$ is multiplied by~$\lambda^{3\over2}$
by \RlambdaDef~(1) and~(4).
\qed\enddemo

Let
	$$\widehat\Xx_1\dg0
	=-{1\over2}\,\(\hat c_j\nabla^F_{e_j}+c_{\Cal R^+-\Cal R^-}\)$$
be the homogeneous part of horizontal degree~$0$
with respect to~$\Omega^*\bigl(\widehat C\bigr)$.
Then
	$$-\(\widehat\Xx_1\dg0\)^2
	={1\over4}\Bigl(\Delta^E+\abs{\Cal R}^2\Bigr)
		+{N^+-N\over2}-{n^+-n^-\over4}$$
is a fibre-wise harmonic oscillator
with spectrum~$\bigl\{0,{1\over2},1,\dots\bigr\}$.
Let~$o(E^-)=\Lambda^{\max}E^-$ denote the orientation bundle of~$E^-$.
Then the map
$$\split
	\widehat J_1\dg0
	&\;\colon\;o(E^-)\otimes F
	\to\ker\Bigl(\widehat\Xx_1\dg0\Bigr)^2\subset\Omega^*\(E/C;F\)\\
		\Ltext{with}
	\widehat J_1\dg0(v)
	&=\pi^{-\textstyle{n\over4}}
		\,e^{-\textstyle{\abs\punkt^2\over2}}
		\,\pi^*v
\endsplit\Formel\JhatNullDef$$
is an isometry
with respect to~$g^F$ and~$g^{\Omega^*(E/C;F)}$.

Because~$\widehat\Xx_1^2-\(\widehat\Xx_1\dg0\)^2$ is nilpotent,
the operators~$\widehat\Xx_1\dg0$ and~$\widehat\Xx_1$ have the same spectrum.
By \XhatScalProp,
the operator~$\widehat\Xx^2$
has spectrum~$\Spec\(\widehat\Xx^2\)_{C\times\{T\}}
=\bigl\{0,{T\over2},T,\dots\bigr\}$.
Let~$\delta\subset\C$ be circle of radius~$1$ around~$0$
and consider the map
$$\align
	\widehat\Pp
	&={1\over2\pi i}\int_{{T\over4}\,\delta}
		{d\lambda\over\lambda-\widehat\Xx^2}\\
	&=\widehat\Pp_T+\widehat\Qq_T\,dT
	\quad\in\Omega^*\Bigl(\widehat C;
		\End\Omega^*\(\widehat E/\widehat C;F\)\Bigr)\;.
\endalign$$
We define another map~$\widehat J
\colon\Lambda^*T\widehat C\otimes o(E^-)\otimes F
\to\Lambda^*T\widehat C\otimes\Omega^*\bigl(\widehat E/\widehat C;F\bigr)$
by
	$$\widehat J
	=\widehat\Pp\circ\biggl(\Bigl(r_T^{\Lambda^*T\widehat C
			\otimes\Omega^*\(\widehat E/\widehat C;F\)}\Bigr)^{-1}
		\circ J_1\dg0
		\circ r_T^{\Lambda^*T\widehat C\otimes o(E^-)
			\otimes F}\biggr)\;.
	\Formel\JhatDef$$
Then we may rewrite Theorem~10.53 in~\BGzwei\ as follows,
where for any vector bundle~$V\to\widehat C$,
we set
	$$\Omega^*\(\widehat C/(0,\infty);V\)
	=\bigl\{\,\omega\in\Omega^*\(\widehat C;V\)
		\bigm|\iota_{\del\over\del T}\omega=0\,\bigr\}\;.$$

\Theorem\JhatThm.
The map~$\widehat\Pp$ is a spectral projection
that commutes with~$\widehat\Aa'$ and~$\widehat\Aa''$,
is orthogonal with respect to~$g^{\Omega^*\(E/C;F\)}$,
and we have
	$$\widehat\Pp\in\Omega^\even\Bigl(\widehat C/(0,\infty);
		\End^\even\Omega^*\(\widehat E/\widehat C\)
			\oplus\End^\odd\Omega^*\(\widehat E/\widehat C\)\,dT
		\Bigr)\;.\tag1$$
For any positive integer~$k$,
	$$\widehat\Pp
	={1\over2\pi i}
		\,\int_{{\textstyle(}{T\over4}{\textstyle)^{k\over2}}\delta}
			\,{d\lambda\over\lambda-\widehat\Xx^k}\;.\tag2$$
\parr
Let~$\widehat\Ff$ be the range of~$\widehat\Pp$,
then~$\widehat\Ff$ is a finite dimensional bundle over~$\widehat C$
which is preserved by the action of~$r_\lambda$,
and~$\widehat\Ff=\ker\bigl(\widehat\Xx^k\bigr)$ for~$k$ sufficiently large.
\parr
The map~$\widehat J$ maps~$\Lambda^*T\widehat C\otimes o(E^-)\otimes F$
isomorphically to~$\widehat\Ff$,
and~$\widehat J\dg0\colon o(E^-)\otimes F\to\widehat\Ff\dg0
=\widehat\Ff\cap\Omega^*\bigl(\widehat E/\widehat C;F\bigr)$ is an isometry.
We may write
	$$\widehat J(\alpha\otimes o\otimes f)
	=\alpha\otimes\Biggl(\biggl({T\over\pi}\biggr)^{\textstyle{n\over4}}
		\,e^{-\textstyle{T\abs{\Cal R}^2\over2}}
		\,\(1+R_{\widehat E}\)\(\pi^*o\)\Biggr)
		\otimes\pi^*f\tag3$$
for~$\alpha\in\Lambda^\even T\widehat C$, $o\in o(E^-)$,
and~$f\in F$,
where
	$$R_{\widehat E}\in\Omega^\even\Bigl(\widehat C/(0,\infty);
		\End^\even\Omega^*\(\widehat E/\widehat C\)
			\oplus\End^\odd\Omega^*\(\widehat E/\widehat C\)\,dT
		\Bigr)\tag4$$
is an $r_\lambda$-invariant polynomial in~$\Cal R^\pm$
with coefficients depending only on~$F^{E^\pm}$ and~$T$.
\qed

\Proof
It follows from \XhatScalProp\ that~$\widehat\Pp$ and~$\widehat\Ff$
are $r_\lambda$-invariant.
The other assertions on~$\widehat\Pp$ and~$\widehat\Ff$
follow as in Theorem~10.53 in~\BGzwei.
The fact that the only term in~\XhatSquareFormel\ containing~$dT$ is
	$${1\over2}\,\hat c_{\Cal R^+-\Cal R^-}\,dT
	\in\End^\odd\Omega^*(\widehat E/\widehat C)\,dT$$
accounts for the form of~(1) compared to the corresponding statement
in Theorem~10.53 in~\BGzwei.
From~\BZzwei,
we know that~$\widehat J\dg0$ takes the form
	$$\widehat J\dg0(v)
	=\biggl({T\over\pi}\biggr)^{\textstyle{n\over4}}
		\,e^{-\textstyle{T\abs{\Cal R}^2\over2}}
		\,\pi^*(v)\;.$$
Then~(3) follows immediately from~(1),
and~$R_{\widehat E}$ is a polynomial by~(10.191) in~\BGzwei,
which is $r_\lambda$-invariant
since~$\widehat\Ff$ and~$\widehat\Pp$ are $r_\lambda$-invariant,
and of the form~(4) by~(1).
\qed\enddemo

Let~$\widehat J^*$ be the adjoint of~$\widehat J$
with respect to~$g^{o(E^-)\otimes F}$
and~$g^{\Omega^*(\widehat E/\widehat C;F)}$
in the sense of \AdjointDef.
Integration over the fibres of~$\widehat E^-\to\widehat C$
defines a map
	$$\widehat I\colon\Omega^*\(\widehat E;F\)
	\to\Omega^*\(\widehat C;o(E^-)\otimes F\)
		\Mtext{with}
	\widehat I(\alpha)
	=\int_{\widehat E^-/\widehat C}\alpha\;.$$
Then we have the following version of~\BGzwei,
Proposition~10.58 and~Theorem~10.59 for our model situation.

