tiny[table]font=tiny

# Counterexamples in $4$-manifold topology

Daniel Kasprowski  Rheinische Friedrich-Wilhelms-Universität Bonn,
Mathematisches Institut,
Endenicher Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS>,  Mark
Powell  Department of Mathematical Sciences, Durham University, Upper
Mountjoy,
Stockton Road, Durham, United Kingdom, DH1 3LE<EMAIL_ADDRESS>and
Arunima Ray  Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn,
Germany<EMAIL_ADDRESS>
###### Abstract.

We consider several of the most commonly studied equivalence relations on
$4$-manifolds and how they are related to one another. We explain implications
e.g. that $h$-cobordant manifolds are stably homeomorphic, and we provide
examples illustrating the failure of other potential implications. The
information is conveniently organised in a flowchart and a table.

###### Key words and phrases:

4-manifolds, equivalence relations

###### 1991 Mathematics Subject Classification:

57K40.

##  1\. Introduction

The goal of this paper is to organise various equivalence relations in
$4$-manifold topology, and to understand the connections between them. We
consider closed, connected $4$-manifolds, unless otherwise specified, and we
work in both the smooth and topological settings. Much work on 4-manifolds
focusses on exotic behaviour, e.g. $4$-manifolds that are homeomorphic but not
diffeomorphic. We aim to illustrate, more broadly, the wealth of $4$-manifold
topology that has been discovered. The flowchart in Figure 1 shows the
relationships between the equivalence relations we study. We will recall their
definitions in Section 2, and prove the nontrivial implications in Section 3.

Figure 1. Equivalence relations on $4$-manifolds. The implications shown in
green are immediate. The black and blue implications are proven in Section 3.
The blue implications hold when the corresponding condition is true, e.g.
homotopy equivalent $4$-manifolds are $\mathbb{CP}^{2}$-stably homeomorphic if
their Kirby-Siebenmann invariants coincide. Where necessary for an implication
to make sense, we assume that the manifolds are smooth. For example, the
bottom arrow means that closed, _smooth_ $\mathbb{CP}^{2}$-stably homeomorphic
$4$-manifolds are $\mathbb{CP}^{2}$-stably diffeomorphic, since the latter
notion is only defined for smooth $4$-manifolds.

We collect counterexamples to the converses of the majority of the
implications shown. Most of the results we discuss are known in the
literature, although there are some original observations and results. First
among these is the following theorem, comparing the equivalence relations of
simple homotopy equivalence, $h$-cobordism, and $s$-cobordism.

######  Theorem 1.1.

For every $n\geq 1$, there is a collection $\\{N_{i}\\}_{i=1}^{n}$ of closed,
orientable, topological 4-manifolds, that are all simple homotopy equivalent
and $h$-cobordant to one another, but which are pairwise not $s$-cobordant.

It would be interesting to have smooth examples, or a corresponding infinite
collection. Our proof makes use of a braid of exact sequences due to
Hambleton-Kreck [Hambleton-Kreck-HSEs] which enables one to estimate the size
of the group of homotopy automorphisms of the $4$-manifold $L_{p,q}\times
S^{1}$, where $L_{p,q}$ is a lens space. We combine this with the surgery
exact sequence for 4-manifolds with fundamental group
$\mathbb{Z}/p\times\mathbb{Z}$ to construct our families of examples.

While investigating the failure of the converses of the implications in the
flowchart, we will progressively impose restrictions on the counterexamples,
e.g. that they be smooth, or orientable. As an example of our approach,
consider the equivalence relations of homeomorphism and homotopy equivalence.
It is straightforward to see that homeomorphism implies homotopy equivalence,
for general spaces. That the converse is false can be seen in the context of
$4$-manifolds by the well-known pair $\mathbb{CP}^{2}$ and
$\mathop{}\\!*\mathbb{CP}^{2}$ [F] (see Section 5.3). The latter manifold,
sometimes called the _Chern manifold_ , was constructed by Freedman and is
homotopy equivalent to $\mathbb{CP}^{2}$, but not homeomorphic to it. Indeed,
$\mathbb{CP}^{2}$ and $\mathop{}\\!*\mathbb{CP}^{2}$ have unequal Kirby-
Siebenmann invariants, implying that $\mathop{}\\!*\mathbb{CP}^{2}$ is not
smoothable. The natural question then arises whether there exists a pair of
smooth, closed $4$-manifolds that are homeomorphic but not homotopy
equivalent. Freedman’s classification result [F, FQ] implies that there is no
such pair of simply connected $4$-manifolds. A pair that satisfies our new
demand consists of $\mathbb{RP}^{4}\\#\mathbb{CP}^{2}$ and
$\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$ [ruberman84, HKT-nonorientable,
Ruberman-Stern] (see Section 5.4), where $\mathcal{R}$ is a 4-manifold
homotopy equivalent to $\mathbb{RP}^{4}$ but with nontrivial Kirby-Siebenmann
invariant. However, $\mathbb{RP}^{4}\\#\mathbb{CP}^{2}$ and
$\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$ are nonorientable. So we can then
escalate further to ask for a pair of smooth, closed, orientable $4$-manifolds
that are homotopy equivalent but not homeomorphic. In this case, the simplest
example we know of comes from Turaev [Turaev1988] (see Section 5.7), who
showed that for lens spaces $L$ and $L^{\prime}$ that are homotopy equivalent
but not homeomorphic, the same holds for the $4$-manifolds $L\times S^{1}$ and
$L^{\prime}\times S^{1}$. Finally, one may ask for an infinite collection of
closed, smooth, orientable $4$-manifolds that are homotopy equivalent but not
homeomorphic. To our knowledge, this is an open question. However, the
following theorem answers the question for topological 4-manifolds.

######  Theorem 1.2 (Kwasik-Schultz [Kwasik-Schultz]*Theorem 1.2).

Let $M:=L\times S^{1}$, where $L$ is a lens space $L_{p,q}$ with $p\geq 2$,
$1\leq q<p$, and $(p,q)=1$. Then there is an infinite collection of closed,
orientable, topological $4$-manifolds $\\{M_{i}\\}_{i=1}^{\infty}$, that are
all simple homotopy equivalent to $M$ but pairwise not homeomorphic.

The proof of Kwasik and Schultz relies on higher $\rho$-invariants. In Section
5.13 we provide a different argument via the surgery exact sequence that
enables us to establish other properties of these manifolds. For example one
can assume that they are all stably homeomorphic and are pairwise not
$h$-cobordant. We will also show (or give citations showing) that all of the
pairs of 4-manifolds discussed in the paragraph preceding Theorem 1.2 are
simple homotopy equivalent, and that all these pairs apart from
$\mathbb{CP}^{2}$ and $\mathop{}\\!*\mathbb{CP}^{2}$ are stably homeomorphic.

Here is a summary of some of the properties of the other counterexamples, not
yet mentioned, that we will discuss in Section 5.

  1. (1)

There are infinitely many smooth, orientable, simply connected 4-manifolds
that are all smoothly $s$-cobordant and homeomorphic to one another, but not
diffeomorphic (see e.g. [GompfStip, akbulut-book]). Since exotic behaviour of
this sort is not our primary focus, we only present the first pair of such
manifolds discovered, due to Donaldson [Donaldson:1987-1], in Section 5.8.

  2. (2)

The pairs of 4-manifolds presented in Sections 5.5, 5.6, and 5.12, due to
Kreck, Cappell-Shaneson, and Akbulut respectively, are smooth, nonorientable
$4$-manifolds that are homeomorphic but not stably diffeomorphic. By a result
of Gompf [Gompf-stable] (see Theorem 3.8) smooth, orientable $4$-manifolds
that are (stably) homeomorphic are stably diffeomorphic, so it is inevitable
that these examples are nonorientable.

  3. (3)

The $4$-manifolds $S^{2}\times S^{2}$ and
$S^{2}\mathbin{\widetilde{\times}}S^{2}$ are smooth, simply connected, have
equal Euler characteristic, and are $\mathbb{CP}^{2}$-stably diffeomorphic but
are not stably homeomorphic and not homotopy equivalent. See Section 5.2.

  4. (4)

The examples of Kreck-Schafer discussed in Section 5.10 are smooth, orientable
$4$-manifolds that are stably diffeomorphic, but not homotopy equivalent. They
also have isometric intersection pairings.

  5. (5)

The examples of Teichner from Section 5.11 provide smooth, orientable
$4$-manifolds that are simple homotopy equivalent but not stably homeomorphic.

In Section 6 we highlight a series of open questions related to the
equivalence relations in Figure 1. Here is the most prominent.

######  Question 6.2.

Does there exist a pair of closed $4$-manifolds that are homotopy equivalent
but not simple homotopy equivalent?

Extra kudos is available if the examples are (i) smoothable, (ii) orientable,
(iii) topologically $h$-cobordant, or (iv) smoothly $h$-cobordant.

We hope that readers will be motivated to answer the questions we could not,
or to follow the paradigm of progressively imposing restrictions to discover
new unanswered questions of their own.

### Outline

In Section 2 we define the equivalence relations we consider. In Section 3 we
justify the implications shown in Figure 1. Section 4 provides a brief review
of the surgery exact sequence. In Section 5 we describe various constructions
of $4$-manifolds and present a table summarising the properties of our
examples.

### Conventions

We write $\mathbb{Z}/2=\\{0,1\\}$ for the integers modulo 2, a group under
addition, and $C_{2}=\\{\pm 1\\}$ for the cyclic group of order 2, with
multiplication as the group operation. The symbol $\simeq$ denotes homotopy
equivalence, while $\simeq_{s}$ denotes simple homotopy equivalence. Depending
on the context the symbol $\cong$ denotes either homeomorphism or
diffeomorphism.

### Acknowledgements

The authors thank the Max Planck Institute for Mathematics in Bonn, where much
of the research leading to this article occurred. We also thank Jim Davis,
Kent Orr, Patrick Orson, and Peter Teichner for several helpful discussions.

