$h$--cobordisms between 1--connected 4--manifolds
Matthias Kreck
In this note we classify the diffeomorphism classes rel.\ boundary of smooth $h$--cobordisms between two fixed 1--connected 4--manifolds in terms of isometries between the intersection forms.
2001-1.tex
%%%%%%%%%%%%%%%%   Geometry and Topology: 2001-1.tex  %%%%%%%%%%%%%
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%%%%            h-cobordisms between 1-connected 4-manifolds
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%%%%                          Matthias Kreck   
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%%%%                Published in Volume 1(2001) pages 1-6
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%%%%                   Publication date 14 January 2001
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%%%%                        This is a LaTeX file
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\lognumber{145}
\volumenumber{5}\papernumber{1}\volumeyear{2001}
\pagenumbers{1}{6}
\accepted{12 January 2001}
\proposed{Robion Kirby}
\seconded{Ronald Stern, Tomasz Mrowka}
\received{20 October 2000}
\revised{9 January 2001}
\published{14 January 2001}


\usepackage{amssymb}
\let\Bbb\mathbb
\newtheorem*{thm}{Theorem}

\title{$h$--cobordisms between 1--connected 4--manifolds}
\asciititle{h-cobordisms between 1-connected 4-manifolds}
\author{Matthias Kreck}


\address{Mathematisches Institut, Universit\"at Heidelberg\\
69120 Heidelberg, Federal Republic of Germany\\\smallskip\\
{\rm and} \\\smallskip\\
Mathematisches Forschungsinstitut Oberwolfach\\
77709 Oberwolfach, Federal Republic of Germany}
\asciiaddress{Mathematisches Institut, Universitat Heidelberg\\
69120 Heidelberg, Federal Republic of Germany\\
and\\
Mathematisches Forschungsinstitut Oberwolfach\\
77709 Oberwolfach, Federal Republic of Germany}

\email{kreck@mathi.uni-heidelberg.de}

\begin{abstract}
In this note we classify the diffeomorphism classes rel.\
boundary of smooth $h$--cobordisms between two fixed 1--connected 4--manifolds
in terms of isometries between the intersection forms.
\end{abstract}


\asciiabstract{
In this note we classify the diffeomorphism classes rel.
boundary of smooth h-cobordisms between two fixed 1-connected 4-manifolds
in terms of isometries between the intersection forms.
}


\keywords{4--manifolds, smooth $h$--cobordisms, surgery}
\asciikeywords{4-manifolds, smooth h-cobordisms, surgery}

\primaryclass{57R80}

\secondaryclass{57N13, 57Q20, 55N45}



\begin{document}

\maketitlepage



In this note we prove the following result.


\begin{thm}
Let $M_0$ and $M_1$ be fixed closed oriented smooth 1--connected
4--manifolds. Then the set of diffeomorphism classes rel.\ boundary of
smooth $h$--cobordisms between $M_0$ and $M_1$ is isomorphic to the set
of isometries between the intersection forms of $M_0$ and $M_1$.

The same result holds in the topological category if $M_0$ and $M_1$
are topological manifolds with same Kirby--Siebenmann invariant $k$
(otherwise there is no $h$--cobordism between them at all), if we
classify up to homeomorphism.
\end{thm}


The motivation for our Theorem comes from the fact that the
$h$--cobordism theorem does not hold for smooth $h$--cobordisms between
$4$--manifolds \cite {D}. During a discussion with S Donaldson and
R Stern about 12 years ago about additional invariants whose
vanishing implies that such an $h$--cobordism is diffeomorphic to the
cylinder we wondered how many $h$--cobordisms exist. The answer above
is simpler than in higher dimensions where, due to the existence of
exotic spheres, the above Theorem is in general wrong, even if $M_0$ and
$M_1$ are spheres. The result above implies that a smooth
$h$--cobordism between smooth $1$--connected $4$--manifolds is the
cylinder if and only if there is a diffeomorphism $f\co  M_0 \to M_1$
inducing $(j_*)^{-1}i_*$, where $i$ and $j$ are the inclusions from
$M_0$ and $M_1$ to $W$ resp. This is of course not the answer one is
looking for. A good answer would be that $W$ is a cylinder if and only
if the Seiberg--Witten invariants for $M_0$ and $M_1$ agree. More
precisely the Seiberg Witten invariants (assuming for simplicity
$b_2^+(M_i) >1$) are maps from $\{ \alpha \in H^2(M_i) \, | \, \alpha
= w_2(M_i) \,{\rm mod} \, 2\}$ to the integers. Thus, using the isometry
between the intersection forms given by the $h$--cobordism to identify
the cohomology groups, one can compare the Seiberg--Witten invariants of
$M_0$ and $M_1$. The challenge is to relate the critical values of a
Morse function on an $h$--cobordism to the Seiberg--Witten invariants
and to show that the equality of these invariants implies that there
is a Morse function without critical values. A relation between the
critical values (which is not yet enough to prove the existence of a
Morse function without critical values) was recently found by Morgan
and Szabo \cite {M-S} (in the first paragraph of this paper they
state that the smooth $h$--cobordisms are classified by the set of
homotopy equivalences, which is not correct, since not every homotopy
equivalence between $M_0$ and $M_1$ can be realized by an
$h$--cobordism, see below).