\Proposition\IhatJhatProp.
There exist $r_\lambda$-invariant characteristic differential forms~$R_{J^*J}$
and~$R_{IJ}\in\Omega^{\even,\ge2}\bigl(\widehat C/(0,\infty)\bigr)$
of~$R^{E^\pm}$,
such that
$$\align
	\widehat J^*\circ\widehat J
	&=1+R_{J^*J}\tag1\\
		\Land
	\widehat I\circ e^{Th^E}\circ\widehat J
	&=e^{Th^C}\,\biggl({T\over\pi}\biggr)^{\textstyle{n^+-n^-\over4}}
		\,\(1+R_{IJ}\)\;.\tag2
\endalign$$
In particular, $R_{J^*J}$, $R_{IJ}$
are polynomials in~$1\over T$ without constant term.

Note in particular that the right hand sides of both formulas
are closed differential forms that do not contain any multiple of~$dT$.


\Proof
These claims follow by elementary calculations using $r_\lambda$-invariance,
see also~\BGzwei, Section~10.15.
Since~$R_{\widehat E}$ is $r_\lambda$-invariant,
	$$R_{\widehat E}|_{(p,T,e)}
	=T^{-\textstyle{N^{\widehat C}\over2}}
		\,R_{\widehat E}|_{(p,1,\sqrt T\,e)}\;.$$
We will use the symbol~$\abs\punkt^2$ in the sense of~\TorSignMetric.
Let~$v\in\(o(E^-)\otimes F\)_{p,T}$, then
$$\align
	\(\widehat J^*\circ\widehat J\,\)(v)
	&=\int_{E_p}\biggl({T\over\pi}\biggr)^{\textstyle{n\over2}}
		\,e^{-T\abs e^2}
		\,\abs{\(1+R_{\widehat E}|_{(p,T,e)}\)(\pi^*v)}^2
		\,d\Vol^E(e)\\
	&=T^{-\textstyle{N^{\widehat C}\over2}}
		\int_{E_p}\pi^{-\textstyle{n\over2}}\,e^{-\abs e^2}
		\,\abs{\(1+R_{\widehat E}|_{(p,1,e)}\)(\pi^*v)}^2
		\,d\Vol^E(e)\;,\\
\endalign$$
which proves~(1).
Note that the part of~$R_{\widehat E}$ that contains~$dT$
does not contribute because it is odd
as an endomorphism of~$\Omega^*\bigl(\widehat E/\widehat C\bigr)$.

Similarly, we get~(2) because
$$\align
	\(\widehat I\circ e^{Th^E}\circ\widehat J\,\)
	&=e^{Th^C}\,\biggl({T\over\pi}\biggr)^{\textstyle{n\over4}}\\
	&\qquad
		\int_{\widehat E^-/\widehat C|_p}e^{T(h^E-h^C)(e^-)}
		\,e^{-\textstyle{T\abs{e^-}^2\over2}}
		\,\(1+R_{\widehat E}|_{(p,T,e^-)}\)\(\pi^*v\)
\allowdisplaybreak
	&=e^{Th^C}\biggl({T\over\pi}\biggr)^{\textstyle{n^+-n^-\over4}}
		\,\pi^{-\textstyle{n\over2}}\\
	&\qquad
		\,T^{-\textstyle{N^{\widehat C}\over2}}
		\int_{\widehat E^-/\widehat C|_p}e^{-\abs{e^-}^2}
		\,\(1+R_{\widehat E}|_{(p,1,e^-)}\)\(\pi^*v\)\;.\\
\endalign$$
Note that if~$v=o\otimes f\in o(E^-)\otimes F$,
then~$\pi^*o$ is a volume form on the fibres of~$\widehat E^-$,
so only the component of~$R_{\widehat E}$
in~$\Omega^*\bigl(\widehat C;\bigl\{\id_{\Omega^*(\widehat E/\widehat C;F)}\bigr\}\bigr)$ contributes,
and this component again does not contain~$dT$ since it is even.

Then both~$R_{J^*J}$ and~$R_{IJ}$ are $r_\lambda$-invariant,
even forms of degree~$\ge 2$,
so they are polynomials in~${1\over T}$ without constant terms.
\qed\enddemo


\Abschnitt\SmallevSubsection Small eigenvalues and spectral projections

Let
	$$\overline M=M\times(0,\infty)\times\R
	\to\overline B=B\times(0,\infty)\times\R$$
be the pull-back of~$M\to B$ to~$\overline B$.
By an abuse of notation,
let~$TX\to\overline M$ and~$F\to\overline M$ denote the pull-backs
of the vector bundles~$TX\to M$ and~$F\to M$ to~$\overline M$.
Let~$g\TX$ and~$g^F$ denote the pull-back metrics on these bundles,
then these metrics induce a fibre-wise $L_2$-metric~$g^{TX,F}$
on the infinite dimensional vector bundle~$\Omega^*(M/B;F)\to B$.
For~$t\in(0,\infty)$, $T\in\R$ consider the metrics
	$$g\TX_t={1\over t}\,g\TX
		\Mand
	g^F_T=e^{-2Th}\,g^F\;.$$
These metrics induce a fibre-wise $L_2$-metric~$\bar g^{TX,F}$
on~$\Omega^*\bigl(\overline M/\overline B;F\bigr)$
such that
	$$\bar g^{TX,F}
	=g^{TX,F}\,\Bigl(t^{N^X-\textstyle{n\over2}}\,e^{-2Th}\punkt
			,\punkt\Bigr)\;.$$

Let~$\Aa'=d^M$ denote the total derivative on~$\Omega^*(M;F)$.
We extend~$\Aa'$ to a superconnection~$\overline\Aa'$
on~$\Omega^*\bigl(\overline M;F\bigr)$ by
	$$\overline\Aa'
	=\Aa'+{\del\over\del t}\,dt+{\del\over\del T}\,dT\;.$$
Let~$\overline\Aa''$ be the adjoint superconnection of~$\overline\Aa'$
with respect to~$\bar g^{TX,F}$.
Then we have
	$$\Aa''
	=t^{-N^X}\,e^{2Th}\,\biggl(\Aa''
			+{\del\over\del t}\,dt+{\del\over\del T}\,dT\biggr)
		\,e^{-2Th}\,t^{N^X}
	\Formel\AaxxtTDefinition$$

Then we define a self-adjoint superconnection~$\overline\Aa$
on~$\Omega^*(M/B;F)$
and a skew-adjoint operator~$\overline\Xx\in\Omega^*\(B;\End\Omega(M/B);F\)$%
---both with respect to~$\bar g^{TX,F}$---%
by
	$$\overline\Aa={1\over2}\,\(\overline\Aa''+\overline\Aa'\)
		\Mand
	\overline\Xx={1\over2}\,\(\overline\Aa''-\overline\Aa'\)\;.$$
We will denote the restrictions of~$\overline\Aa'$, $\overline\Aa''$
and~$\bar g^{TX,F}$ to~$M\times\{t\}\times\{T\}$ by~$\Aa'\tT$, $\Aa''\tT$
and~$g^{TX,F}\tT$, respectively.
We have
	$$\overline\Xx=\Xx\tT+N^X\,{dt\over2t}-h\,dT\;.$$


If~$\Aa\tT\dg0$ and~$\Xx\tT\dg0\in\Omega^0\(B;\End\Omega^*(M/B;F)\)$
denote the homogeneous part of horizontal degree~$0$,
then~$\overline\Aa^2|\tT-(\Aa\tT\dg0)^2$
and~$\overline\Xx^2|\tT-(\Aa\tT\dg0)^2$
are elements of~$\Omega^{>0}\(B;\End\Omega^*(M/B;F)\)$
and thus nilpotent in~$\Omega^*\(B;\End\Omega^*(M/B;F)\)$.
This implies that
$$\align
	\Spec\overline\Aa^2|\tT&=\Spec\Aa\tT^2=\Spec\(\Aa\tT\dg0\)^2\\
	=-\Spec\overline\Xx^2|\tT&=-\Spec\Xx^2\tT=-\Spec\(\Xx\tT\dg0\)^2\;.
\endalign$$

Recall that a vector bundle~$V\to B$ was defined in \MorseBundleDef\ by
$$\align
	V^j&=\hat p^j_*F|_{C^j}\otimes o(N^u)\;,\\
		\Land
	V&=\hat p_*F|_C\otimes o(N^u)=\bigoplus_jV^j\;,
\endalign$$
and that we have constructed a map~$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$
in \PatchSuperconnectionDefinition,
which is an $\Omega^*(B)$-linear combination of maps
that arise by integration
over the fibres of certain subbundles of~$M\to B$.
Let~$I\dg0\colon\Omega^*(M/B;F)\to V$ denote the homogeneous part
of horizontal degree~$0$.
Then we have the following result of \BZeins, Theorems~7.8 and~7.9,
and~\BZzwei, Theorem~6.11,
cf.~\BGzwei, Theorems~10.2 and~10.59.