DK was supported by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. MP was partially
supported by EPSRC New Investigator grant EP/T028335/1 and EPSRC New Horizons
grant EP/V04821X/1.

###### Contents

  1. 1 Introduction
  2. 2 Equivalence relations on 4-manifolds
  3. 3 Justification of implications
  4. 4 Review of surgery exact sequences
  5. 5 Counterexamples
    1. 5.1 $S^{2}\times S^{2}$ and $S^{4}$
    2. 5.2 $S^{2}\times S^{2}$ and $S^{2}\mathbin{\widetilde{\times}}S^{2}$
    3. 5.3 $\mathbb{CP}^{2}$ and Freedman’s $\mathop{}\\!*\mathbb{CP}^{2}$
    4. 5.4 $\mathbb{RP}^{4}\\#\mathbb{CP}^{2}$ and $\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$
    5. 5.5 Kreck’s examples $K3\\#\mathbb{RP}^{4}$ and $\\#^{11}(S^{2}\times S^{2})\\#\mathbb{RP}^{4}$
    6. 5.6 $\mathbb{RP}^{4}$ and the Cappell-Shaneson fake $\mathbb{RP}^{4}$
    7. 5.7 $L\times S^{1}$ and $L^{\prime}\times S^{1}$
    8. 5.8 Donaldson’s examples $E(1)$ and the Dolgachev surface $E(1)_{2,3}$
    9. 5.9 $\,\,{\scalebox{1.2}{\\#}^{3}}E_{8}$ and the Leech manifold
    10. 5.10 Kreck-Schafer manifolds
    11. 5.11 Teichner’s $E\\#E$ and $*E\\#\mathop{}\\!*E$
    12. 5.12 Akbulut’s fake $(S^{1}\mathbin{\widetilde{\times}}S^{3})\\#(S^{2}\times S^{2})$
    13. 5.13 Kwasik-Schultz manifolds homotopy equivalent to $L\times S^{1}$
    14. 5.14 Simple homotopy equivalent, $h$-cobordant $4$-manifolds that are not $s$-cobordant
  6. 6 Open problems

##  2\. Equivalence relations on 4-manifolds

Recall that we implicitly assume throughout that $4$-manifolds are closed and
connected. We assume that the reader is familiar with homotopy equivalence,
homeomorphism, and diffeomorphism of manifolds, and so we shall not define
them. The classification of manifolds with respect to these three notions, and
their comparison, is a central area of research. For example, the Poincaré
conjecture, which has occupied topologists for over a century, asks for each
$n$ whether every homotopy equivalence from an $n$-manifold to the $n$-sphere
is homotopic to a homeomorphism, or even to a diffeomorphism.111The Poincaré
conjecture is true in the topological category for all $n$, due to Perelman
[Perelman:2002-1, Perelman:2003-1, Perelman:2003-2], Freedman [F], and Newman
[Newman]. It is true in the PL category for all $n\neq 4$ due to Perelman
[loc. cit.], Smale [smale-poincare-bulletin, Smale-h-cob-thm], Stallings
[Stallings-BAMS], and Zeeman [zeeman-poincare]. In the smooth category it is
known to hold in dimensions $1,2,3,5,6,12,56,61$, due to Perelman [loc. cit.],
Kervaire-Milnor [Kervaire-Milnor:1963-1], Isaksen [Isaksen], and Wang-Xu
[Wang-Xu]; is false in all odd dimensions other than $1,3,5,61$; and is false
in all even dimensions $8\leq n\leq 200$ other than $12$, $56$, $142$, $166$,
$176$, $188$. At the time of writing the smooth version is open in dimensions
$4$, $142$, $166$, $176$, $188$, and for infinitely many even dimensions
$n>200$; see [Behrens-Hill-Hopkins-Mahowald, Isaksen-Wang-Xu-PNAS] for the
published state of the art. In dimensions at least five, surgery theory
provides a concrete, effective framework within which one can try to improve a
classification of manifolds up to homotopy equivalence to a classification up
to homeomorphism or diffeomorphism. The programme can be applied to
topological 4-manifolds under a restriction on the fundamental group; see
Section 4 for an overview.

Next we discuss the various notions of stable equivalence.

######  Definition 2.1.

The $4$-manifolds $M$ and $N$ are said to be _stably homeomorphic_ if there
are integers $s,t$ such that $M\\#\,\,{\scalebox{1.2}{\\#}^{s}}(S^{2}\times
S^{2})$ and $N\\#\,\,{\scalebox{1.2}{\\#}^{t}}(S^{2}\times S^{2})$ are
homeomorphic. They are said to be _$\mathbb{CP}^{2}$ -stably homeomorphic_ if
there are integers $s,t$ such that
$M\\#\,\,{\scalebox{1.2}{\\#}^{s}}\mathbb{CP}^{2}$ and
$N\\#\,\,{\scalebox{1.2}{\\#}^{t}}\mathbb{CP}^{2}$ are homeomorphic, for some
choices of connected sum.

######  Definition 2.2.

The smooth $4$-manifolds $M$ and $N$ are said to be _stably diffeomorphic_ if
there are integers $s,t$ such that
$M\\#\,\,{\scalebox{1.2}{\\#}^{s}}(S^{2}\times S^{2})$ and
$N\\#\,\,{\scalebox{1.2}{\\#}^{t}}(S^{2}\times S^{2})$ are diffeomorphic. They
are said to be _$\mathbb{CP}^{2}$ -stably diffeomorphic_ if there are integers
$s,t$ such that $M\\#\,\,{\scalebox{1.2}{\\#}^{s}}\mathbb{CP}^{2}$ and
$N\\#\,\,{\scalebox{1.2}{\\#}^{t}}\mathbb{CP}^{2}$ are diffeomorphic, for some
choices of connected sum.

Note that $S^{2}\times S^{2}$ admits an orientation reversing self-
diffeomorphism, so there is essentially only one choice of connected sum. On
the other hand $\mathbb{CP}^{2}$ does not admit any such diffeomorphism (nor
homeomorphism), so for oriented manifolds there are two possible connected
sums up to diffeomorphism/homeomorphism, usually denoted $M\\#\mathbb{CP}^{2}$
and $M\\#\overline{\mathbb{CP}^{2}}$. Therefore, the definitions above say
that $M$ and $N$ are $\mathbb{CP}^{2}$-stably diffeomorphic (resp.
homeomorphic) if there are integers $s_{1},s_{2},t_{1},t_{2}$ such that
$M\\#\,\,{\scalebox{1.2}{\\#}^{s_{1}}}\mathbb{CP}^{2}\\#\,\,{\scalebox{1.2}{\\#}^{s_{2}}}\overline{\mathbb{CP}^{2}}$
and
$N\\#\,\,{\scalebox{1.2}{\\#}^{t_{1}}}\mathbb{CP}^{2}\\#\,\,{\scalebox{1.2}{\\#}^{t_{2}}}\overline{\mathbb{CP}^{2}}$
are diffeomorphic (resp. homeomorphic). Note that for nonorientable manifolds,
there is a unique connected sum $N\\#\mathbb{CP}^{2}$.

We emphasise that “stably” refers by default to connected sum with copies of
$S^{2}\times S^{2}$, and only “$\mathbb{CP}^{2}$-stably” refers to connected
sum with copies of $\mathbb{CP}^{2}$.

To motivate Definitions 2.1 and 2.2, consider an alternative strategy to
classify manifolds, based on Kreck’s modified surgery [surgeryandduality] and
realised by Hambleton-Kreck for e.g. 4-manifolds with finite cyclic
fundamental group in [Hambleton-Kreck:1988-1, HK88b, Hambleton-Kreck-93,
HK93b]: first classify manifolds up to stable homeomorphism, and then
investigate the homeomorphism types within each stable class. In the latter
step one attempts to prove that $S^{2}\times S^{2}$ summands can be cancelled,
and this is what Hambleton and Kreck achieved for finite cyclic fundamental
groups and also for more general finite groups under some additional
hypotheses. This strategy in principle also applies to diffeomorphism
classifications, but the cancellation step is much harder. Similarly, another
approach to classification is to first classify manifolds up to
$\mathbb{CP}^{2}$-stable equivalence, and then attempt to blow down extraneous
$\mathbb{CP}^{2}$ summands.

Next we discuss $h$-cobordisms, simple homotopy equivalences, and
$s$-cobordisms.

######  Definition 2.3.

The $4$-manifolds $M$ and $N$ are _topologically $h$-cobordant_ if there is a
$5$-dimensional compact topological cobordism $(W;M,N)$ where the inclusion
maps $M\hookrightarrow W$ and $N\hookrightarrow W$ are homotopy equivalences.
The manifold $W$ is called an _$h$ -cobordism_. If $M$ and $N$ are smooth,
they are _smoothly $h$-cobordant_ if they cobound a smooth $h$-cobordism.

Associated with a homotopy equivalence $f\colon X\to Y$ between CW complexes
$X$ and $Y$ is an algebraic invariant called the _Whitehead torsion_
$\tau(f)\in\operatorname{Wh}(\pi_{1}(X))$, with values in the _Whitehead
group_ of $\pi_{1}(X)$, which we define next. Let
$\operatorname{GL}(\mathbb{Z}[\pi_{1}(X)])$ be the stable general linear
group, and let $\operatorname{E}(\mathbb{Z}[\pi_{1}(X)])$ be the subgroup of
elementary matrices, i.e. consisting of products of the matrices that produce
row and column operations. By definition

$\operatorname{K}_{1}(\mathbb{Z}[\pi_{1}(X)]):=\operatorname{GL}(\mathbb{Z}[\pi_{1}(X)])/\operatorname{E}(\mathbb{Z}[\pi_{1}(X)])$

and
$\operatorname{Wh}(\pi_{1}(X)):=\operatorname{K}_{1}(\mathbb{Z}[\pi_{1}(X)])/\\{\pm(g)\mid
g\in\pi_{1}(X)\\}$. For example $\operatorname{Wh}(\\{e\\})=0$, essentially
because of the Euclidean algorithm. See [Cohen] for an accessible introduction
to simple homotopy theory, including more examples of Whitehead groups and the
definition of Whitehead torsion.