The theorem also follows from \cite [Proposition 1]{L}, where T.  
Lawson classifies invertible bordisms, and Stalling's result \cite{S}  
that invertible bordisms and $h$-cobordisms agree. The proof of  
Lawson's proposition uses also Stalling's result as well as \cite  
[Proposition 2.1]{Q}. The proof of this result is not correct as  
pointed out and corrected in \cite {C-H}. Our proof is more direct  
and elementary. 


\begin{proof}
We will give the proof in the smooth category and discuss the
necessary modifications for the topological result at each point.

It is clear that the composition of the inclusion of $M_0$ into an
$h$--cobordism $W$ between $M_0$ and $M_1$ and the homotopy inverse of
the inclusion from $M_1$ is an orientation preserving homotopy
equivalence and thus induces an isometry between the intersection
forms. This way one obtains a map from the set of diffeomorphism
classes rel.\ boundary of $h$--cobordisms between $M_0$ and $M_1$ to
the set of isometries from $H_2(M_0) \to H_2(M_1)$. It is known that
this map is surjective. Namely, each isometry can be realized by a
homotopy equivalence \cite {M}. And each homotopy equivalence can
after composition with a self equivalence of $M_1$ which operates
trivially on $H_2(M_1)$ be realized by a smooth $s$--cobordism (\cite
[Theorem 16.5]{W} and the correction in \cite {C-H} --- the proof of
this result implies that not every homotopy equivalence can be
realized by an $h$--cobordism). If $M_0$ and $M_1$ are topological
manifolds with $k(M_0) = k(M_1)$, then it is known that each isometry
can be realized by a homeomorphism \cite [Theorem 10.1] {F-Q}. This
implies surjectivity in the topological case. A different argument for
surjectivity both in the smooth and topological category can be found
in the proof of \cite [Theorem C]{H-K}.  Thus we only have to show
injectivity.

Let $W$ and $W'$ be two smooth $h$--cobordisms between $M_0$ and $M_1$
inducing the same isometry between the intersection forms. We will use
\cite [Theorem 3]{K2} to show that $W$ and $W'$ are diffeomorphic
rel.\ boundary. For this we first determine the normal 1--type of an
$h$--cobordism $W$. By \cite [Proposition 2]{K2} this is the fibration
$B = BSO \to BO$, if $w_2(W) = w_2(M_0) \ne 0$, the non-spin case, and
$B = BSpin \to BO$, if $w_2(W) = w_2(M_0) = 0$, the spin case. In the
topological case we have to take instead $B = BSTop$ or $B =
BSTopSpin$. If we want to apply \cite [Theorem 3]{K2} we have as a
first step to check that normal 1--smoothings of $W$ and $W'$ exist
which coincide on the common boundary $M_0 + M_1$. A normal
1--smoothing is in the non-spin case equivalent to an orientation and
in the spin case to a spin-structure. Thus, since $M_i$ are simply
connected, compatible choices exist.

The next step is to decide if $X = W \cup_{\partial W = \partial W'}
W'$ is $B$--zero-bordant. In the smooth spin case the $B$--bordism
group is spin-bordism which vanishes in dimension $5$. In the smooth
non-spin case the $B$--bordism group is oriented bordism which is
$\mathbb Z /2$ detected by $w_2 \cdot w_3$. One has the same answer in
the topological case.  One can argue that all $5$--manifolds can be
made $1$--connected by surgery and then they admit a smooth structure
since the Kirby--Sibenmann obstruction for the existence of a
$PL$--structure in the 4--th cohomology with $\mathbb Z
/2$--coefficients vanishes, and in dimension $5$ the $PL$ and the
smooth categories are equivalent. In the rest of the argument there is
no difference between the smooth and topological case.

Now and later on we need information about the (co)homology of
$X$. For this we choose a fibre homotopy equivalence between $X$ and
the mapping torus of the homotopy equivalence on $M_0$ given by $ f =
j_0 \cdot (j_1)^{-1}\cdot j'_1 \cdot (j'_0)^{-1}$, where $j_i$ and
$j'_i$ are the inclusions from $M_i$ to $W$ resp.\ $W'$. If $W$ and
$W'$ induce the same isometry between the intersection forms of $M_0$
and $M_1$, then $f$ induces the identity map in second
(co)homology. Thus by the Wang sequence for the mapping torus of $f$
we obtain, for arbitrary coefficients, isomorphisms $i^\star \co
H^2(X) \to H^2(M_0)$, where $i$ is the inclusion, and $\delta \co
H^0(M_0) \to H^1(X)$ and $\delta \co H^2(M_0) \to H^3(X)$.