\Theorem\SmallEvThm.
There exists~$T_0\ge 0$ such that for all~$T\ge T_0$,
	$$\Spec\(\Aa^2\tT\)=-\Spec\(\Xx^2\tT\)
	\subset\biggl[0,{t\over4}\biggr]\cup(4t,\infty)\;,$$
and the direct sums of eigenspaces associated to eigenvalues
in~$\bigl[0,{t\over4}\bigr]$
form a vector bundle~$W\to B\times(0,\infty)\times[T_0,\infty)
\subset\overline B$,
such that
	$$I\dg0|_W\colon W\to V$$
is a linear isomorphism.

\Proof
We can copy the proof of~\BZzwei, Theorem~6.11.
However, since~$I\dg0$ is not simply integration over the unstable cell
of the gradient flow,
we have to use \PatchThm.
\qed\enddemo

Let~$S^1(t)\subset\C$ denote a circle around the origin of radius~$t$,
and note that if~${t\over4}<t_0$, $t_1<4t$, then
	$${1\over2\pi i}\,\int_{S^1(t_0)}
		{d\lambda\over\lambda-\overline\Aa^2}
	={1\over2\pi i}\,\int_{S^1(t_1)}
		{d\lambda\over\lambda-\overline\Aa^2}\;.$$
This allows us to patch spectral projectors
that are defined on~$B\times\bigl({t_0\over4},4t_0\bigr)\times[T_0,\infty)$
together as in the following definition,
cf.\ Definition~10.33 in~\BGzwei.


\Definition\PsmallDef
Define an endomorphism~$\overline\Pp$
of the bundle~$\Lambda^*T\(\overline B\)
\otimes\Omega^*\bigl(\overline M/\overline B;F\bigr)$
for~$T\ge T_0$ in a neighbourhood of~$B\times\{(t_0,T)\}$
in~$B\times(0,\infty)\times[T_0,\infty)$ by
	$$\overline\Pp
	={1\over2\pi i}\,\int_{S^1(t_0)}
		{d\lambda\over\lambda-\overline\Aa^2}\;.$$
\enddefinition


We recall parts of Theorem~10.9 from~\BGzwei.

\Theorem\PsmallThm.
For~$T\ge T_0$,
the operator~$\overline\Pp
\in\Omega^*\bigl(\overline B;\End\Omega^*(M/B;F)\bigr)$
is an even self-adjoint projector
with range a finite dimensional
vector bundle~$\overline\Ww\to B\times(0,\infty)\times[T_0,\infty)$,
which commutes with the action
of~$\Lambda^*T\(\overline B\)$
and with~$\overline\Aa'$ and~$\overline\Aa''$.
For any integer~$k>0$ and~$t\in\bigl({t_0\over4},4t_0\bigr)$,
we have
	$$\overline\Pp
	={1\over2\pi i}\int_{S^1{\textstyle\bigl(}t_0^{k\over2}
			{\textstyle\bigr)}}
		{d\lambda\over\lambda-\overline\Xx^k}\;.$$

Let~$\bold g^\Ww_T$ denote the restriction of the $L_2$-metric~$g^{TX,F}_{1,T}$
on~$\Omega^*\bigl(\overline M/\overline B;F\bigr)$
to~$\Ww_T=\overline\Ww|_{B\times\{(1,T)\}}$,
which is a generalized metric on~$\Ww$ in the sense of \GenMetDef.
Similarly,
let~$g^H_T$ denote the $L_2$-metric on~$H$ induced by~$g^{TX,T}_{1,T}$.

\Theorem\LargeEvThm.
There exists a constant~$\delta\in(0,1)$ such that for~$T\ge T_0$,
	$$\int_1^\infty\biggl(\strace\Bigl(N^X\,f'\(\Xx\tT\)\Bigr)
			-\strace\Bigl(N^X\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)
		\biggr)\,{dt\over2t}
	=O\(T^{-\delta}\)\;,\tag1$$
and for all~$T_1\ge T_0$,
$$\multline
	\tilde f\(\nabla^H,g^H,g^H_{T_1}\)
		-(2\pi i)^{-\textstyle{N^B\over2}}\int_1^\infty\biggl(
			\strace\Bigl(N^X\,f'\(\Xx_{t,T_1}\)\Bigr)
			-\chi'(H)\,f'(0)\biggr)\,{dt\over2t}\\
	=\tilde f\(\nabla^H,g^H,g^H_{T_0}\)
		-(2\pi i)^{-\textstyle{N^B\over2}}\int_1^\infty\biggl(
			\strace\Bigl(N^X\,f'\(\Pp_{t,T_0}\,\Xx_{t,T_0}
				\,\Pp_{t,T_0}\)\Bigr)
			-\chi'(H)\,f'(0)\biggr)\,{dt\over2t}\\
	+\tilde f\(\Aa'_{1,\punkt},\bold g^\Ww_{T_0}
		,\bold g^\Ww_{T_1}\)
%	-(2\pi i)^{-\textstyle{N^B\over2}}\int_{T_0}^{T_1}
%		\strace\Bigl(h\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)\,dT
	+O\(T_1^{-\delta}\)
\endmultline\tag2$$
modulo an exact form on~$B$.

\Proof
Equation~(1) is a consequence of \SmallEvThm,
cf.~\BGzwei, Theorem~10.5.

We recall that
	$$\overline\Xx_T
	=\Xx\tT+\(g^{TX,F}\tT\)^{-1}\,{\del g^{TX,F}\tT\over\del t}\,dt
	=\Xx\tT+{N^X\over2t}\,dt\;.$$
Let~$\overline\Pp_T=\Pp\tT+\Pp'\tT\,dt$
and note that~$\overline\Pp_T^2=\overline\Pp_T$ implies
	$$\Pp'\tT=\Pp\tT\,\Pp'\tT+\Pp'\tT\,\Pp\tT\;,$$
so~$\Pp'\tT$ interchanges the images of~$\Pp\tT$ and~$\id-\Pp\tT$.
This implies that
$$%\multline
\split\kern-2em
	\strace\Bigl(f\(\overline\Pp_T\,\overline\Xx_T\,\overline\Pp_T\)\Bigr)
	&=\strace\Bigl(f\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)\\
	&\qquad
	+\strace\Biggl(\biggl(-\Pp'\tT\,\Xx\tT\,\Pp\tT
				+{N^X\over2t}+\Pp\tT\,\Xx\tT\,\Pp'\tT\biggr)\\
	&\kern6em
			\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Biggr)\,dt\\
	&=\strace\Bigl(f\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)
%\endsplit\\
		+\strace\Bigl(N^X
			\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)\,{dt\over2t}\;.
	\kern-2em
%\endmultline\tag\FormelNummer\SmallTProjectionFormel
\endsplit\Formel\SmallTProjectionFormel$$
A similar argument with fixed~$t$ and variable~$T$ shows that
$$%\multline
	\strace\Bigl(f\(\overline\Pp_t\,\overline\Xx_t
			\,\overline\Pp_t\)\Bigr)
	=\strace\Bigl(f\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)%\\
		-\strace\Bigl(h
			\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)\,dT\;.
%\endmultline\tag\FormelNummer
\Formel\LargeTProjectionFormel$$

Now consider the closed differential form
on~$B\times(0,\infty)\times\{T\}$,
which is given by
$$\multline
	\strace\Bigl(f\(\overline\Pp\,\overline\Xx\,\overline\Pp\)\Bigr)
		-\chi'(H)\,f'(0)\,{dt\over t}\\
	=\strace\Bigl(f\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)
		+\biggl(\strace\Bigl(N^X\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)
			-\chi'(H)\,f'(0)\biggr)\,{dt\over2t}\\
	-\strace\Bigl(h\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)\,dT
	+R\,dt\,dT\;,
\endmultline$$
with~$R\in\Omega^*\(B\times(0,\infty)\times[T_0,\infty)\)$.