######  Definition 2.4.

A homotopy equivalence $f\colon X\to Y$ between CW complexes $X$ and $Y$ is a
_simple homotopy equivalence_ if its Whitehead torsion $\tau(f)$ vanishes.

It is an open question whether every topological $4$-manifold is homeomorphic
to a $4$-dimensional CW complex. However, we can define the notion of simple
homotopy equivalence of $4$-manifolds as follows. Embed $M$ in high-
dimensional Euclidean space. By [Kirby-Siebenmann:1977-1]*Essay III, Section
4, there is a normal disc bundle $D(M)$ admitting a triangulation. The
inclusion map $z_{M}\colon M\to D(M)$ of the $0$-section is a homotopy
equivalence. Let $z_{M}^{-1}$ denote the homotopy inverse of $z_{M}$.

######  Definition 2.5.

We say that a homotopy equivalence $f\colon M\to N$ between topological
manifolds (not necessarily of the same dimension) is a _simple homotopy
equivalence_ if the composition $z_{N}\circ f\circ z_{M}^{-1}\colon D(M)\to
D(N)$ is a simple homotopy equivalence.

The Whitehead torsion $\tau(W;M)$ of an $h$-cobordism $(W;M,N)$ is by
definition the Whitehead torsion of the inclusion map $M\hookrightarrow W$.
This also coincides with the Whitehead torsion of the relative chain complex
$C_{*}(\widetilde{W},\widetilde{M})$, where $\widetilde{W}$ and
$\widetilde{M}$ are the universal covers.

######  Definition 2.6.

The $4$-manifolds $M$ and $N$ are _topologically $s$-cobordant_ if they
cobound a topological $h$-cobordism $W$ with trivial Whitehead torsion. The
manifold $W$ is called an _$s$ -cobordism_. If in addition $M$, $N$, and $W$
are smooth, then $M$ and $N$ are _smoothly $s$-cobordant_ and $W$ is called a
_smooth $s$-cobordism_.

An $h$\- or $s$-cobordism approximates a product, in the eyes of homotopy
equivalence and simple homotopy equivalence respectively. One of the most
spectacular results of the 20th century was Smale’s $h$-cobordism theorem
[smale-poincare-bulletin, Smale-h-cob-thm], which states that smooth, simply
connected $h$-cobordisms between $n$-manifolds with $n\geq 5$ are indeed
homeomorphic to products. This was later extended to other categories and to
the case of $s$-cobordisms [Smale:1962-1, Barden, Mazur:1963-1, Stallings-
Tata, Kirby-Siebenmann:1977-1]. Consequences include the high-dimensional
Poincaré conjecture in the piecewise-linear category in dimension at least
five.

In dimension four, the celebrated work of Freedman and Quinn [F, FQ] includes
an $s$-cobordism theorem, with a restriction on fundamental groups. This is
the principal method for establishing the existence of a homeomorphism between
$4$-manifolds. We state the result in the next section as Theorem 3.5 and we
outline the proof.

##  3\. Justification of implications

The implications given in green in Figure 1 are immediate from the
definitions. Now we justify the other implications.

######  Proposition 3.1.

Stably homeomorphic manifolds are $\mathbb{CP}^{2}$-stably homeomorphic.
Stably diffeomorphic manifolds are $\mathbb{CP}^{2}$-stably diffeomorphic.

###### Proof.

Both statements follow from the diffeomorphism

$(S^{2}\times
S^{2})\\#\mathbb{CP}^{2}\cong\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}\\#\mathbb{CP}^{2}.\qed$

To establish the relationship between homotopy equivalence and
$\mathbb{CP}^{2}$-stable homeomorphism, we will use the following theorem of
Kreck. Recall that an identification of the fundamental group $\pi_{1}(M)$ of
a $4$-manifold $M$ with a group $\pi$ determines a map $c_{M}\colon M\to
B\pi$, up to homotopy, classifying the universal cover, where $B\pi\simeq
K(\pi,1)$ is the classifying space.

######  Theorem 3.2 (Kreck [surgeryandduality]).

  1. (i)

Two closed, smooth $4$-manifolds $M$ and $N$ with fundamental group isomorphic
to $\pi$ and orientation character $w\colon\pi\to C_{2}$ are
$\mathbb{CP}^{2}$-stably diffeomorphic if and only if $c_{M*}[M]=c_{N*}[N]\in
H_{4}(\pi;\mathbb{Z}^{w})/\pm\operatorname{Aut}(\pi,w)$.

  2. (ii)

Two closed, topological $4$-manifolds $M$ and $N$ with fundamental group
isomorphic to $\pi$ and orientation character $w\colon\pi\to C_{2}$ are
$\mathbb{CP}^{2}$-stably homeomorphic if and only if their Kirby-Siebenmann
invariants in $\mathbb{Z}/2$ agree and $c_{M*}[M]=c_{N*}[N]\in
H_{4}(\pi;\mathbb{Z}^{w})/\pm\operatorname{Aut}(\pi,w)$.

Here $\operatorname{Aut}(\pi,w)$ denotes the set of automorphisms of $\pi$
compatible with the map $w$. We have to factor out by the action of
$\pm\operatorname{Aut}(\pi,w)$ in order to account for the choice of
identifications $\pi_{1}(M)\cong\pi$ and $\pi_{1}(N)\cong\pi$, and for the
choice of (twisted) fundamental classes in $H_{4}(-;\mathbb{Z}^{w})$.

We will also need the _Kirby-Siebenmann invariant_
$\operatorname{ks}(M)\in\mathbb{Z}/2$ of a $4$-manifold $M$, which by
definition is the unique obstruction for the stable tangent microbundle of $M$
to admit a lift to a piecewise linear bundle. See [Kirby-Siebenmann:1977-1]*p.
318[FQ]*Section 10.2B[guide]*Section 8.2 for further details on the
definition. In general it will suffice for us to know that the Kirby-
Siebenmann invariant satisfies strong additivity properties, in particular
under gluing and connected sum, and that it vanishes for a $4$-manifold $M$ if
and only if $M\times\mathbb{R}$ admits a smooth structure, if and only if
$M\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times S^{2})$ admits a smooth
structure for some $k$.

######  Theorem 3.3.

  1. (1)

Homotopy equivalent 4-manifolds with equal Kirby-Siebenmann invariants are
$\mathbb{CP}^{2}$-stably homeomorphic.

  2. (2)

Smooth $4$-manifolds that are $\mathbb{CP}^{2}$-stably homeomorphic are also
$\mathbb{CP}^{2}$-stably diffeomorphic.

###### Proof.

For the first implication, let $M$ and $N$ be homotopy equivalent
$4$-manifolds with equal Kirby-Siebenmann invariants. Fix a map $c_{N}\colon
N\to B\pi$ as mentioned above the statement of Theorem 3.2 and let $f\colon
M\to N$ be the claimed homotopy equivalence. Then define $c_{M}:=c_{N}\circ
f$. This ensures that $c_{M*}[M]=c_{N*}[N]\in
H_{4}(\pi;\mathbb{Z}^{w})/\pm\operatorname{Aut}(\pi,w)$, and so by Theorem 3.2
(ii), we see that $M$ and $N$ are $\mathbb{CP}^{2}$-stably homeomorphic.

For the second implication, let $M$ and $N$ be $\mathbb{CP}^{2}$-stably
homeomorphic and smooth. By Theorem 3.2 (ii) we see that
$c_{M*}[M]=c_{N*}[N]\in
H_{4}(\pi;\mathbb{Z}^{w})/\pm\operatorname{Aut}(\pi,w)$. Then apply Theorem
3.2 (i) to see that $M$ and $N$ are $\mathbb{CP}^{2}$-stably diffeomorphic. ∎

Next we show the relationship between $h$-cobordism and stable diffeomorphism.
The case of simply connected $4$-manifolds was addressed by Wall in [Wall-on-
simply-conn-4mflds]*Theorem 3. A similar argument also applies in the general
setting as explained by Lawson [Lawson-decomposing-5-mnfs]*Proposition. We
sketch the proof.

######  Theorem 3.4 ([Wall-on-simply-conn-4mflds]*Theorem 3[Lawson-
decomposing-5-mnfs]*Proposition).

Smoothly $h$-cobordant 4-manifolds are stably diffeomorphic. Similarly,
topologically $h$-cobordant 4-manifolds are stably homeomorphic.

###### Proof.

The proof is the same in both cases, by using the fact that $5$-dimensional
topological cobordisms $(W;M,N)$ admit handle decompositions, i.e. $W$ can be
built by attaching $5$-dimensional handles to $M\times[0,1]$ along topological
embeddings of the attaching regions [Quinn-annulus]*Theorem 2.3.1[FQ]*Theorem
9.1. In the case of a smooth $h$-cobordism, we get a smooth handle
decomposition relative to $M$ by Morse theory.

In either case, we can perform handle trading to ensure that the handle
decomposition has only $2$\- and $3$-handles since the boundary inclusions
$M\hookrightarrow W$ and $N\hookrightarrow W$ are $1$-connected. Consider the
middle level $M_{1/2}$ of the cobordism, obtained after attaching $2$-handles
to $M$. Since $W$ is an $h$-cobordism, the $2$-handles are attached along
trivial circles, and so

$M_{1/2}\cong M\\#\big{(}\,\,{\scalebox{1.2}{\\#}^{t_{1}}}S^{2}\times
S^{2}\big{)}\\#\big{(}\,\,{\scalebox{1.2}{\\#}^{t_{2}}}S^{2}\mathbin{\widetilde{\times}}S^{2}\big{)},$

where $t_{1}+t_{2}$ is the number of $2$-handles. Here
$S^{2}\mathbin{\widetilde{\times}}S^{2}$ is the twisted $S^{2}$-bundle over
$S^{2}$.