By the Wu-formulas we have $w_3(X) = Sq^1(w_2(X)) = 0$, since $Sq^1 =
0$ in $H^2(X) \cong H^2(M_0)$. Thus the characteristic number $w_2
\cdot w_3(X)$ vanishes and also in the non-spin case $X$
bounds. Choose in both cases a zero bordism $Y$ and use surgery to
make the map $Y \to B$ 3--connected \cite [Proposition 4]{K2}.

The next step is to analyze the surgery obstruction $\theta(Y) \in
l^\sim_6(1)$. Note that in both cases $\langle w_4(B), \pi_4(B)\rangle
\ne 0$ implying that the obstruction is contained in $l^ \sim_6(1)$
instead of $l_6 (1)$ making life easier since we do not have to
consider quadratic refinements. The obstruction is given by the
equivalence class
$$
[H_3(Y,W) \leftarrow {\rm im}(d\co \pi_{4}(B,Y) \to \pi_3(Y)) \rightarrow
H_3(Y,W'), \lambda]
$$
where the maps are induced by inclusion and $\lambda$ is the
intersection pairing between $(Y,W)$ and $(Y,W')$. We will show that
this obstruction is elementary, ie, there is a submodule $U \subset
{\rm im}(d\co \pi_{4}(B,Y) \to \pi_3(Y))$ such that under both maps $U$ maps
to a half rank direct summand and $\lambda$ vanishes on $U$. We first
note that since $\pi_3(B) = 0$, we can replace ${\rm im}(d\co \pi_{4}(B,Y)
\to \pi_3(Y))$ by $\pi_3(Y)$ and since $\pi_3(Y) \to H_3(Y)$ is
surjective we can work with $H_3(Y)$ instead. The situation is here
particularly easy since by our homological information both $H_3(Y,W)$
and $H_3(Y,W')$ are isomorphic to $H_3(Y,M_0)$. Thus we have to find
$U \subset H_3(Y)$ such that, under inclusion, $U$ maps to a half rank
direct summand of $H_3(Y,M_0)$ and $\lambda$ vanishes on $U$. Looking
at the exact sequence $H_3(Y) \to H_3(Y, M_0)\to H_2(M_0)$ and using
that the latter group is free we can pass to rational
coefficients. Here we make use of the fact that we do not have to take
quadratic refinements into account. Thus the obstruction is
elementary if there is $U \subset H_3(Y;\Bbb Q)$ such that, under
inclusion, $U$ maps to a half rank summand of $H_3(Y,M_0; \Bbb Q)$ and
$\lambda$ vanishes on $U$. Namely, for such a $U$ choose $U' \subset
H_3(Y)$ such that $U'$ is a direct summand in $H_3(Y)$ and $U' \otimes
\mathbb Q = U$. Since $H_2(M_0)$ is torsion free $U'$ maps to a direct
summand in $H_3(Y, M_0)$. If $\lambda$ vanishes for $U$ the same holds
for $U'$ and thus our obstruction is elementary.

Using that $H_4(Y,X;\Bbb Q) = H^2(Y;\Bbb Q) \cong H^2(B;\Bbb Q) = 0$
and $H_2(X,M_0;\Bbb Q) = 0$ by the homology information above we
obtain an exact sequence
$$
0 \to H_3(X,M_0 ;\Bbb Q)
 \to H_3(Y,M_0 ;\Bbb Q)
 \to H_3( Y,X ;\Bbb Q) \to 0.
$$
By the homological information above we have isomorphisms
$$
H_2(M_0;\Bbb Q) \cong H_3(X;\Bbb Q) \cong H_3(X, M_0; \Bbb Q).
$$
Together with the exact sequence
$$
0 \to H_3(X; \Bbb Q) \to H_3(Y; \Bbb Q) \to H_3(Y,X;\Bbb Q) \to H_2(X;
\Bbb Q) = H_2(M_0;\Bbb Q) \to 0
$$
this implies
$$
{\rm rank} H_3(Y,M_0;\Bbb Q) = 2\cdot {\rm rank} H_2(M_0;\Bbb Q) +
{\rm rank} ({\rm coker} (H_3(X;\Bbb Q) \to H_3(Y;\Bbb Q))).
$$ 
Since the intersection form on ${\rm coker} (H_3(X;\Bbb Q) \to
H_3(Y;\Bbb Q))$ is unimodular and skew symmetric there is a submodule
$U_1 \subset H_3(Y;\Bbb Q)$ of half rank of this cokernel, on which
the intersction pairing vanishes. Finally the intersection form on the
image $U_2$ of $H_3(X;\Bbb Q)$ in $ H_3(Y;\Bbb Q)$ is contained in the
radical and has rank equal to ${\rm rank}(H_2(M_0)$. Thus $U = U_1
\oplus U_2$ is the desired submodule in $H_3(Y;\Bbb Q)$ implying that
the obstruction $\theta(Y)$ is elementary. Then $W$ and $W'$ are
diffeomorphic rel.\ boundary by \cite [Theorem 3]{K2}.\end{proof}

I would like to finish the paper with two remarks suggested by the
referees. Both concern applications of the theorem above to known
results. In the paper \cite [Theorem 5.2] {C-H} the authors show that
the map associating to a self equivalence of a smooth (or PL) simply
connected closed $4$--manifold $X$ the normal invariant is an
injection whose image is the kernel of the map into the $L$--group
$L_4$. We used the latter fact to argue that each self equivalence is
induced from an $h$--cobordism. The injectivity can be derived from
the theorem above and the surgery exact sequence.