Let~$\bar g^H$ be the metric on the pull-back of~$H$
to~$B\times[T_0,\infty)$,
let~$\overline\nabla^{H,*}$ denote the adjoint
of the pull-back~$\overline\nabla^H$ of~$\overline\nabla^H$
with respect to~$\bar g^H$,
and let
	$$\overline\omega^H=\overline\nabla^{H,*}-\overline\nabla^H\;.$$
Then we know that as~$t\to\infty$,
	$$\strace\Bigl(f\(\overline\Pp\,\overline\Xx\,\overline\Pp\)\Bigr)
		-\chi'(H)\,f'(0)\,{dt\over t}
	=\strace\Bigl(f\(\overline\omega^H/2\)\Bigr)
		+O\(t^{-\textstyle{1\over2}}\)
		+O\(t^{-\textstyle{3\over2}}\)\,dt\;,$$
cf.~\BL, Theorems~3.16 and~3.21.
Now, equation~(2) follows easily.
\qed\enddemo



\Abschnitt\SmallTSubsection
The limit~$t\to\infty$ and the class~$S_f\(A',g^V\)$

We identify the bundle~$\Ww\to B$ with~$\Lambda^*TB\otimes V$.
This allows us to compare the term
	$$-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_1^\infty\biggl(\strace\Bigl(N^X
		\,f'\(\Pp_{t,T_0}\,\Xx_{t,T_0}\,\Pp_{t,T_0}\)\Bigr)
			-\chi'(H)\,f'(0)\biggr)\,{dt\over2t}$$
of \LargeEvThm~(2) with a class~$S_f\(A',\bold g^V,g^H\)$
of the bundle~$V\to B$.

Let~$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$
be the integration map of \PatchSuperconnectionDefinition,
then we have an obvious extension
	$$\overline I
	\colon\Omega^*\(\overline M;F\)
	\to\Omega^*\(\overline B;V\)\;,$$
for which the analogue of \PatchThm\ still holds.
By \SmallEvThm,
$I\dg0$ provides us with an isomorphism of the bundle~$W\tT\to B$
of the small eigenspaces with the bundle~$V$ of \MorseBundleDef.
By~$\Omega^*(B)$-linearity,
this implies that
	$$\overline\Ii
	=\overline I|_{\overline\Ww}
	\colon\overline\Ww
	\to\Lambda^*T{\overline B}\otimes V
	\Formel\IiDef$$
is an isomorphism as well.
Define
	$$\bar{\bold g}^V\tT
	=\(\overline\Ii^{-1}\)^*\(\bold g^{\overline\Ww}\)
	=g^{TX,F}\Bigl(\overline\Ii^{-1}\punkt
		,\overline\Ii^{-1}\,\punkt\Bigr)
	\Formel\BgVDef$$
Let~$N^M$ denote the total number operator on~$\Omega^*(M;F)$.
Recall that we used the subscript~$(\punkt)_T$
as an abbreviation for~$(\punkt)_{1,T}$.
We collect a few properties of the maps~$\overline\Pp$
and~$\overline\Ii$ and of~$\bar{\bold g}^V$.

\Proposition\SmallTProp.
For~$T\ge T_0$,
the map~$\Pp\tT$ satisfies
	$$\Pp\tT
	=t^{-\textstyle{N^M\over2}}\,\Pp_T\,t^{\textstyle{N^M\over2}}\;.
	\tag1$$
Moreover, $\Pp\tT$ is even in~$\Omega^*\(B;\End\Omega^*(M/B;F)\)$
and polynomial in~$1\over t$,
and~$\Pp\tT\dg0=\Pp_T\dg0$ independent of~$t$.
\parr
The inverse of~$\Ii\tT$ is even
in~$\Omega^*\bigl(B;\Hom\(V,\Omega^*(M/B;F)\)\bigr)$
and polynomial in~$1\over t$
with~$\(\Ii\tT^{-1}\)\dg0=(I\dg0|_{W_T})^{-1}$ independent of~$t$,
and satisfies
	$$\Ii\tT^{-1}
	=t^{-\textstyle{N^M\over2}}\,\Ii_T^{-1}
		\,t^{\textstyle{N^B+N^V\over2}}\;.\tag2$$
\parr
The map~$\bold g^V\tT$ is a generalized metric on~$V$
that depends polynomially on~$1\over t$.
If we set~$g^V_T=\(\bold g^V_T\)\dg0$,
then there exists an even~$g'_T\in\Omega^*\(B;\End V\)$
such that~\UMetricInftyCondition\ holds for each~$T\ge T_0$.
\parr
Let~$A''\tT$ denote the adjoint of~$A'$ with respect to~$\bold g^V\tT$
and set~$X\tT={1\over2}\,\(A''\tT-A'\)$,
then
	$$\Ii\tT\,\Xx\tT=X\tT\,\Ii\tT\;.\tag3$$

\Proof
Since~$\Aa'$ is of total degree~$1$ and~$\Aa''$ is of total degree~$-1$,
we have
	$$t^{\textstyle{1-N^M\over2}}\,\Aa'\,t^{\textstyle{N^M\over2}}=\Aa'
		\Mand
	t^{\textstyle{1-N^M\over2}}\,\Aa''_T\,t^{\textstyle{N^M\over2}}
	=\Aa''\tT\;.$$
By \PsmallDef, this implies~(1).

Next, we note that~$\Aa'\tT$ raises the total degree by~$1$,
while~$\Aa^{\prime\prime[k]}\tT$ raises it by~$2k-1$.
This implies that
	$$\Aa^2\tT=\Aa'\tT\Aa''\tT+\Aa''\tT\Aa'\tT$$
does not lower the total degree.
Moreover
	$$\(\Aa\tT\dg0\)^2
	=\Aa^{\prime[0]}\tT\Aa^{\prime\prime[0]}\tT
		+\Aa^{\prime\prime[0]}\tT\Aa^{\prime[0]}\tT
	=t\,\(\Aa_T\dg0\)^2$$
preserves the total degree,
and~$\Aa\tT^2-\(\Aa\tT\dg0\)^2$ is nilpotent.
In particular,
$$\split
	\Pp\tT
	&={1\over2\pi i}\int_{S^1(t)}
		{1\over\lambda-\(\Aa\tT\dg0\)^2}
		\,\sum_{k\ge0}\biggl(\Bigl(\Aa\tT^2-\(\Aa\tT\dg0\)^2\Bigr)
			\,{1\over\lambda-\(\Aa\tT\dg0\)^2}\biggr)^k
		\,d\lambda\\
	&={1\over2\pi i}\int_{S^1(1)}
		{1\over\lambda-\(\Aa_T\dg0\)^2}
		\,\sum_{k\ge0}\Biggl(\biggl({1\over t}\,\Aa\tT^2
				-\(\Aa_T\dg0\)^2\biggr)
			\,{1\over\lambda-\(\Aa_T\dg0\)^2}\Biggr)^k
		\,d\lambda\;,
\endsplit$$
where the summation index~$k$ is bounded by~$\dim B$.
Since~${1\over t}\,\Aa\tT^2-\(\Aa_T\dg0\)^2$ is even,
it is polynomial in~$1\over t$ by~(1),
which proves the remaining statements on~$\Pp\tT$.