If $t_{2}=0$ we are done. In case $t_{2}\neq 0$, then there is an embedded
$2$-sphere in $M_{1/2}$ with odd framing of its normal bundle. Via the
homotopy equivalence $W\to M$, we see there is a map of a sphere to $M$ with
odd framing of its normal bundle, implying that the universal cover of $M$ is
non-spin. In this case we have that
$M\\#(S^{2}\mathbin{\widetilde{\times}}S^{2})\cong M\\#(S^{2}\times S^{2})$
(see e.g. [GompfStip]*Exercise 5.2.6(b)), and so we may assume $t_{2}=0$
again.

We have now argued that $M_{1/2}$ is a stabilisation of $M$. Applying the same
argument to the upside down handlebody, we see further that $M_{1/2}$ is a
stabilisation of $N$. Thus $M$ and $N$ are stably homeomorphic or
diffeomorphic, depending on whether the handle decomposition was smooth or
merely topological to begin with. ∎

Next we state and prove the $s$-cobordism theorem in dimension four. Below, we
say a group is _good_ if it satisfies the $\pi_{1}$-null disc property
[Freedman-Teichner:1995-1] (see also [Freedman-book-goodgroups]). We do not
repeat the definition here. In practice, it generally suffices to know that
virtually solvable groups, and more generally groups of subexponential growth
are good, and the class of good groups is closed under taking subgroups,
quotients, extensions, and colimits [Freedman-Teichner:1995-1, Krushkal-
Quinn:2000-1].

######  Theorem 3.5 ($s$-cobordism theorem).

Let $M$ be a topological $4$-manifold with $\pi:=\pi_{1}(M)$ a good group.

  1. (1)

Let $(W;M,M^{\prime})$ be an $h$-cobordism over $M$. Then $W$ is trivial over
$M$, i.e. $W\cong M\times[0,1]$, via a homeomorphism restricting to the
identity on $M$, if and only if its Whitehead torsion
$\tau(W;M)\in\operatorname{Wh}(\pi)$ vanishes.

  2. (2)

For any $\varsigma\in\operatorname{Wh}(\pi)$ there exists an $h$-cobordism
$(W;M,M^{\prime})$ with $\tau(W;M)=\varsigma$.

  3. (3)

The function assigning to an $h$-cobordism $(W;M,M^{\prime})$ its Whitehead
torsion $\tau(W;M)$ yields a bijection from the homeomorphism classes relative
to $M$ of $h$-cobordisms over $M$ to the Whitehead group
$\operatorname{Wh}(\pi)$.

######  Remark 3.6.

It was asserted in [Rourke-Sanderson]*p. 90 that Theorem 3.5 (2) holds in
dimension 4 in the piecewise-linear category, and without the assumption on
$\pi_{1}(M)$. But the proof there does not take into account the need for
geometrically dual spheres to control the fundamental group of $M^{\prime}$.

###### Proof of Theorem 3.5.

The statement (1) is [FQ]*Theorem 7.1A, and relies on Freedman’s disc
embedding theorem [F] (see also [FQ, Freedman-notes]). We prove (2), which
follows the high dimensional argument from [Milnor-whitehead-torsion]*Theorem
11.1.

Let $A\in\operatorname{GL}_{k}(\mathbb{Z}\pi)$ represent $\varsigma$. Attach
$k$ trivial $2$-handles to $M\times\\{1\\}\subseteq M\times[0,1]$. This yields
a bordism $W^{\prime}$ from $M$ to
$M\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times S^{2})$, where we assume that
the ascending spheres of the $2$-handles are $S^{2}\times\\{s_{0}\\}$ in each
copy of $S^{2}\times S^{2}$. For every $1\leq i\leq k$, we can realize the
element
$(0,(0,A_{ij})_{j=1}^{k})\in\pi_{2}(M)\oplus\bigoplus_{j=1}^{k}(\mathbb{Z}\pi)^{2}\cong\pi_{2}(M\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times
S^{2}))$ by an embedded framed sphere by tubing together parallel copies of
the embedded framed spheres $\\{s_{0}\\}\times S^{2}$ in $S^{2}\times S^{2}$.
Let $\\{f_{i}\\}_{i=1}^{k}$ be the resulting collection of spheres. These
spheres admit pairwise disjoint algebraically dual spheres
$\\{g_{i}\\}_{i=1}^{k}$, obtained by tubing together parallel copies of the
embedded framed spheres $S^{2}\times\\{s_{0}\\}$ in order to realise the rows
of the matrix $(\overline{A}^{T})^{-1}$. By the sphere embedding theorem
[FQ]*Theorem 5.1B[PRT20]*Theorem B there is a collection
$\\{\overline{f}_{i}\\}_{i=1}^{k}$ of topologically flat embedded spheres with
$\overline{f}_{i}$ homotopic to $f_{i}$ and such that the collection
$\\{\overline{f}_{i}\\}_{i=1}^{k}$ admits a geometrically dual collection of
(immersed) spheres $\\{\overline{g}_{i}\\}_{i=1}^{k}$.

Attach $3$-handles to $W^{\prime}$ along neighbourhoods of the spheres
$\\{\overline{f}_{i}\\}_{i=1}^{k}$ to obtain a bordism $W$ from $M$ to
$M^{\prime}$. Since we attached the handles of index 2 along trivial circles,
$\pi_{1}(M)\cong\pi_{1}(W)$. Since the $\\{\overline{f}_{i}\\}$ have
geometrically dual spheres, surgery along them does not change the fundamental
group and so we have $\pi_{1}(M^{\prime})\cong\pi_{1}(W)$. The handle chain
complex $C_{*}(\widetilde{W},\widetilde{N})$ is

$0\to
C_{3}(\widetilde{W},\widetilde{M})\cong\oplus^{k}\mathbb{Z}\pi\xrightarrow{A}C_{2}(\widetilde{W},\widetilde{M})\cong\oplus^{k}\mathbb{Z}\pi\to
0.$

Since $A$ is invertible, $H_{*}(\widetilde{W},\widetilde{M})=0$, and then by
duality $H_{*}(\widetilde{W},\widetilde{M^{\prime}})=0$ too. Therefore $W$ is
an $h$-cobordism. The torsion can be read off from the handle chain complex as
$\tau(W;M)=[A]=\varsigma$.

Finally, (3) is a consequence of (1) and (2). The proof follows [Milnor-
whitehead-torsion]*Theorem 11.3. Let $(W;M,M^{\prime})$ and
$(W^{\prime};M,M^{\prime\prime})$ be $h$-cobordisms over $M$ with torsion
$\varsigma$. By (2), there is a 4-manifold $N$ and an $h$-cobordism
$(W^{\prime\prime};M^{\prime},N)$ with torsion $-\varsigma$. By the additivity
of Whitehead torsion, $(W\cup_{M^{\prime}}W^{\prime\prime},M,N)$ is an
$s$-cobordism. Since $\pi$ is good, $W\cup_{M^{\prime}}W^{\prime\prime}$ is
homeomorphic to $M\times[0,1]$ relative to $M$ by (1). In particular $N$ is
homeomorphic to $M$. We can thus form
$(W^{\prime\prime}\cup_{M}W^{\prime};M^{\prime},M^{\prime\prime})$, which
again is an $s$-cobordism and is thus homeomorphic to $M^{\prime}\times[0,1]$,
relative to $M^{\prime}$ by (1). We obtain a homeomorphism

$W\cong W\cup_{M^{\prime}}(M^{\prime}\times[0,1])\cong
W\cup_{M^{\prime}}W^{\prime\prime}\cup_{M}W^{\prime}\cong(M\times[0,1])\cup_{M^{\prime}}W^{\prime}\cong
W^{\prime},$

relative to $M$, as claimed. ∎

######  Remark 3.7.

Every $5$-dimensional $s$-cobordism $(W;M,N)$, with no restriction on
fundamental groups, becomes homeomorphic to a product, relative to $M$, after
sufficiently many connected sums with $S^{2}\times S^{2}\times[0,1]$ along
arcs joining $M$ and $N$ [quinn-1983]*Theorem 1.1. This gives another proof of
Theorem 3.4 that $s$-cobordant $4$-manifolds are stably
diffeomorphic/homeomorphic.

The last remaining implication in Figure 1 is given by the following result of
Gompf.

######  Theorem 3.8 ([Gompf-stable]).

Smooth, orientable $4$-manifolds that are stably homeomorphic are also stably
diffeomorphic.

In the same paper Gompf also showed that smoothings of a nonorientable
$4$-manifold become diffeomorphic after connected sum with sufficiently many
copies of $S^{2}\mathbin{\widetilde{\times}}S^{2}$.

##  4\. Review of surgery exact sequences

In the next section we will appeal on several occasions to surgery exact
sequences. We refer to [Freedman-book-surgery] for an account of the
4-dimensional case, and e.g. [Wall-surgery-book] and [CLM] for detailed
treatments of general surgery theory in dimensions at least $5$.

The surgery exact sequences are centred on the _structure sets._ For the
remainder of this section, let $M$ be a closed, connected, topological
4-manifold.

######  Definition 4.1.

The _homotopy structure set_ of $M$, denoted $\mathcal{S}^{h}(M)$, is by
definition the set of pairs $(N,f\colon N\xrightarrow{\simeq}M)$, where $N$ is
a closed topological 4-manifold and $f$ is a homotopy equivalence, considered
up to $h$-cobordism over $M$. That is,
$[(N,f)]=[(N,f^{\prime})]\in\mathcal{S}^{h}(M)$ if and only if there is an
$h$-cobordism $(W;N,N^{\prime})$, with inclusion maps $i\colon N\to W$ and
$i^{\prime}\colon N^{\prime}\to W$, together with a map $F\colon W\to M$ such
that $F\circ i=f$ and $F\circ i^{\prime}=f^{\prime}$.