The other remark concerns pseudo-isotopy classes of closed
$1$--connected topological $4$--manifolds. The theorem above implies
that two self homeomorphisms which agree on $H_2$ are pseudo-isotopic,
a result which previously had been proven by Quinn \cite {Q} and the
author (for diffeomorphisms) \cite {K1}. Quinn and independently
Perron \cite {P} have shown that pseudo-iosotopy implies isotopy (in
the topological category). Thus the group of isotopy classes of
homeomorphisms is isomorphic to the isometries of $H_2$.


\np

\begin{thebibliography}


\let\oldbibitem\bibitem
\def\bibitem#1]#2{\oldbibitem{#2}}


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On the homotopy theory of simply connected four manifolds.} 
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\bibitem[D]{D} {\bf S\,K Donaldson}, {\em Irrationality and the
$h$--cobordism conjecture. } J. Differential Geom. 26 (1987) 141--168

\bibitem[F-Q]{F-Q} {\bf Michael H Freedman}, {\bf Frank S Quinn}, {\em
Topology of 4--manifolds. } Princeton Mathematical Series,
39, Princeton University Press, Princeton NJ (1990)

\bibitem[H-K]{H-K} {\bf Ian Hambleton}, {\bf M Kreck}, {\em
Cancellation, elliptic surfaces and the topology of certain
four-manifolds}, J. Reine Angew. Math. 444 (1993) 79--100

\bibitem[K1]{K1} {\bf M Kreck}, {\em Isotopy classes of
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1978'', Lect. Notes Math. 763, Springer--Verlag, Berlin (1979) 643--663

\bibitem[K2]{K2}{\bf M Kreck}, { \em Surgery and Duality}, 
Annals of Math. 149 (1999) 707--754

\bibitem [L]{L} {\bf T Lawson}, {\em $h$--cobordisms between simply  
connected $4$-manifolds},
Topology Appl. 28 (1988) 75--82

\bibitem[M]{M} {\bf J Milnor}, {\em On simply-connected 4--manifolds.}
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\bibitem[M-S]{M-S} {\bf John W Morgan}, {\bf  Zoltan Szabo}, {\em
Complexity of 4--dimensional $h$--cobordisms},
Invent. Math. 136 (1999) 273--286 

\bibitem[P]{P} {\bf B Perron}, {\em Pseudo-isotopies et isotopies en
dimension quatre dans la categorie topologique, (Pseudo-isotopies and
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25 (1986) 381--397

\bibitem[Q]{Q} {\bf Frank Quinn}, {\em Isotopy of 4--manifolds.}
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\bibitem [S]{S} {\bf J Stallings}, {\em On infinite processes leading
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\bibitem[W]{W} {\bf C\,T\,C Wall}, { \em Surgery on Compact
Manifolds}, London Math. Soc. Monographs~1, Academic Press (1970)








\end{thebibliography}

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\immediate\write\gtoutfile{Subject: put or rep NNNNN:pppp}
\immediate\write\gtoutfile{--text follows this line--}
\immediate\write\gtoutfile{Proxy-for: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi\s<\ifx\theasciiemail\relax\theemail\else\theasciiemail\fi>}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{Authors: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi}
\immediate\write\gtoutfile{Title: \ifx\theasciititle\relax
\thetitle\else\theasciititle\fi}
\immediate\write\gtoutfile{Subj-class: GT or SG or MG etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Geom. Topol. \thevolumenumber
(\thevolumeyear) \startpage-\finishpage}
\immediate\write\gtoutfile{Comments: Published by Geometry and Topology at}
\immediate\write\gtoutfile{\s\s http://www.maths.warwick.ac.uk/gt/GTVol\thevolumenumber/paper\thepapernumber.abs.html}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\ifx\theasciiabstract\relax
\immediate\write\gtoutfile{\theabstract}\else
\immediate\write\gtoutfile{\theasciiabstract}\fi
\immediate\write\gtoutfile{}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\immediate\closeout\gtoutfile}}  %%% end of definition of \makeheadfile

\def\maketitlepage{\maketitlep\makeheadfile}
\let\maketitle\maketitlepage

\endinput
%%%
%%%  This version of  gtoutput.tex  is intended to finish formatting
%%%  papers published in Geometry & Topology and stored in the
%%%  arXiv.   All versions of  gtoutput.tex  are copyright 
%%%  GT Publications and are to be used _only_ for formatting
%%%  the officially published version of G&T papers.
%%%
%%%
%%%                                             Colin Rourke  14.9.2000
%%%
%%%  To create header file  head.xxx  comment out the first \endinput