Taking the inverse of~$I\dg0|_{W_T}$,
and noting that~$\Ii\tT=I\,\Pp\tT$
and~$\bigl(I\,\Pp\tT\,\(I\dg0|_{W_T}\)^{-1}\bigr)\dg0=\id$,
we may write
	$$\Ii\tT^{-1}
	=\Pp\tT\,\(I\dg0|_{W_T}\)^{-1}
		\,\Bigl(I\,\Pp\tT\,\(I\dg0|_{W_T}\)^{-1}\Bigr)^{-1}\;.$$
This explains why~$\Ii\tT$ is even and polynomial in~$1\over t$.
Claim~(2) follows from the analogous statement for~$\Pp\tT$,
because~$I\dg0$ preserves the total degree.

Since~$(I\dg0|_W)^{-1}\colon V\to\Omega^(M/B;F)$ is clearly injective
and preserves the total degree,
we know that~$\(\bold g^V\tT\)\dg0=\bigl((I\dg0|_W)^{-1}\bigr)^*g^{TX,F}\tT$
is a Hermitian metric on~$V$ that respects the $\Z$-grading.
Moreover,
$\bold g^V\tT$ is even because~$\Ii\tT^{-1}$ is.
Finally, \GenMetDef~(3), (4) are clearly satisfied by \TorSignMetric,
so~$\bold g^V\tT$ is a generalised metric.
It is also clear that~$\bold g^V\tT$ is polynomial in~$1\over t$.

In order to check~\UMetricInftyCondition,
note that
$$\split
	\bold g^V\tT
	&=t^{-\textstyle{n\over 2}}
		\,g^{TX,F}_T\Bigl(t^{\textstyle{N^X\over2}}\,\Ii\tT^{-1}\punkt,
			t^{\textstyle{N^X\over2}}\,\Ii\tT^{-1}\punkt\Bigr)\\
	&=t^{-N^B-\textstyle{n\over 2}}\,g^{TX,F}_T\Bigl(
			\Ii_T^{-1}\,t^{\textstyle{N^B+N^V\over2}}\punkt,
			\Ii_T^{-1}\,t^{\textstyle{N^B+N^V\over2}}\punkt\Bigr)\\
	&=t^{-N^B-\textstyle{n\over 2}}
		\,\bold g^V_T\Bigl(t^{\textstyle{N^B+N^V\over2}}\punkt,
			t^{\textstyle{N^B+N^V\over2}}\punkt\Bigr)\;.
\endsplit$$
Since~$\Ii_T^{-1}$ is $\Omega^*(B)$-linear,
it cannot lower horizontal degrees,
which proves that~$t^{\textstyle{n-N^V\over2}}\,\bold g^V\tT
\,t^{-\textstyle{N^V\over2}}$
is a polynomial in~${1\over\sqrt t}$ with leading term~$g^V_T$.
This implies the first equation of~\UMetricInftyCondition.
It also implies that
$$\multline
	t^{\textstyle{N^V\over2}}\,\(\bold g^V\tT\)^{-1}
		\,{\del\bold g^V\tT\over\del t}\,t^{-\textstyle{N^V\over2}}
	=\Bigl(t^{\textstyle{n-N^V\over2}}\,\bold g^V\tT
		\,t^{-\textstyle{N^V\over2}}\Bigr)^{-1}
		\,{\del\bigl(t^{n-N^V\over2}\,\bold g^V\tT
			\,t^{-{N^V\over2}}\bigr)\over\del t}\\
	+{N^V\over2t}
	+\Bigl(t^{\textstyle{n-N^V\over2}}\,\bold g^V\tT
		\,t^{-\textstyle{N^V\over2}}\Bigr)^{-1}
		\,{N^V-n\over2t}
		\,\Bigl(t^{\textstyle{n-N^V\over2}}\,\bold g^V\tT
		\,t^{-\textstyle{N^V\over2}}\Bigr)
\endmultline$$
is a polynomial in~${1\over\sqrt t}$
with leading term~${N^V\over t}-{n\over2t}$,
which implies the second equation of~\UMetricInftyCondition.

Finally,
the adjoint of~$A'$ with respect to~$\bold g^V\tT$
can be written as
$$\align
	A''\tT
	&=\Bigl(\overline{\Ii\tT^{-1}}^*\,g^{TX,F}\tT\,\Ii\tT^{-1}\Bigr)^{-1}
		\,\overline{A'}^*
		\,\Bigl(\overline{\Ii\tT^{-1}}^*\,g^{TX,F}\tT
			\,\Ii\tT^{-1}\Bigr)\\
	&=\Ii\tT\,\(g^{TX,F}\tT\)^{-1}\,\overline{\Aa'}^*
		\,g^{TX,F}\tT\,\Ii\tT^{-1}\\
	&=\Ii\tT\,\Aa''\tT\,\Ii\tT^{-1}\;,
\endalign$$
which implies~(3).
\qed\enddemo

Let~$\overline A'$, $\overline A''_T$ and~$\overline X_T$
denote the obvious extensions of~$A'$, $A''\tT$ and~$X\tT$
to the pullback of~$V$ to~$B\times(0,\infty)\times\{T\}$\;.
Note that
	$$\strace\Bigl(f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)
	=\chi(V)\,f'(0)$$
because~$f'$ is even and~$\Pp\tT\,\Xx\tT\,\Pp\tT$
is odd.
With this observed,
we can now calculate
$$\align
	&\kern-6em
	-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_1^\infty\biggl(\strace\Bigl(N^X
		\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\Bigr)
			-\chi'(H)\,f'(0)\biggr)\,{dt\over2t}\\
	&=-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_1^\infty\Biggl(\strace\biggl(\Bigl(N^X-{n\over2}\Bigr)
		\,f'\(\Pp\tT\,\Xx\tT\,\Pp\tT\)\biggr)\\
	&\kern8em
			-\Bigl(\chi'(H)-{n\over2}\,\chi(V)\Bigr)
				\,f'(0)\Biggr)\,{dt\over2t}
\allowdisplaybreak
	&=-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_{(B\times(1,\infty)\times\{T\})/B\times\{T\}}
		\biggl(\strace\Bigl(f\(\overline\Pp_T
			\,\overline\Xx_T\,\overline\Pp_T\)\Bigr)\\
	&\kern8em
			-\Bigl(\chi'(H)-{n\over2}\,\chi(V)\Bigr)
				\,f'(0)\,{dt\over2t}\biggr)
\allowdisplaybreak
	&=-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_{(B\times(1,\infty)\times\{T\})/B\times\{T\}}
		\biggl(\strace\Bigl(f\(\overline X_T\)\Bigr)\\
	&\kern8em
			-\Bigl(\chi'(H)-{n\over2}\,\chi(V)\Bigr)
				\,f'(0)\,{dt\over2t}\biggr)
\allowdisplaybreak
	&=-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_1^\infty\Biggl(\strace_V\biggl(\(\bold g^V\tT\)^{-1}
		\,{\del\bold g^V\tT\over\del t}\,f'\(X\tT\)\biggr)\\
	&\kern8em
			-\Bigl(\chi'(H)-{n\over2}\,\chi(V)\Bigr)
				\,f'(0)\Biggr)\,{dt\over2t}\\
	&=S_f\(A',\bold g^V_T,g^H_T\)\;.
\endalign$$

Using \UVThm\ and~\LargeTProjectionFormel,
we can now rewrite the equation in \LargeEvThm~(2) as
$$\multline
	\tilde f\(\nabla^H,g^H,g^H_T\)
		+S_f\(T^HM,g\TX,\nabla^F,g^F_T\)\\
%		-(2\pi i)^{-\textstyle{N^B\over2}}\int_1^\infty\biggl(
%			\strace\Bigl(N^X\,f'\(\Xx\tT\)\Bigr)
%			-\chi'(H)\biggr)\,{dt\over2t}\\
\split
	&=\tilde f\(\nabla^H,g^H,g^H_{T_0}\)
		+S_f\(A',\bold g^V_{T_0},g^H_{T_0}\)
		+\tilde f\(A',\bold g^V_{T_0},\bold g^V_T\)
		+O\(T^{-\delta}\)\\
	&=\tilde f\(\nabla^H,g^H,g^H_{V}\)
		+S_f\(A',g^V\)
		+\tilde f\(A',g^V,\bold g^V_T\)
		+O\(T^{-\delta}\)
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;.
\endsplit\endmultline\tag\FormelNummer\SmallEvFormelEins$$


\Abschnitt\LargeTSubsection
The limit~$T\to\infty$ and the class~$T_f\(A',g^V,h^V\)$

We state some estimates for the generalised metric~$\bold g_T^V$
as~$T\to\infty$ that were established in~\BGzwei,
and we use these estimates to identify the right hand side
of~\SmallEvFormelEins\ with the torsion form~$T_f\(A',g^V,h^V\)$,
up to divergences and correction terms.