Note that $F$ is necessarily also a homotopy equivalence in the definition
above.

######  Definition 4.2.

The _simple structure set_ of $M$, denoted $\mathcal{S}^{s}(M)$, is by
definition the set of pairs $(N,f\colon N\xrightarrow{\simeq_{s}}M)$, where
$N$ is a closed topological 4-manifold and $f$ is a simple homotopy
equivalence, considered up to $s$-cobordism over $M$. That is,
$[(N,f)]=[(N,f^{\prime})]\in\mathcal{S}^{s}(M)$ if and only if there is an
$s$-cobordism $(W;N,N^{\prime})$, with inclusion maps $i\colon N\to W$ and
$i^{\prime}\colon N^{\prime}\to W$, together with a map $F\colon W\to M$ such
that $F\circ i=f$ and $F\circ i^{\prime}=f^{\prime}$.

Note that $F$ is necessarily a simple homotopy equivalence. Suppose that
$\pi_{1}(M)$ is a good group. Then every $s$-cobordism is a product, and we
can alternatively describe the equivalence relation in the definition of the
simple structure set without reference to $s$-cobordisms, by instead requiring
a homeomorphism $G\colon N\to N^{\prime}$ such that there is a homotopy
$f^{\prime}\circ G\sim f\colon N\to M$.

It follows that one approach to the classification of manifolds simple
homotopy equivalent to $M$, up to homeomorphism, is to first compute the
simple structure set of $M$, and then to compute the set of orbits of the
post-composition action on it by the group $\operatorname{hAut}^{s}(M)$ of
homotopy classes of simple self-homotopy equivalences of $M$. The set of
orbits is then the set of homeomorphism classes $\mathcal{M}(M)$ of manifolds
simple homotopy equivalent to $M$.

Similarly, the set $\mathcal{M}^{h}(M)$ of $h$-cobordism classes of closed
4-manifolds homotopy equivalent to $M$ is in bijective correspondence with the
orbits of $\mathcal{S}^{h}(M)$ under the action of the group
$\operatorname{hAut}(M)$ of homotopy classes of self-homotopy equivalences of
$M$.

The surgery sequences are exact sequences of abelian groups, whose underlying
sets are given as follows; the group structures arise via the theory of
spectra and are hard to define geometrically. We will explain the terms other
than the structure sets after stating the sequences. Let $\pi:=\pi_{1}(M)$ and
let $w\colon\pi\to C_{2}$ be the orientation character of $M$. Assume that
$\pi$ is a good group. For the homotopy structure set, we have an exact
sequence:

${\mathcal{N}(M\times[0,1],M\times\\{0,1\\})}$${L_{5}^{h}(\mathbb{Z}\pi,w)}$${\mathcal{S}^{h}(M)}$${\mathcal{N}(M)}$${L_{4}^{h}(\mathbb{Z}\pi,w),}$$\scriptstyle{\sigma}$$\scriptstyle{W}$$\scriptstyle{\eta}$$\scriptstyle{\sigma}$

while for the simple structure set we have an exact sequence:

${\mathcal{N}(M\times[0,1],M\times\\{0,1\\})}$${L_{5}^{s}(\mathbb{Z}\pi,w)}$${\mathcal{S}^{s}(M)}$${\mathcal{N}(M)}$${L_{4}^{s}(\mathbb{Z}\pi,w),}$$\scriptstyle{\sigma}$$\scriptstyle{W}$$\scriptstyle{\eta}$$\scriptstyle{\sigma}$

The _degree one normal maps_ , the terms involving $\mathcal{N}$, are
independent of the $h$ and $s$ decorations. For $(X,\partial X)$ equal to
either $(M,\emptyset)$ or $(M\times[0,1],M\times\\{0,1\\})$, the set
$\mathcal{N}(X,\partial X)$ consists of the set of manifolds $(N,\partial N)$
with a degree one map $N\to X$ that restricts to a homeomorphism on $\partial
N\to\partial X$, together with some normal bundle data that we will not define
here, up to an analogous notion of degree one normal bordism. Details can be
found in the references provided at the start of this section.

The _$L$ -groups_ $L_{5}^{h}(\mathbb{Z}\pi,w)$, $L_{5}^{s}(\mathbb{Z}\pi,w)$,
$L_{4}^{h}(\mathbb{Z}\pi,w)$, and $L_{4}^{s}(\mathbb{Z}\pi,w)$ have purely
algebraic definitions, in terms of $\pi$, $w$, and the decoration $h$ or $s$.
We will also not go into the details of the definitions here. Roughly
speaking, the $h$-decorated groups are defined in terms of nonsingular,
sesquilinear, hermitian forms on free $\mathbb{Z}\pi$-modules. For the
$s$-decorations the $\mathbb{Z}\pi$-modules must be based, and certain
isomorphisms between based modules are required to be simple, meaning that
they represent the trivial element of the Whitehead group
$\operatorname{Wh}(\pi)$. When the orientation character $w$ is trivial, we
often suppress $w$ from the notation of the $L$-group.

The homomorphisms $\sigma$ in the surgery sequences from degree one normal
maps to the $L$-groups are called the _surgery obstruction maps_. The maps
marked $W$ are given by the _Wall realisation_ actions of the $L_{5}$ groups
on the structure sets.

##  5\. Counterexamples

Figure 2 shows what we know about the converses of the implications in Figure
1. In this section we collect the counterexamples indicated in Figure 2,
explaining their construction and properties. The properties are also
collected in Table 1 at the end of the paper.

Figure 2. What is known about the converses of the implications in Figure 1.
The symbol $\nRightarrow$ denotes the cases where we know an implication does
not hold, indicating which subsections contain corresponding counterexamples.
The $\xRightarrow{?}$ symbol denotes the three cases where it is unknown
whether an implication holds.

###  5.1. $S^{2}\times S^{2}$ and $S^{4}$

  * •

Both manifolds are smooth, orientable, and simply connected.

  * •

As $S^{2}\times S^{2}\cong S^{4}\\#(S^{2}\times S^{2})$, they are stably
diffeomorphic, and therefore stably homeomorphic and $\mathbb{CP}^{2}$-stably
diffeomorphic.

  * •

$\chi(S^{2}\times S^{2})=4\neq 2=\chi(S^{4})$. Therefore, they are neither
(simple) homotopy equivalent, $s$\- nor $h$-cobordant (in either category),
homeomorphic, nor diffeomorphic.

This straightforward example shows that to meaningfully ask for homotopically
inequivalent 4-manifolds that are stably diffeomorphic, one should also
require that the Euler characteristics coincide. Equivalently, one should
require that the number of copies of $S^{2}\times S^{2}$ added is the same for
both manifolds, that is $s=t$ in Definition 2.2.

###  5.2. $S^{2}\times S^{2}$ and $S^{2}\mathbin{\widetilde{\times}}S^{2}$

As before $S^{2}\mathbin{\widetilde{\times}}S^{2}$ denotes the nontrivial
$S^{2}$-bundle over $S^{2}$. It can be constructed by gluing two copies of
$D^{2}\times S^{2}$ together using the Gluck twist on their common boundary
$S^{1}\times S^{2}$. Alternatively, recall that oriented $3$-plane bundles
over $S^{2}$ are classified up to isomorphism by homotopy classes of maps
$[S^{2},BSO(3)]\cong[S^{1},SO(3)]$, and there are two such homotopy classes.
The nontrivial map gives a $3$-plane bundle whose sphere bundle is
$S^{2}\widetilde{\times}S^{2}$.

  * •

Both $S^{2}\times S^{2}$ and $S^{2}\mathbin{\widetilde{\times}}S^{2}$ are
smooth, orientable, and simply connected.

  * •

$\chi(S^{2}\times S^{2})=4=\chi(S^{2}\mathbin{\widetilde{\times}}S^{2})$.

  * •

By the diffeomorphisms $(S^{2}\times
S^{2})\\#\mathbb{CP}^{2}\cong\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}\\#\mathbb{CP}^{2}\cong(S^{2}\mathbin{\widetilde{\times}}S^{2})\\#\mathbb{CP}^{2}$,
the manifolds are $\mathbb{CP}^{2}$-stably diffeomorphic and
$\mathbb{CP}^{2}$-stably homeomorphic.

  * •

The second Stiefel-Whitney classes are distinct, since $S^{2}\times S^{2}$ is
spin but $S^{2}\mathbin{\widetilde{\times}}S^{2}$ is not. So they are neither
(stably) diffeomorphic, nor (stably) homeomorphic, nor (simple) homotopy
equivalent, $s$\- nor $h$-cobordant (in either category).

This example shows that it is easy to find manifolds that are
$\mathbb{CP}^{2}$-stably homeomorphic or $\mathbb{CP}^{2}$-stably
diffeomorphic but do not satisfy any of the other equivalence relations.

###  5.3. $\mathbb{CP}^{2}$ and Freedman’s $\mathop{}\\!*\mathbb{CP}^{2}$

Freedman [F]*p. 370 constructed the manifold $\mathop{}\\!*\mathbb{CP}^{2}$,
which he called the _Chern manifold_ , as follows. Attach a $2$-handle to
$D^{4}$ along a $+1$-framed trefoil $K$, to obtain the _$1$ -trace_ of the
trefoil. The boundary is a homology sphere $\Sigma$, which bounds a compact,
contractible manifold $C$ [F]*Theorem 1.4’[FQ]*Corollary 9.3C. Cap off the
$1$-trace with $C$. The resulting closed $4$-manifold is
$\mathop{}\\!*\mathbb{CP}^{2}$. The same construction with any knot $K$ with
$\operatorname{Arf}(K)=1$ gives rise to a homeomorphic manifold.