%  test for latex or plain tex
\def\ifplaintex{\expandafter\ifx\csname documentclass\endcsname\relax}

%  get print centerpage:

\ifplaintex 
\hoffset 14truemm
\voffset 31truemm
\else
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
\fi


%  load pictex if not already loaded :
\expandafter\ifx\csname beginpicture\endcsname\relax
\expandafter\ifx\csname documentclass\endcsname\relax
\input pictex \else\font\fiverm=cmr5
\input prepictex \input pictex \input postpictex \fi\fi

\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology}}        %  journal title in recommended style

\def\gtp{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal P\!$ublications}}  % GT publications

%  define the various new ingredients of the title page 

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\samplenumber#1{\def\thesamplenumber{#1}\def\thevolumenumber{#1}
   \def\thepapernumber{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\proposed#1{\def\theproposer{#1}}
\def\seconded#1{\def\theseconders{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\def\accepted#1{\def\accepteddate{#1}}
\def\asciititle#1{\def\theasciititle{#1}}
\def\covertitle#1{\def\thecovertitle{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\shortauthors#1{\def\theshortauthors{#1}}
\def\shorttitle#1{\def\theshorttitle{#1}}

%  initialise

\let\\\par\let\thelognumber\relax
\let\thevolumenumber\relax\let\thepapernumber\relax
\let\thevolumeyear\relax\let\thesamplenumber\relax\let\startpage\relax
\let\finishpage\relax\let\publishdate\relax\let\receiveddate\relax
\let\reviseddate\relax\let\accepteddate\relax\let\theasciititle\relax
\let\thecovertitle\relax\let\theasciiauthors\relax\let\theasciiaddress\relax
\let\theasciiabstract\relax\let\theasciikeywords\relax
\let\theasciiemail\relax\let\theshortauthors\relax\let\theshorttitle\relax
\let\thecoverauthors\relax

\long\def\maketitlep{   % start of definition of \maketitlep

\count0=\startpage

\gt\hfill      %   Journal title (top left) 
%    Logo (top right) :
\beginpicture
\setcoordinatesystem units <0.33truein, 0.33truein> point at 2.2 0.9
\setplotsymbol ({$\cal G$})
\plotsymbolspacing=9truept
\circulararc 315 degrees from 0 1 center at 0 0
\setplotsymbol ({$\cal T$})
\circulararc 315 degrees from 1 -1 center at 1 0
\endpicture
%   end of logo
%
\break
{\small\ifx\thesamplenumber\relax % sample?  
Volume \else Sample
\fi\thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}
\vglue 0.5truein plus 0.4fil minus 0.1truein

% title
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\ifplaintex\large
\else\Large\fi\bf\thetitle}\par\medskip}   

\vglue 0pt plus 0.1fil 

% authors
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
\par\medskip}

 
\vglue 0pt plus 0.1fil 

%address(es)
{\small\parskip=0pt\let\newline\\
{\leftskip 0pt plus 1fil\def\\{\par}{\sl\theaddress}\par}
\expandafter\ifx\theemail\relax    % email address?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace} 
\cl{Email:\stdspace\tt\theemail}\fi
\ifx\theurl\relax                  % URL given?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace}
\cl{URL:\stdspace\tt\theurl}\fi\par}

\vglue 7pt plus 0.3fil minus 3pt

{\bf Abstract}
\vglue 5pt plus 0.1fil minus 2pt

\theabstract

\vglue 7pt plus 0.3fil minus 3pt

{\bf AMS Classification numbers}\quad Primary:\quad \theprimaryclass

Secondary:\quad \thesecondaryclass

\vglue 5pt plus 0.3fil minus 2pt

{\bf Keywords:}\quad \thekeywords

\vglue 10pt plus 0.5fil minus 5pt

{\small  Proposed: \theproposer\hfill Received: \receiveddate\nl
Seconded: \theseconders\hfill 
\ifx\reviseddate\relax                         % paper revised?
Accepted: \accepteddate                        % no
\else
Revised: \reviseddate                          % yes
\fi}
\eject
}       %  end of definition of \maketitlep

\let\maketitlepage\maketitlep
\let\maketitle\maketitlepage

%%% Headers and footers

\font\phead=cmsl9 scaled 950
\font\lhead=cmsl9 scaled 1050
\font\pnum=cmbx10 scaled 913
\font\lnum=cmbx10 
\font\pfoot=cmsl9 scaled 950
\font\lfoot=cmsl9 scaled 1050
\ifplaintex
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (printed) \hfill {\pnum\folio}\else\ifodd\count0\def\\{ }% 
\ifx\theshorttitle\relax\thetitle\else\theshorttitle\fi\hfill{\pnum\folio}
\else\def\\{ and }{\pnum\folio}\hfill\ifx\theshortauthors\relax\theauthors
\else\theshortauthors\fi\fi\fi}\vss}}
\footline{\vbox to 0pt{\vglue 0mm\line{\small\pfoot\ifnum\count0=\startpage
\copyright\ \gtp\hfill\else
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss
}}
\else
\makeatletter
\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (printed) \hfill {\lnum\number\count0}\else\ifodd\count0
\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
\hfill\ifx\theshortauthors\relax 
\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage\copyright\ \gtp\hfill\else
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
\def\@evenfoot{\@oddfoot}
\makeatother
\fi