Recall that we have defined an $\Omega^*(C)$-linear map~$\widehat J
\colon\Omega^*\(C;o(E^-)\otimes F\)\to\Omega^*\bigl(\widehat E;F\bigr)$
in~\JhatDef\
that maps~$\Lambda^*T\widehat C\otimes o(E^-)\otimes F$
isomorphically onto the bundle~$\Ff\to\widehat C$
of the fibre-wise generalised eigenspaces of the model operator~$\widehat\Xx$
for the generalised eigenvalue~$0$.
We will apply this definition to~$E=TX|_C=T^sX\oplus T^uX$.
Restriction to~$T\ge T_0$ and push-down to~$B\times[T_0,\infty)$
gives a map
	$$\overline J
	\colon\Omega^*\(B\times[T_0,\infty),V\)
	\to\Omega^*\(TX|_C\times[T_0,\infty),\pi^*F\)\;.
	\Formel\JbarDef$$

Recall that by the simplifying assumptions of \SimplifyingAssumptions,
the fibre-wise exponential map at~$C\subset M$ maps
the $2\eps$-disc bundle of~$TX|_C$ fibre-wise isometrically
onto a tubular neighbourhood of~$C$ in~$M$.
Let us fix a $C^\infty$ cut-off function~$\phy\colon\R\to[0,1]$
with
	$$\phy(r)
	=\cases
		1	&\text{if~$r\le\eps$, and}\\
		0	&\text{if~$r\ge2\eps$.}\\
	\endcases$$
Then we may regard~$\phy(\abs{\Cal R})\,\overline J$ as a map
	$$\phy(\abs{\Cal R})\,\overline J
	\colon\Omega^*\(B\times[T_0,\infty),V\)
	\to\Omega^*\(M\times[T_0,\infty),F\)\;.
	\Formel\PhyJbarDef$$
This allows to define
	$$\bold e_T
	\colon\Lambda^*TB\otimes V
	\to\Ww_T
		\Mtext{by}
	\bold e_T
	=\Pp_T\,\(e^{Th}\,\phy(\abs{\Cal R})\,J_T\)\;.
	\Formel\EbarDef$$

We recall a fundamental result from~\BGzwei.

\Theorem\FirstApproxThm{(\BGzwei, Theorem 10.56,
cf.~\BZzwei, Theorem~6.7)}.
There exists a constant~$c>0$ such that as~$T\to\infty$ for any~$k\ge 0$
and any~$v\in\Omega^*(B;V)$,
	$$\(\bold e_T-e^{Th}\,\phy(\abs{\Cal R})\,J_T\)(v)
	=e^{Th}\,O\(e^{-cT}\,\abs v\)$$
uniformly in $C^k$-topology over compact subsets of~$B$.
In particular,
$\bold e_T$ is an $\Omega^*(B)$-linear vector bundle isomorphism
for sufficiently large~$T$.\qed

This theorem was applied to prove
an estimate for~$\bold g^V_T$ in~\BGzwei, Section~10.16,
which we now recall.

\Theorem\SecondApproxThm.
There exists an $r_\lambda$-invariant and~$g^V$-self-adjoint
$R_T\in\Omega^{\even,\ge2}(B;\End^\even V)$
commuting with~$\nabla^V$, $h^V$ and~$N^V$,
which is a polynomial in~$1\over T$ without constant term,
and another $g^V$-self-adjoint~$R'_T\in\Omega^*(B;\End V)$
with~$\abs{R'_T}=O\(e^{-cT}\)$ uniformly in $C^k$-topology,
such that
	$$\bold g^V_T
	=\Biggl(\(\id_V+R_T+R'_T\)\,e^{\textstyle-Th^V}
		\,\biggl({T\over\pi}\biggr)^{\textstyle{N^V\over2}-{n\over4}}
		\Biggr)^*g^V\;.$$

\Proof
By definition of~$\bold g^V_T$,
$$\split
	\bold g^V_T
	&=g^{TX,F}_T\(\Ii_T^{-1}\punkt,\Ii_T^{-1}\punkt\)\\
	&=g^{TX,F}_T\(\bold e_T\circ(I\circ\bold e_T)^{-1}\punkt,
		\bold e_T\circ(I\circ\bold e_T)^{-1}\punkt\)\;.
\endsplit\Formel\SecondApproxFormelNull$$
We will prove \SecondApproxThm\
by giving suitable approximations for~$I\circ\bold e_T$
and~$\bold e_T^*\,g^{TX,F}_T$
using \PatchThm\ and \IhatJhatProp.

Let~$c\in C_\beta$ be a fibre-wise critical point of~$h$,
and let~$v\in F_c\otimes o(T^u_cX)$.
Then
$$\split
	\(I\circ\bold e_T\)_\alpha(v)
	&=e^{Th_\alpha}\,I\(e^{-Th_\alpha}\bold e_T(v)\)\\
	&=e^{Th_\alpha}\,I\(e^{T(h-h_\alpha)}\,\phy(\abs{\Cal R})\,J(v)\)
	+e^{Th_\alpha}\,I\Bigl(e^{T(h-h_\alpha)}\,O\(e^{-cT}\,\abs v\)\Bigr)\;.
\endsplit\Formel\SecondApproxFormelEins$$

Clearly,
	$$e^{T(h-h_\alpha)}|_{h^{-1}\(-\infty,h_\alpha-\eps^2\)}
	=O\(e^{-\eps^2T}\)\;.$$
Together with \PatchThm~(2),
this allows us to replace~$M$ near~$C$ by the total space
of the vertical tangent bundle~$E=TX|_C\to C$,
and thus to approximate~$e^{-Th_\alpha}\,I_T\circ\bold e_T
\in$
by~$\widehat I\circ e^{Th^E}\circ\widehat J$
considered in \IhatJhatProp~(2).
Here~$h^E$ replaces~$h-h_\alpha$,
and we set~$h^C=0$.
In particular,
if~$\alpha\ne\beta$,
then~\SecondApproxFormelEins\ gives
	$$\(I_T\circ\bold e_T\)_\alpha(v)
	=e^{Th_\alpha}\,O\(e^{-cT}\abs v\)$$
uniformly in $C^k$-topology,
possibly with a new constant~$c$.

If we assume that~$\beta=\alpha$,
then \IhatJhatProp~(2) and~\SecondApproxFormelEins\ imply that
$$\align
	\(I_T\circ\bold e_T\)_\alpha(v)
	&=e^{Th_\alpha}\,\Bigl(\widehat I\circ e^{T(h-h_\alpha)}
			\circ\widehat J(v)\)
		+O\(e^{-cT}\,\abs v\)\Bigr)\\
	&=e^{Th_\alpha}\,\Biggl(\biggl({T\over\pi}\biggr)^{\textstyle
			{n^+-n^-\over4}}\,\(1+R_{IJ,T}\)\,v
		+O\(e^{-cT}\,\abs v\)\Biggr)\;,
\endalign$$
where~$R_{IJ}$ is an~$r_\lambda$-invariant characteristic differential
form of~$\nabla\TX$ on~$C$.
We can summarize our results so far by
	$$I_T\circ\bold e_T
	=e^{\textstyle Th^V}\,\biggl({T\over\pi}\biggr)^{\textstyle
			{n\over4}-{N^V\over2}}
		\,\Bigl(1+\hat p_*R_{IJ,T}+O\(e^{-cT}\)\Bigr)
	\Formel\SecondApproxFormelZwei$$
with~$O\(e^{-cT}\)\in\Omega^*(B;\End V)$ for some~$c>0$,
cf.~\BGzwei, Theorem~10.59, and~\BZzwei, Theorem~6.11.
Here, $\hat p_*R_{IJ,T}\in\Omega^{\ge2}(B,\End V)$
clearly commutes with~$\nabla^V$, $h^V$, and~$N^V$.
By \IhatJhatProp, $R_{IJ,T}$ is a polynomial in~$1\over T$
without constant term.