  * •

The manifolds $\mathbb{CP}^{2}$ and $\mathop{}\\!*\mathbb{CP}^{2}$ are
orientable and simply connected.

  * •

$\mathbb{CP}^{2}$ is smooth. The Kirby-Siebenmann invariant
$\operatorname{ks}(\mathop{}\\!*\mathbb{CP}^{2})=\operatorname{ks}(C)=\mu(\Sigma)=\operatorname{Arf}(K)=1$,
where $\mu(\Sigma)$ is the Rochlin invariant of $\Sigma$, by [FQ]*page 165
[Gonzalez-Acuna]. Therefore, $\mathop{}\\!*\mathbb{CP}^{2}$ is not smoothable,
even stably.

  * •

The manifolds are homotopy equivalent [Whitehead-4-complexes, Milnor-simply-
connected-4-manifolds] and therefore have equal Euler characteristic
$\chi(\mathbb{CP}^{2})=3=\chi(\mathop{}\\!*\mathbb{CP}^{2})$.

  * •

Since the Kirby-Siebenmann invariants are different, they are not
$\mathbb{CP}^{2}$-stably homeomorphic and therefore not homeomorphic, not
$s$\- or $h$-cobordant, and not stably homeomorphic.

  * •

The smooth questions are not applicable to this pair.

This example shows that one must restrict to manifolds with the same Kirby-
Siebenmann invariant, and moreover ideally smooth manifolds, to find really
interesting examples of homotopy equivalent but not homeomorphic manifolds.

###  5.4. $\mathbb{RP}^{4}\\#\mathbb{CP}^{2}$ and
$\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$

Here $\mathop{}\\!*\mathbb{CP}^{2}$ is the Chern manifold from Section 5.3.
The manifold $\mathcal{R}$ was first constructed by Ruberman [ruberman84], as
follows. By [F]*Theorem 1.4’[FQ]*Corollary 9.3C, the Brieskorn sphere
$\Sigma(5,9,13)$ bounds a compact, contractible, topological $4$-manifold
$\mathcal{U}$. As a Seifert fibered manifold, $\Sigma(5,9,13)$ admits an
orientation preserving order two self-diffeomorphism $t$ given by the
antipodal map on the generic fibres. Since the parameters $5$, $9$, and $13$
are all odd, $t$ is also the antipodal map on the exceptional fibres and $t$
has no fixed points. The manifold $\mathcal{R}$ is then defined as

$\mathcal{R}:=\mathcal{U}/x\sim t(x)\text{ for }x\in\partial\mathcal{U}.$

Compare with the earlier construction of Fintushel-Stern [FS-fake], given in
Section 5.6. Ruberman’s proof that $\mathcal{R}$ is not homeomorphic to
$\mathbb{RP}^{4}$ utilises Rochlin’s theorem [Rochlin] (see also [freedman-
kirby][Kirby_lecturenotes]*Chapter XI and the fact that the Rochlin invariant
$\mu(\Sigma(5,9,13))=1$. That $\mathcal{R}$ is a homotopy $\mathbb{RP}^{4}$
follows from the same principle as in [FS-fake]. Namely, write $S^{4}$ as
$\mathcal{U}\cup_{t}\mathcal{U}$, and observe that $\mathcal{R}$ is the
quotient under a free involution. The same construction can be applied to any
integer homology sphere $\Sigma$ admitting a free orientation preserving
involution and with $\mu(\Sigma)=1$.

In the literature $\mathcal{R}$ is sometimes denoted
$\mathop{}\\!*\mathbb{RP}^{4}$. We prefer not to use this notation to avoid
confusion with the star construction, defined in Section 5.11. The manifold
$\mathcal{R}$ also arises via a surgery construction, which we now outline
following Hambleton-Kreck-Teichner [HKT-nonorientable]*pp. 650–1. There is a
degree one normal map, namely the collapse map $E_{8}\to S^{4}$, with domain
the $E_{8}$ manifold constructed by Freedman [F]*Theorem 1.7. Connect sum with
$\mathbb{RP}^{4}$ in both domain and codomain to obtain a degree one normal
map

$F\colon\mathbb{RP}^{4}\\#E_{8}\to\mathbb{RP}^{4}.$

The surgery kernel is the image of the $E_{8}$ form under the map
$L_{4}(\mathbb{Z})\to L_{4}(\mathbb{Z}[\mathbb{Z}/2],w)$, where
$w\colon\mathbb{Z}/2\to C_{2}$ is the nontrivial character. But this is the
zero map by [Wall-surgery-book]*Chapter 13A, pp. 172-3. This implies that,
perhaps after stabilising with copies of $S^{2}\times S^{2}$, the surgery
kernel is a hyperbolic form. Then by [PRT20]*Corollary 1.4 there is a
homeomorphism

$\mathbb{RP}^{4}\\#E_{8}\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times
S^{2})\cong X\\#\,\,{\scalebox{1.2}{\\#}^{4+k}}(S^{2}\times S^{2})$

for some integer $k$ and for some $4$-manifold $X$. By additivity of the
Kirby-Siebenmann invariant (see e.g. [guide]*Theorem 8.2)

$\operatorname{ks}(X)=\operatorname{ks}(X\\#\,\,{\scalebox{1.2}{\\#}^{4+k}}(S^{2}\times
S^{2}))=\operatorname{ks}(\mathbb{RP}^{4}\\#E_{8}\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times
S^{2}))=\operatorname{ks}(E_{8})=1,$

whereas $\mathbb{RP}^{4}$ is smooth and so
$\operatorname{ks}(\mathbb{RP}^{4})=0$. We can then define

$\mathcal{R}:=X.$

By [HKT-nonorientable]*Theorem 3 the manifold $X$ is determined up to
homeomorphism by this construction, and in particular coincides with
Ruberman’s construction above. Since we built $\mathcal{R}$ by killing the
surgery kernel of a degree one normal map, we have constructed an element of
the structure set,
$[\mathcal{R}\to\mathbb{RP}^{4}]\in\mathcal{S}(\mathbb{RP}^{4})$, and in
particular $\mathcal{R}\simeq\mathbb{RP}^{4}$.

  * •

The manifolds $\mathbb{RP}^{4}\\#\mathbb{CP}^{2}$ and
$\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$ are nonorientable with nontrivial
fundamental group isomorphic to $\mathbb{Z}/2$.

  * •

$\mathbb{RP}^{4}\\#\mathbb{CP}^{2}$ is smooth by construction. Ruberman-Stern
[Ruberman-Stern] showed that $\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$ is
also smoothable. Namely, let $K$ be a knot with $S^{3}_{1}(K)=\partial
X_{1}(K)=\Sigma(5,9,13)$ a Brieskorn sphere and let $t$ be a free orientation
preserving involution of $\Sigma(5,9,13)$ as described at the beginning of
this section. Construct the smooth $4$-manifold

$X:=X_{1}(K)/x\sim t(x)\text{ for }x\in\partial X_{1}(K).$

To see that $X$ is homeomorphic to
$\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$, let $\mathcal{U}_{K}$ be the
compact, contractible, topological $4$-manifold with boundary $\Sigma(5,9,13)$
provided by [F]*Theorem 1.4’[FQ]*Corollary 9.3C. By Freedman [F] and Boyer
[Boyer1], there is a homeomorphism

$\mathop{}\\!*\mathbb{CP}^{2}\\#\mathcal{U}_{K}\cong(X_{1}(K)\cup_{\Sigma(5,9,13)}\mathcal{U}_{K})\\#\mathcal{U}_{K}\cong
X_{1}(K).$

This homeomorphism descends to a homeomorphism
$\mathop{}\\!*\mathbb{CP}^{2}\\#\mathcal{R}\cong X$ when quotienting the
boundaries by the involution. Alternatively, one can apply the classification
theorem of [HKT-nonorientable] to show that $X$ is homeomorphic to
$\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$. Different choices of $K$, and
thereby the Brieskorn sphere, might give rise to different smooth structures
on $\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$, but this is currently an open
question.

  * •

The manifolds $\mathbb{RP}^{4}\\#\mathbb{CP}^{2}$ and
$\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2}$ are homotopy equivalent, and
therefore have equal Euler characteristic
$\chi(\mathbb{RP}^{4}\\#\mathbb{CP}^{2})=2=\chi(\mathcal{R}\\#\mathop{}\\!*\mathbb{CP}^{2})$.

  * •

Since they are homotopy equivalent and smoothable, they are
$\mathbb{CP}^{2}$-stably homeomorphic and $\mathbb{CP}^{2}$-stably
diffeomorphic.

  * •

Hambleton-Kreck-Teichner [HKT-nonorientable] showed that they are stably
homeomorphic, but not homeomorphic.

  * •

The manifolds are stably diffeomorphic by [guide]*Theorem 12.3, which states
that smooth, nonorientable, compact 4-manifolds with universal cover non-spin,
that are stably homeomorphic, are stably diffeomorphic.

  * •

They are not $s$-cobordant, in either category, since if they were they would
be homeomorphic by the topological $s$-cobordism theorem (Theorem 3.5). Since
the Whitehead group of $\mathbb{Z}/2$ is trivial, they are also not
$h$-cobordant in either category.