\endinput  %%%comment out to create xxx header file


\newwrite\gtoutfile
\long\gdef\makeheadfile{  %%% start of definition of \makeheadfile
{\def\\{, }\def\s{ }
\immediate\openout\gtoutfile head.xxx
\immediate\write\gtoutfile{To: math@arxiv.org}
\immediate\write\gtoutfile{Subject: put or rep NNNNN:pppp}
\immediate\write\gtoutfile{--text follows this line--}
\immediate\write\gtoutfile{Proxy-for: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi\s<\ifx\theasciiemail\relax\theemail\else\theasciiemail\fi>}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{Authors: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi}
\immediate\write\gtoutfile{Title: \ifx\theasciititle\relax
\thetitle\else\theasciititle\fi}
\immediate\write\gtoutfile{Subj-class: GT or SG or MG etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Geom. Topol. \thevolumenumber
(\thevolumeyear) \startpage-\finishpage}
\immediate\write\gtoutfile{Comments: Published by Geometry and Topology at}
\immediate\write\gtoutfile{\s\s http://www.maths.warwick.ac.uk/gt/GTVol\thevolumenumber/paper\thepapernumber.abs.html}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\ifx\theasciiabstract\relax
\immediate\write\gtoutfile{\theabstract}\else
\immediate\write\gtoutfile{\theasciiabstract}\fi
\immediate\write\gtoutfile{}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\immediate\closeout\gtoutfile}}  %%% end of definition of \makeheadfile

\def\maketitlepage{\maketitlep\makeheadfile}
\let\maketitle\maketitlepage

\endinput
%%%
%%%  This version of  gtoutput.tex  is intended to finish formatting
%%%  papers published in Geometry & Topology and stored in the
%%%  arXiv.   All versions of  gtoutput.tex  are copyright 
%%%  GT Publications and are to be used _only_ for formatting
%%%  the officially published version of G&T papers.
%%%
%%%
%%%                                             Colin Rourke  14.9.2000
%%%
%%%  To create header file  head.xxx  comment out the first \endinput

%  test for latex or plain tex
\def\ifplaintex{\expandafter\ifx\csname documentclass\endcsname\relax}

%  get print centerpage:

\ifplaintex 
\hoffset 14truemm
\voffset 31truemm
\else
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
\fi


%  load pictex if not already loaded :
\expandafter\ifx\csname beginpicture\endcsname\relax
\expandafter\ifx\csname documentclass\endcsname\relax
\input pictex \else\font\fiverm=cmr5
\input prepictex \input pictex \input postpictex \fi\fi

\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology}}        %  journal title in recommended style

\def\gtp{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal P\!$ublications}}  % GT publications

%  define the various new ingredients of the title page 

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\samplenumber#1{\def\thesamplenumber{#1}\def\thevolumenumber{#1}
   \def\thepapernumber{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\proposed#1{\def\theproposer{#1}}
\def\seconded#1{\def\theseconders{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\def\accepted#1{\def\accepteddate{#1}}
\def\asciititle#1{\def\theasciititle{#1}}
\def\covertitle#1{\def\thecovertitle{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\shortauthors#1{\def\theshortauthors{#1}}
\def\shorttitle#1{\def\theshorttitle{#1}}

%  initialise

\let\\\par\let\thelognumber\relax
\let\thevolumenumber\relax\let\thepapernumber\relax
\let\thevolumeyear\relax\let\thesamplenumber\relax\let\startpage\relax
\let\finishpage\relax\let\publishdate\relax\let\receiveddate\relax
\let\reviseddate\relax\let\accepteddate\relax\let\theasciititle\relax
\let\thecovertitle\relax\let\theasciiauthors\relax\let\theasciiaddress\relax
\let\theasciiabstract\relax\let\theasciikeywords\relax
\let\theasciiemail\relax\let\theshortauthors\relax\let\theshorttitle\relax
\let\thecoverauthors\relax

\long\def\maketitlep{   % start of definition of \maketitlep

\count0=\startpage

\gt\hfill      %   Journal title (top left) 
%    Logo (top right) :
\beginpicture
\setcoordinatesystem units <0.33truein, 0.33truein> point at 2.2 0.9
\setplotsymbol ({$\cal G$})
\plotsymbolspacing=9truept
\circulararc 315 degrees from 0 1 center at 0 0
\setplotsymbol ({$\cal T$})
\circulararc 315 degrees from 1 -1 center at 1 0
\endpicture
%   end of logo
%
\break
{\small\ifx\thesamplenumber\relax % sample?  
Volume \else Sample
\fi\thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}
\vglue 0.5truein plus 0.4fil minus 0.1truein

% title
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\ifplaintex\large
\else\Large\fi\bf\thetitle}\par\medskip}   