We regard the generalised metric~$\bold e_T^*\,g^{TX,F}_T$ on~$V$.
Taking adjoints with respect to~$g^{TX,F}$ and~$g^V$,
\FirstApproxThm\ implies
$$\split\bold e_T^*\,g^{TX,F}_T
	&=g^{TX,F}\(e^{-Th}\,\bold e_T\punkt,e^{-Th}\,\bold e_T\punkt\)\\
	&=g^V\biggl(\punkt,\Bigl(J_T^*\,\phy({\Cal R})+O\(e^{-cT}\)\Bigr)
		\,\Bigl(\phy({\Cal R})\,J_T+O\(e^{-cT}\)\Bigr)\punkt\biggr)\;.
\endsplit$$
The presence of the cut-off function~$\phy({\Cal R})$ allows us again
to replace~$M$ by~$\hat p_*(TX|_C)$.
By \JhatThm\ and~\IhatJhatProp, we find
$$%\split
	\bold e_T^*\,g^{TX,F}_T%&
	=g^V\biggl(\punkt,\Bigl(\hat p_*\(\widehat J_T^*\,\widehat J_T\)
		+O\(e^{-cT}\)\Bigr)\punkt\biggr)%\\%&
	=g^V\biggl(\punkt,\Bigl(\id_V+\hat p_*R_{J^*J,T}
		+O\(e^{-cT}\)\Bigr)\punkt\biggr)
%\endsplit
\Formel\SecondApproxFormelDrei$$
uniformly in $C^k$-topology,
cf~\BGzwei, Proposition~10.58.
Moreover,
$\hat p_*R_{J^*J,T}\in\Omega^{\ge2}(B,\End V)$ again
commutes with~$\nabla^V$, $h^V$, and~$N^V$,
and~$\hat p_*R_{J^*J,T}$ is a polynomial in~$1\over T$
without constant term.

Let~$\bigl(\id_V+\hat p_*R_{J^*J,T}+O\(e^{-cT}\)\bigr)^{1\over2}$
denote the square-root
of~$\bigl(\id_V+\hat p_*R_{J^*J,T}+O\(e^{-cT}\)\bigr)$,
which is self adjoint in the sense of \AdjointDef.
Combining~\SecondApproxFormelNull, \SecondApproxFormelZwei,
and~\SecondApproxFormelDrei,
we finally find that
$$\split\bold g^V_T
	&=\Biggl(\Bigl(\id_V+\hat p_*R_{J^*J,T}+O\(e^{-cT}\)\Bigr)
			^{\textstyle{1\over2}}\\
	&\kern3em
		\,\Bigl(1+\hat p_*R_{IJ,T}+O\(e^{-cT}\)\Bigr)^{-1}
		\,\biggl({T\over\pi}\biggr)^{\textstyle{N^V\over2}-{n\over4}}
		\,e^{\textstyle-Th^V}\Biggr)^*g^V\\
	&=\Biggl(\Bigl(\id_V+R_T+O\(e^{-cT}\)\Bigr)
		\,\biggl({T\over\pi}\biggr)^{\textstyle{N^V\over2}-{n\over4}}
		\,e^{\textstyle-Th^V}\Biggr)^*g^V\;.
	\quad\qed
\endsplit$$
\enddemo

We can now compare the generalised metric~$\bold g^V_T$
with the deformed metric~$e^{-2Th}\,g^V$.

\Theorem\ThirdApproxThm.
As~$T\to\infty$,
	$$\tilde f\(A',e^{-2Th}\,g^V,\bold g^V_T\)
	=\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,(\log T-\log\pi)\,f'(0)+O\biggl({1\over T}\biggr)\;.$$

\Proof
We set
	$$E_{s,T}
	=\Bigl(\id_V+s\,\(R_T+R'_T\)\Bigr)\,e^{\textstyle-Th^V}
		\,\biggl({T\over\pi}\biggr)
			^{\textstyle s\,\bigl({N^V\over2}-{n\over4}\bigr)}
	\quad\in\Omega^*\(B;\End V\)\;,$$
and construct a family of metrics
	$$\tilde\bold g^V_{s,T}
	=E_{s,T}^*g^V\;,$$
so~$g^V=\tilde\bold g^V_{0,T}$ and~$\bold g^V_T=\tilde\bold g^V_{1,T}$.
Let~$\widetilde A''_{s,T}$ denote the adjoint of~$A'$
with respect to~$\tilde\bold g^V_{s,T}$,
and let~$\widetilde X_{s,T}={1\over2}\,\bigl(\widetilde A''_{s,T}-A'\bigr)$,
then we have
	$$\tilde f\(A',e^{-2Th}\,g^V,\bold g^V_T\)
	=(2\pi i)^{-\textstyle{N^B\over2}}
		\,\int_0^1\strace_V\biggl({1\over2}
			\(\tilde\bold g^V_{s,T}\)^{-1}
			\,{\del\tilde\bold g^V_{s,T}\over\del s}
			\,f'\(\widetilde X_{s,T}\)\biggr)\,ds\;.
	\Formel\LargeTFslFormel$$

Using \SecondApproxThm, we can estimate the various terms
appearing in~\LargeTFslFormel.
First of all,
$$%\multline
\split
	E_{s,T}\,\(\tilde\bold g^V_{s,T}\)^{-1}
		\,{\del\tilde\bold g^V_{s,T}\over\del s}\,E_{s,T}^{-1}%\\
%\split
	&=\Bigl(\id_V+s\(R_T+R'_T\)\Bigr)^{-1}
		\,\Biggl(\(R_T+R'_T\)\\
	&\kern4em
			+\biggl({N^V\over2}-{n\over4}\biggr)\,(\log T-\log\pi)
			\,\Bigl(\id_V+s\(R_T+R'_T\)\Bigr)\Biggr)\\
	&\qquad
	+\Biggl(\biggl({N^V\over2}-{n\over4}\biggr)\,(\log T-\log\pi)
			\,\Bigl(\id_V+s\(R_T+R'_T\)\Bigr)\\
	&\kern4em
		+\(R_T+R'_T\)\Biggr)
		\,\Bigl(\id_V+s\(R_T+R'_T\)\Bigr)^{-1}\\
	&=\biggl(N^V-{n\over2}\biggr)\,(\log T-\log\pi)
		+2R_T\,\(\id+s\,R_T\)^{-1}
		+O\(e^{-cT}\)
\endsplit%\endmultline
$$
uniformly in~$s$,
since~$R_T$ and~$N^V$ commute.