###  5.5. Kreck’s examples $K3\\#\mathbb{RP}^{4}$ and $\\#^{11}(S^{2}\times
S^{2})\\#\mathbb{RP}^{4}$

These are a pair of relatively easy to understand exotic 4-manifolds
discovered by Kreck [Kreck-exotic]. Indeed this was the first known exotic
pair of closed 4-manifolds; the examples of Cappell-Shaneson discussed in
Section 5.6 were constructed earlier, but they were not shown to be
homeomorphic until much later [HKT-nonorientable]. The $K3$ surface is a well-
known smooth, simply connected 4-manifold. One way to construct it is to first
consider $E(1):=\mathbb{CP}^{2}\\#9\overline{\mathbb{CP}^{2}}$, which comes
with an _elliptic fibration_ $F\colon E(1)\to S^{2}$ (be warned that this is
not a Serre or Hurewicz fibration). Generically the point inverse images are
tori with trivial normal bundles $T^{2}\times D^{2}$. The _fibre sum_ of two
copies of $E(1)$ is the $K3$ surface:

$K3:=E(2):=E(1)\\#_{T^{2}}E(1)=(E(1)\setminus
T^{2}\times\mathring{D}^{2})\cup_{T^{2}\times S^{1}}(E(1)\setminus
T^{2}\times\mathring{D}^{2}).$

We can then construct the manifolds $K3\\#\mathbb{RP}^{4}$ and
$\\#^{11}(S^{2}\times S^{2})\\#\mathbb{RP}^{4}$.

  * •

The manifolds $K3\\#\mathbb{RP}^{4}$ and $\\#^{11}(S^{2}\times
S^{2})\\#\mathbb{RP}^{4}$ are smooth, nonorientable, and have nontrivial
fundamental group isomorphic to $\mathbb{Z}/2$. They have equal Euler
characteristics $\chi(K3\\#\mathbb{RP}^{4})=23=\chi(\\#^{11}(S^{2}\times
S^{2})\\#\mathbb{RP}^{4})$.

  * •

As we will explain below, Kreck showed that $K3\\#\mathbb{RP}^{4}$ and
$\\#^{11}(S^{2}\times S^{2})\\#\mathbb{RP}^{4}$ are homeomorphic but not
stably diffeomorphic.

  * •

Since they are homeomorphic, they are stably homeomorphic,
$\mathbb{CP}^{2}$-stably homeomorphic, homotopy equivalent, simple homotopy
equivalent, and topologically $h$\- and $s$-cobordant.

  * •

Since they are not stably diffeomorphic, the two 4-manifolds are not
diffeomorphic, and neither smoothly $h$-cobordant nor smoothly $s$-cobordant.

######  Remark 5.1.

More generally, Kreck [Kreck-exotic]*Theorem 1 showed that there is at least
one such example of a pair of homeomorphic but not stably diffeomorphic smooth
4-manifolds for each _1-type_ $(\pi,w\colon\pi\to C_{2})$ with $\pi$ a
finitely presented group and $w$ nontrivial. By Gompf’s result (Theorem 3.8),
orientable (stably) homeomorphic 4-manifolds are stably diffeomorphic, so this
phenomenon only arises for nonorientable manifolds.

Now we argue why $K3\\#\mathbb{RP}^{4}$ and $\\#^{11}(S^{2}\times
S^{2})\\#\mathbb{RP}^{4}$ are homeomorphic. The intersection form of $K3$ is
isometric to the orthogonal sum of two $E_{8}$ forms and a rank 6 hyperbolic
form. By the classification of closed, simply connected 4-manifolds up to
homeomorphism [F, FQ], it follows that $K3$ is homeomorphic to
$E_{8}\\#E_{8}\\#^{3}(S^{2}\times S^{2})$, where $E_{8}$ as before denotes the
$E_{8}$ manifold constructed by Freedman [F]*Theorem 1.7. Next, observe that
there is a unique connected sum of an orientable manifold with a nonorientable
manifold such as $\mathbb{RP}^{4}$, because any two embeddings of $D^{4}$ in
$\mathbb{RP}^{4}$ are isotopic. Using this we have homeomorphisms

$\displaystyle\mathbb{RP}^{4}\\#K3$
$\displaystyle\cong\mathbb{RP}^{4}\\#E_{8}\\#E_{8}\\#^{3}(S^{2}\times S^{2})$
$\displaystyle\cong\mathbb{RP}^{4}\\#\overline{E_{8}}\\#E_{8}\\#^{3}(S^{2}\times
S^{2})$ $\displaystyle\cong\mathbb{RP}^{4}\\#^{11}(S^{2}\times S^{2}).$

The last homeomorphism uses the classification of closed, simply connected
4-manifolds again. Here the 4-manifolds
$\overline{E_{8}}\\#E_{8}\\#^{3}(S^{2}\times S^{2})$ and $\\#^{11}(S^{2}\times
S^{2})$ have isometric intersection forms, since they are indefinite and have
the same rank, parity, and signature [MH]*Theorem 5.3.

Next we explain the obstruction to stable diffeomorphism. Heuristically, the
construction of a homeomorphism above does not work smoothly, because one
cannot split the $K3$ surface smoothly, because of Rochlin’s theorem that
smooth, spin, closed, 4-manifolds have signature divisible by 16 [Rochlin]
(see also [freedman-kirby][Kirby_lecturenotes]*Chapter XI. When necessary we
use $\mathbb{RP}^{\infty}\simeq B\mathbb{Z}/2=K(\mathbb{Z}/2,1)$ as a model
for the Eilenberg-Maclane space. Define

$B:=B\mathbb{Z}/2\times\operatorname{BSpin}$

and let $\xi\colon B\to BO$ be the composition

$B=B\mathbb{Z}/2\times\operatorname{BSpin}\xrightarrow{\gamma^{\perp}\times
p}BO\times BO\xrightarrow{\oplus}BO,$

where $\gamma^{\perp}$ is the orthogonal complement to the tautological line
bundle $\gamma$ over $B\mathbb{Z}/2\simeq\mathbb{RP}^{\infty}$,
$p\colon\operatorname{BSpin}\to BO$ is the standard projection, and $\oplus$
is the map corresponding to the Whitney sum of stable bundles. Replace $\xi$
by a fibration, and by an abuse of notation denote the resulting homotopy
equivalent domain by the same letter $B$ and the new map to $BO$ again by
$\xi$.

Let $\Omega_{4}(B,\xi)$ denote the group of bordism classes of closed, smooth
4-manifolds, equipped with a lift to $B$ of the classifying map $\nu_{M}\colon
M\to BO$ of the stable normal bundle. In other words, $\Omega_{4}(B,\xi)$ has
elements represented by pairs $(M,\widetilde{\nu}_{M})$, where $M$ is a
closed, smooth $4$-manifold and $\nu_{M}$ classifies its stable normal bundle,
such that the following diagram commutes.

${B}$${M}$${BO.}$$\scriptstyle{\xi}$$\scriptstyle{\widetilde{\nu}_{M}}$$\scriptstyle{\nu_{M}}$

A bordism between $(M,\widetilde{\nu}_{M})$ and $(N,\widetilde{\nu}_{N})$ is a
compact, smooth 5-manifold $(W,\widetilde{\nu}_{W})$ with a corresponding lift
of the stable normal bundle $\nu_{W}\colon W\to BO$ to $B$, and a
diffeomorphism $i_{M}\sqcup i_{N}\colon M\sqcup N\xrightarrow{\cong}\partial
W$ such that $\widetilde{\nu}_{W}\circ i_{M}=\widetilde{\nu}_{M}$ and
$\widetilde{\nu}_{W}\circ i_{N}=\widetilde{\nu}_{N}$.

For $M\in\\{K3\\#\mathbb{RP}^{4},\\#^{11}(S^{2}\times
S^{2})\\#\mathbb{RP}^{4}\\}$, Kreck showed that $M$ admits a 2-connected lift
$\widetilde{\nu}_{M}\colon M\to B$. Roughly speaking, this is because
$(B,\xi)$ was chosen to be compatible with the Stiefel-Whitney classes
$w_{1}(\nu_{M})$ and $w_{2}(\nu_{M})$, and also with the fundamental group
$\pi_{1}(M)$. Such a lift $\widetilde{\nu}_{M}$ is called a _normal
1-smoothing_. The following theorem is a special case of
[surgeryandduality]*Theorem C.

######  Theorem 5.2 (Kreck [Kreck-exotic]).

Let $(M,\widetilde{\nu}_{M}),(N,\widetilde{\nu}_{N})\in\Omega_{4}(B,\xi)$ with
$\widetilde{\nu}_{M}$ and $\widetilde{\nu}_{N}$ 2-connected. Then $M$ and $N$
are stably diffeomorphic if and only if

$[(M,\widetilde{\nu}_{M})]=[(N,\widetilde{\nu}_{N})]\in\Omega_{4}(B,\xi)/\operatorname{hAut}(B,\xi),$

where $\operatorname{hAut}(B,\xi)$ denotes the group of fibre homotopy classes
of fibre homotopy equivalences of the fibration $\xi\colon B\to BO$.

Kreck showed [Kreck-exotic]*Proposition 2 that there is an isomorphism

$\alpha\colon\Omega_{4}(B,\xi)\xrightarrow{\cong}\mathbb{Z}/16,$ (5.1)

with $\alpha(\\#^{11}(S^{2}\times S^{2}),s)=0$ and $\alpha(K3,s)=8$. Here
these are simply-connected 4-manifolds, and the maps $s$ factor through
$\\{*\\}\times\operatorname{BSpin}$, and correspond to the unique spin
structures on the respective 4-manifolds. The automorphism group
$\operatorname{hAut}(B,\xi)$ acts by isomorphisms of
$\Omega_{4}(B,\xi)\cong\mathbb{Z}/16$ and so preserves these two elements.
Since $\alpha$ is a homomorphism and the connected sum is bordant to the
disjoint union, it follows that $K3\\#\mathbb{RP}^{4}$ and
$\\#^{11}(S^{2}\times S^{2})\\#\mathbb{RP}^{4}$ are distinct in
$\Omega_{4}(B,\xi)/\operatorname{hAut}(B,\xi)$ and are therefore not stably
diffeomorphic. Next we proceed to explain the computation of
$\Omega_{4}(B,\xi)$ and the isomorphism $\alpha$ in a little more detail.