\vglue 0pt plus 0.1fil 

% authors
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
\par\medskip}

 
\vglue 0pt plus 0.1fil 

%address(es)
{\small\parskip=0pt\let\newline\\
{\leftskip 0pt plus 1fil\def\\{\par}{\sl\theaddress}\par}
\expandafter\ifx\theemail\relax    % email address?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace} 
\cl{Email:\stdspace\tt\theemail}\fi
\ifx\theurl\relax                  % URL given?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace}
\cl{URL:\stdspace\tt\theurl}\fi\par}

\vglue 7pt plus 0.3fil minus 3pt

{\bf Abstract}
\vglue 5pt plus 0.1fil minus 2pt

\theabstract

\vglue 7pt plus 0.3fil minus 3pt

{\bf AMS Classification numbers}\quad Primary:\quad \theprimaryclass

Secondary:\quad \thesecondaryclass

\vglue 5pt plus 0.3fil minus 2pt

{\bf Keywords:}\quad \thekeywords

\vglue 10pt plus 0.5fil minus 5pt

{\small  Proposed: \theproposer\hfill Received: \receiveddate\nl
Seconded: \theseconders\hfill 
\ifx\reviseddate\relax                         % paper revised?
Accepted: \accepteddate                        % no
\else
Revised: \reviseddate                          % yes
\fi}
\eject
}       %  end of definition of \maketitlep

\let\maketitlepage\maketitlep
\let\maketitle\maketitlepage

%%% Headers and footers

\font\phead=cmsl9 scaled 950
\font\lhead=cmsl9 scaled 1050
\font\pnum=cmbx10 scaled 913
\font\lnum=cmbx10 
\font\pfoot=cmsl9 scaled 950
\font\lfoot=cmsl9 scaled 1050
\ifplaintex
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (printed) \hfill {\pnum\folio}\else\ifodd\count0\def\\{ }% 
\ifx\theshorttitle\relax\thetitle\else\theshorttitle\fi\hfill{\pnum\folio}
\else\def\\{ and }{\pnum\folio}\hfill\ifx\theshortauthors\relax\theauthors
\else\theshortauthors\fi\fi\fi}\vss}}
\footline{\vbox to 0pt{\vglue 0mm\line{\small\pfoot\ifnum\count0=\startpage
\copyright\ \gtp\hfill\else
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss
}}
\else
\makeatletter
\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (printed) \hfill {\lnum\number\count0}\else\ifodd\count0
\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
\hfill\ifx\theshortauthors\relax 
\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage\copyright\ \gtp\hfill\else
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
\def\@evenfoot{\@oddfoot}
\makeatother
\fi

\endinput  %%%comment out to create xxx header file


\newwrite\gtoutfile
\long\gdef\makeheadfile{  %%% start of definition of \makeheadfile
{\def\\{, }\def\s{ }
\immediate\openout\gtoutfile head.xxx
\immediate\write\gtoutfile{To: math@arxiv.org}
\immediate\write\gtoutfile{Subject: put or rep NNNNN:pppp}
\immediate\write\gtoutfile{--text follows this line--}
\immediate\write\gtoutfile{Proxy-for: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi\s<\ifx\theasciiemail\relax\theemail\else\theasciiemail\fi>}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{Authors: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi}
\immediate\write\gtoutfile{Title: \ifx\theasciititle\relax
\thetitle\else\theasciititle\fi}
\immediate\write\gtoutfile{Subj-class: GT or SG or MG etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Geom. Topol. \thevolumenumber
(\thevolumeyear) \startpage-\finishpage}
\immediate\write\gtoutfile{Comments: Published by Geometry and Topology at}
\immediate\write\gtoutfile{\s\s http://www.maths.warwick.ac.uk/gt/GTVol\thevolumenumber/paper\thepapernumber.abs.html}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\ifx\theasciiabstract\relax
\immediate\write\gtoutfile{\theabstract}\else
\immediate\write\gtoutfile{\theasciiabstract}\fi
\immediate\write\gtoutfile{}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\immediate\closeout\gtoutfile}}  %%% end of definition of \makeheadfile

\def\maketitlepage{\maketitlep\makeheadfile}
\let\maketitle\maketitlepage

\endinput
%%%
%%%  This version of  gtoutput.tex  is intended to finish formatting
%%%  papers published in Geometry & Topology and stored in the
%%%  arXiv.   All versions of  gtoutput.tex  are copyright 
%%%  GT Publications and are to be used _only_ for formatting
%%%  the officially published version of G&T papers.
%%%
%%%
%%%                                             Colin Rourke  14.9.2000
%%%
%%%  To create header file  head.xxx  comment out the first \endinput

%  test for latex or plain tex
\def\ifplaintex{\expandafter\ifx\csname documentclass\endcsname\relax}

%  get print centerpage:

\ifplaintex 
\hoffset 14truemm
\voffset 31truemm
\else
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
\fi