Now,
we investigate the superconnections~$A'$ and~$\widetilde A''_{s,T}$.
Let~$A''$ be the adjoint of~$A'$ with respect to~$g^V$.
Then we have
	$$A'=\nabla^V+\sum_ja_j\Mand A''=\nabla^{V,*}+\sum_ja_j^*\;,$$
where adjoints are taken with respect to~$g^V$,
and we know that~$a_j$ raises eigenvalues of~$h^V$ by at least~$\eps^2$
because of \PatchThm,
while~$a_j^*$ lowers them by at least the same amount.
Since the estimates in \FirstApproxThm\ and \SecondApproxThm\
include derivatives of~$R'_T$,
we find
$$\align
	E_{s,T}\,A'\,E_{s,T}^{-1}
	&=\Bigl(\id_V+s\(R_T+R'_T\)\Bigr)
		\,\Biggl(\nabla^V+T\,\bigl[\nabla^V,h^V\bigr]\\
	&\kern4em
		+\sum_j\biggl({T\over\pi}\biggr)^{\textstyle{s(1-j)\over2}}
			\,e^{-Th^V}\,a_j\,e^{Th^V}\Biggr)
		\,\Bigl(\id_V+s\(R_T+R'_T\)\Bigr)^{-1}\\
	&=\nabla^V+T\,\bigl[\nabla^{V,*},h^V\bigr]+O\(e^{-cT}\)\;.
\endalign$$
Similarly,
we obtain
	$$E_{s,T}\,\widetilde A''_{s,T}\,E_{s,T}^{-1}
	=E_{s,T}^{-1}\,A''\,E_{s,T}
	=\nabla^{V,*}-T\,\bigl[\nabla^V,h^V\bigr]+O\(e^{-cT}\)\;.$$

Now we note that~$h^V$ acts as a scalar function
on each component~$V^\alpha=\hat V|_{C_\alpha}$ of~$V$,
and that~$R_T$, $\nabla^V$ and~$\nabla^{V,*}$ preserve these components.
Writing
	$$\omega^V=\nabla^{V,*}-\nabla^V
		\Mand
	dh^V=\bigl[\nabla^V,h^V\bigr]=\bigl[\nabla^{V,*},h^V\bigr]\;,$$
we finally obtain
$$\multline
	\strace_V\Biggl({1\over2}\,\(\tilde\bold g^V_{s,T}\)^{-1}
		\,{\del\tilde\bold g^V_{s,T}\over\del s}
		\,f'\(\widetilde X_{s,T}\)\Biggr)\\
\split
	&=\strace_V\Biggl({1\over2}
		\,\biggl(\Bigl(N^V+{n\over2}\Bigr)\,(\log T-\log\pi)
			+2R_T\,\(\id+R_T\)^{-1}\Bigr)
		\,f'\biggl({\omega^V\over2}-T\,dh^V\biggr)
	+O\(e^{-cT}\)\\
	&=\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,(\log T-\log\pi)\,f'(0)+O\biggl({1\over T}\biggr)\;,
\endsplit\endmultline$$
because~${\omega^V\over2}+T\,dh^V$ is odd
and commutes with~$N^V+2R_T\,\(\id+R_T\)^{-1}$,
while~$f'$ is an even function,
and supertraces of even powers of odd operators vanish.
By~\LargeTFslFormel,
this finishes our proof of \ThirdApproxThm.
\qed\enddemo

Using this result,
we can summarize the results of this section,
where we now make use of the fact that~$f(z)=ze^{z^2}$
by~\GenFnChoice.

\Theorem\SmallEvTorsionThm.
As~$T\to\infty$,
we have
$$\multline
	\tilde f\(\nabla^H,g^H,g^H_T\)
	+S_f\(T^HM,g\TX,\nabla^F,g^F_T\)
	-\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,\log T
	+T\,\strace_V\(h^V\)\\
\split
	&\to\tilde f\(\nabla^H,g^H,g^H_V\)
		+T_f\(A',g^V,h^V\)\\
	&\kern4em
	-\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,\log\pi
		-\(\chi'(V)-\chi'(H)\)
			\,\int_1^\infty f'\(\sqrt{-t}/2\)\,{dt\over2t}\\
	&\kern4em
		-\(\chi'(V)-\chi'(H)\)
			\,\int_0^1\Bigl(f'\(\sqrt{-t}/2\)
				-1\Bigr)\,{dt\over2t}
\endsplit\endmultline$$
modulo exact forms on~$B$.

This theorem is our analogue of Theorem~9.8 in~\BGzwei.
Note however,
that a different notion of torsion appears in~\BGzwei.

\Proof
From~\SmallEvFormelEins, \ThirdApproxThm\ and the fact that~$f'(0)=1$,
we get
$$\multline
	\tilde f\(\nabla^H,g^H,g^H_T\)
		+S_f\(T^HM,g\TX,\nabla^F,g^F_T\)
		-\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,(\log T-\log\pi)\\
	=\tilde f\(\nabla^H,g^H,g^H_{V}\)
		+S_f\(A',g^V\)
		+\tilde f\(A',g^V,e^{-2Th^V}\,g^V\)
		+O\(T^{-\delta}\)
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;.
\endmultline\tag\FormelNummer\SmallEvFormelZwei$$


We change back to the notation of \MVBSubsection.
In particular, $A''_{1,\tau}$ now refers to the adjoint of~$A'$
with respect to the metric~$g^V\bigl(T^{h^V}\punkt,\punkt\bigr)$,
and of course~$X_{1,T}={1\over2}\,\(A''_{1,T}-A'\)$.
Then
$$\split
	\tilde f\(A',g^V,e^{-2Th^V}\,g^V\)
	&=-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_0^T\strace_V\Bigl(h^V
			\,f'\(X_{1,e^{-2\tau}}\)\Bigr)\,d\tau\\
	&=-(2\pi i)^{-\textstyle{N^B\over2}}
		\int_{e^{-2T}}^1\strace_V\Bigl(h^V
			\,f'\(X_{1,T'}\)\Bigr)\,{dT'\over2T'}\;.
\endsplit\Formel\LargeTFslFormelZwei$$

By \SmallEvFormelZwei\ and \LargeTFslFormelZwei,
$$\multline
	\tilde f\(\nabla^H,g^H,g^H_T\)
		+S_f\(T^HM,g\TX,\nabla^F,g^F_T\)
		-\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,(\log T-\log\pi)
		+T\,\strace_V\(h^V\)\\
	=\tilde f\(\nabla^H,g^H,g^H_{V}\)
		+S_f\(A',g^V\)
		-(2\pi i)^{-\textstyle{N^B\over2}}\,\int_{e^{-2T}}^1\Bigl(
			\strace_V\(h^V\,f'(X_{1,T'})\)\\
			-\strace_V\(h^V\)\Bigr)\,{dT'\over2T'}
		+O\(T^{-\delta}\)
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;.
\endmultline$$
In the limit~$T\to\infty$, \MVBTorsionDef\ gives us
$$\multline
	\tilde f\(\nabla^H,g^H,g^H_T\)
		+S_f\(T^HM,g\TX,\nabla^F,g^F_T\)
		-\biggl({\chi'(V)\over2}-{n\,\chi(V)\over4}\biggr)
		\,(\log T-\log\pi)
		+T\,\strace_V\(h^V\)\\
\split
	&\to\tilde f\(\nabla^H,g^H,g^H_{V}\)
		+T_f\(A',g^V,h^V\)\\
	&\kern4em
		-\(\chi'(V)-\chi'(H)\)
			\,\int_1^\infty f'\(\sqrt{-t}/2\)\,{dt\over2t}\\
	&\kern4em
		-\(\chi'(V)-\chi'(H)\)
			\,\int_0^1\Bigl(f'\(\sqrt{-t}/2\)
				-1\Bigr)\,{dt\over2t}
	\quad\in\Omega^*(B)/d^B\Omega^*(B)\;,
\endsplit\endmultline$$
which finishes the proof.
\qed\enddemo

\Remark\SmallEvEquivRem
Suppose that~$\gamma$ is a fibre-wise isometry of~$M$
that acts on~$F$ and preserves~$T^HM$, $g\TX$, $\nabla^F$ and~$g^F$.
Suppose also that~$\gamma$ acts on the fibres of~$V\to B$
preserving~$A'$, $g^V$ and~$h^V$,
and that the map~$I\colon\Omega^*(M;F)\to\Omega^*(B;V)$
is $\gamma$-equivariant.
Then there are forms~$\tilde f_\gamma\(\nabla^H,g^H,g^H_T\)$,
$S_{f,\gamma}\(T^HM,g\TX,\nabla^F,g^F_T\)$,
$T_{f,\gamma}\(A',g^V,h^V\)$ etc.,
and all considerations in this section still go through
and provide us with a $\gamma$-equivariant version
of \SmallEvTorsionThm,
cf.~\BGzwei, Theorem~9.8
\endremark


\enddocument