%  load pictex if not already loaded :
\expandafter\ifx\csname beginpicture\endcsname\relax
\expandafter\ifx\csname documentclass\endcsname\relax
\input pictex \else\font\fiverm=cmr5
\input prepictex \input pictex \input postpictex \fi\fi

\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology}}        %  journal title in recommended style

\def\gtp{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal P\!$ublications}}  % GT publications

%  define the various new ingredients of the title page 

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\samplenumber#1{\def\thesamplenumber{#1}\def\thevolumenumber{#1}
   \def\thepapernumber{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\proposed#1{\def\theproposer{#1}}
\def\seconded#1{\def\theseconders{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\def\accepted#1{\def\accepteddate{#1}}
\def\asciititle#1{\def\theasciititle{#1}}
\def\covertitle#1{\def\thecovertitle{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\shortauthors#1{\def\theshortauthors{#1}}
\def\shorttitle#1{\def\theshorttitle{#1}}

%  initialise

\let\\\par\let\thelognumber\relax
\let\thevolumenumber\relax\let\thepapernumber\relax
\let\thevolumeyear\relax\let\thesamplenumber\relax\let\startpage\relax
\let\finishpage\relax\let\publishdate\relax\let\receiveddate\relax
\let\reviseddate\relax\let\accepteddate\relax\let\theasciititle\relax
\let\thecovertitle\relax\let\theasciiauthors\relax\let\theasciiaddress\relax
\let\theasciiabstract\relax\let\theasciikeywords\relax
\let\theasciiemail\relax\let\theshortauthors\relax\let\theshorttitle\relax
\let\thecoverauthors\relax

\long\def\maketitlep{   % start of definition of \maketitlep

\count0=\startpage

\gt\hfill      %   Journal title (top left) 
%    Logo (top right) :
\beginpicture
\setcoordinatesystem units <0.33truein, 0.33truein> point at 2.2 0.9
\setplotsymbol ({$\cal G$})
\plotsymbolspacing=9truept
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math0101163.extracted_bib
\begin{thebibliography}


\let\oldbibitem\bibitem
\def\bibitem#1]#2{\oldbibitem{#2}}


\bibitem[C-H]{C-H} {\bf Tim D Cochran}, {\bf Nathan Habegger}, {\em
On the homotopy theory of simply connected four manifolds.} 
Topology, 29 (1990) 419--440 

\bibitem[D]{D} {\bf S\,K Donaldson}, {\em Irrationality and the
$h$--cobordism conjecture. } J. Differential Geom. 26 (1987) 141--168

\bibitem[F-Q]{F-Q} {\bf Michael H Freedman}, {\bf Frank S Quinn}, {\em
Topology of 4--manifolds. } Princeton Mathematical Series,
39, Princeton University Press, Princeton NJ (1990)

\bibitem[H-K]{H-K} {\bf Ian Hambleton}, {\bf M Kreck}, {\em
Cancellation, elliptic surfaces and the topology of certain
four-manifolds}, J. Reine Angew. Math. 444 (1993) 79--100

\bibitem[K1]{K1} {\bf M Kreck}, {\em Isotopy classes of
diffeomorphisms of $(k-1)$--connected almost-parallelizable
$2k$--manifolds}, from: ``Algebraic topology, Proc. Symp. Aarhus
1978'', Lect. Notes Math. 763, Springer--Verlag, Berlin (1979) 643--663

\bibitem[K2]{K2}{\bf M Kreck}, { \em Surgery and Duality}, 
Annals of Math. 149 (1999) 707--754

\bibitem [L]{L} {\bf T Lawson}, {\em $h$--cobordisms between simply  
connected $4$-manifolds},
Topology Appl. 28 (1988) 75--82

\bibitem[M]{M} {\bf J Milnor}, {\em On simply-connected 4--manifolds.}
from: ``International Symposium on Algebraic Topology, Universidad Nacional
Aut\'onoma de M\'exico and UNESCO'', (1958) 122--128

\bibitem[M-S]{M-S} {\bf John W Morgan}, {\bf  Zoltan Szabo}, {\em
Complexity of 4--dimensional $h$--cobordisms},
Invent. Math. 136 (1999) 273--286 

\bibitem[P]{P} {\bf B Perron}, {\em Pseudo-isotopies et isotopies en
dimension quatre dans la categorie topologique, (Pseudo-isotopies and
isotopies of dimension four in the topological category)},  Topology,
25 (1986) 381--397

\bibitem[Q]{Q} {\bf Frank Quinn}, {\em Isotopy of 4--manifolds.}
J. Differential Geom. 24, (1986) 343--372


\bibitem [S]{S} {\bf J Stallings}, {\em On infinite processes leading
to differentiability in the complement of a point}, from:
``Differential and combinatorial topology (a symposium in honour of
Marston Morse)'', Princeton Univ. Press, Princeton NJ (1965) 245--254


\bibitem[W]{W} {\bf C\,T\,C Wall}, { \em Surgery on Compact
Manifolds}, London Math. Soc. Monographs~1, Academic Press (1970)








\end{thebibliography}