The group $\Omega_{4}(B,\xi)$ can in principle be computed via the Atiyah-
Hirzebruch spectral sequence. The key term on the $E^{2}$ page lies in
$H_{0}(B\mathbb{Z}/2;(\Omega_{4}^{\operatorname{Spin}})^{w})\cong
H_{0}(B\mathbb{Z}/2;\mathbb{Z}^{w})\cong\mathbb{Z}/2$, with coefficients
twisted by the isomorphism
$w\colon\mathbb{Z}/2=\pi_{1}(B\mathbb{Z}/2)\to\\{\pm
1\\}=\operatorname{Aut}(\mathbb{Z})\cong\operatorname{Aut}(\Omega_{4}^{\operatorname{Spin}})$.
The aim is to prove that the $K3$ surface is nontrivial in
$H_{0}(B\mathbb{Z}/2;(\Omega_{4}^{\operatorname{Spin}})^{w})$. For this Kreck
constructed the map $\alpha$ as an explicit invariant of elements of
$\Omega_{4}(B,\xi)$, similar in character to the Cappell-Shaneson invariant
discussed in Section 5.6 below.

Represent an element of $\Omega_{4}(B,\xi)$ by a map $f\colon
M\to\mathbb{RP}^{N}\times\operatorname{BSpin}$, for some large $N\geq 5$,
using the cellular approximation theorem and the fact that
$B\mathbb{Z}/2\cong\mathbb{RP}^{\infty}$. Let
$\operatorname{pr}_{1}\colon\mathbb{RP}^{N}\times\operatorname{BSpin}\to\mathbb{RP}^{N}$
be the projection, and make $\operatorname{pr}_{1}\circ f$ transverse to
$\mathbb{RP}^{N-1}\subseteq\mathbb{RP}^{N}$. Let $F\subseteq M$ be the inverse
image $f^{-1}(\mathbb{RP}^{N-1}\times\operatorname{BSpin})$, which is a closed
3-manifold. The restriction of $f$ to $F$ determines, with a little work, an
induced spin structure on $F$ and a map
$F\to\mathbb{RP}^{N-1}\subseteq\mathbb{RP}^{\infty}\cong B\mathbb{Z}/2$. So
$F$ determines an element of
$\Omega_{3}^{\operatorname{Spin}}(B\mathbb{Z}/2)$.

Kreck showed in [Kreck-exotic]*Proposition 3 that
$\Omega_{3}^{\operatorname{Spin}}(B\mathbb{Z}/2)\cong\mathbb{Z}/8$, generated
by the pair consisting of $(\mathbb{RP}^{3},\operatorname{inc}\times
s\colon\mathbb{RP}^{3}\to\mathbb{RP}^{\infty}\times\operatorname{BSpin})$,
where $\operatorname{inc}$ is the inclusion and $s$ is some choice of spin
structure. The 3-manifold $\mathbb{RP}^{3}$ bounds the 4-manifold $V$ obtained
by adding a 2-handle to $D^{4}$ along an unknot with framing coefficient 2. A
generator of $\pi_{2}(V)$ is represented by an embedded 2-sphere. Taking the
2-fold cover of $V$ branched along such an embedding yields a branched
covering $\widehat{V}\to V$ restricting to the standard nontrivial double
cover $S^{3}\cong\partial\widehat{V}\to\mathbb{RP}^{3}\cong\partial V$ on the
boundary. It follows that every element $(F,f)$ of
$\Omega_{3}^{\operatorname{Spin}}(B\mathbb{Z}/2)$ bounds some 4-manifold $W$
which admits a branched double covering $\widehat{W}\to W$ restricting to the
double cover $\widehat{F}\to F$ corresponding to $f$.

Make a choice of such a bounding 4-manifold $W$. Let
$\tau\colon\widehat{W}\to\widehat{W}$ be the deck transformation, and let
$\operatorname{Fix}\tau\subseteq\widehat{W}$ denote the branching set. Also
let $M_{F}:=M\setminus\nu F$ be the complement of a tubular neighbourhood of
$F$. Then we define a precursor of the invariant from (5.1), as

$\overline{\alpha}(M,f):=\operatorname{sign}(M_{F})-\operatorname{sign}(W)-\operatorname{Fix}\tau\cdot\operatorname{Fix}\tau\in\mathbb{Z}/32,$

where $\operatorname{Fix}\tau\cdot\operatorname{Fix}\tau$ denotes the
homological self-intersection number of the $2$-manifold
$\operatorname{Fix}\tau\subseteq\widehat{W}$. Kreck showed that
$\overline{\alpha}$ is a well-defined invariant of $\Omega_{4}(B,\xi)$, namely
it is unaffected by cobordism over $B$ and is independent of the choice of
$(W,\tau)$. This uses the Atiyah-Singer $G$-signature theorem [Atiyah-Singer-
G-sign]*Section 6 (see also [Gordon-G-signature]) and Rochlin’s theorem
[Rochlin] (see also [freedman-kirby][Kirby_lecturenotes]*Chapter XI. Thus as
alluded to above the exotic behaviour can be ultimately be traced back to
Rochlin’s theorem.

Kreck also showed [Kreck-exotic]*p. 256 that
$\operatorname{Im}\overline{\alpha}=2\mathbb{Z}/32\cong\mathbb{Z}/16$,
generated by
$\overline{\alpha}(\mathbb{RP}^{4},\widetilde{\nu}_{\mathbb{RP}^{4}})$ for
some normal smoothing
$\widetilde{\nu}_{\mathbb{RP}^{4}}\colon\mathbb{RP}^{4}\to B$. In this case
$F=\mathbb{RP}^{3}$, $W=V$ from above,
$\operatorname{sign}(M_{F})=\operatorname{sign}(D^{4})=0$,
$\operatorname{sign}(W)=1$, and $\widehat{W}=\widehat{V}$ so
$\operatorname{Fix}\tau\cdot\operatorname{Fix}\tau=1$, so
$\overline{\alpha}(\mathbb{RP}^{4},f)=-2$. Kreck also showed by analysing the
Atiyah-Hirzebruch spectral sequence alluded to above that
$|\Omega_{4}(B,\xi)|\leq 16$. It follows that $\overline{\alpha}$ is an
isomorphism onto $2\mathbb{Z}/32\subseteq\mathbb{Z}/32$. Finally,
$\overline{\alpha}(K3,s)=16$ since $F=\emptyset$ and so $M_{F}=K3$. Since
$\operatorname{Im}\overline{\alpha}=2\mathbb{Z}/32$ we may define
$\alpha:=\overline{\alpha}/2\colon\Omega_{4}(B,\xi)\xrightarrow{}\mathbb{Z}/16$,
which gives the isomorphism $\alpha$ that we have been trying to explain.

###  5.6. $\mathbb{RP}^{4}$ and the Cappell-Shaneson fake $\mathbb{RP}^{4}$

Cappell and Shaneson [CS-new-four-mflds-bams, CS-new-four-mflds-annals]
constructed a smooth manifold $R$ which they showed is homotopy equivalent,
but not stably diffeomorphic, to $\mathbb{RP}^{4}$; later work [HKT-
nonorientable] showed that $R$ is homeomorphic to $\mathbb{RP}^{4}$. Fintushel
and Stern [Fintushel-Stern-exotic-free] constructed a smooth manifold $R_{FS}$
with the same properties as a quotient of $S^{4}$ by an exotic free action of
$\mathbb{Z}/2$ on $S^{4}$. See [Aitchison-Rubinstein]*Theorem 5.1 for the
relationship between the two constructions.

The Fintushel-Stern construction is easier to describe, so we start with that.
Start with two copies of a compact, contractible, smooth $4$-manifold $U$ with
boundary the Brieskorn homology sphere $\partial U\cong\Sigma(3,5,19)$. This
homology sphere is a Seifert-fibred $3$-manifold. The antipodal map on the
$S^{1}$ fibres induces a fixed-point free, order-two self-diffeomorphism
$t\colon\partial U\to\partial U$, as in Section 5.4. Fintushel-Stern showed
that there is a diffeomorphism $S^{4}\cong U\cup_{t}U$. Switching the two $U$
factors of the union gives rise to a smooth free involution
$\widetilde{t}\colon S^{4}\to S^{4}$. Define

$\displaystyle R_{FS}$ $\displaystyle:=S^{4}/\widetilde{t}$
$\displaystyle\equiv U/x\sim t(x)\text{ for }x\in\partial U.$

This inspired the construction of $\mathcal{R}$ in [ruberman84] given in
Section 5.4.

We will recall the construction of the Cappell-Shaneson $R$ presently. Before
doing so, we describe the diffeomorphism obstruction from [CS-new-four-mflds-
annals]. Consider the nonorientable linear $S^{3}$-bundle over $S^{1}$, which
we denote by $S^{1}\mathbin{\widetilde{\times}}S^{3}$. Cappell-Shaneson
defined an invariant

$\alpha\colon\mathcal{N}(S^{1}\mathbin{\widetilde{\times}}S^{3})\to\mathbb{Z}/32$

from the set of _smooth_ degree one normal maps with target
$S^{1}\mathbin{\widetilde{\times}}S^{3}$. Let $f\colon R\to
S^{1}\mathbin{\widetilde{\times}}S^{3}$ represent an element of
$\mathcal{N}(S^{1}\mathbin{\widetilde{\times}}S^{3})$. Define $\Sigma_{R}$ to
be a framed $3$-manifold $f^{-1}(S^{3})$ obtained by making $f$ transverse to
a fibre $S^{3}$ in $S^{1}\mathbin{\widetilde{\times}}S^{3}$, taking the
inverse image, and pulling back the framing. Let
$\mu(\Sigma_{R})\in\mathbb{Z}/16$ be the Rochlin invariant of $\Sigma_{R}$, by
definition the signature mod 16 of a framed 4-manifold with boundary
$\Sigma_{R}$. In addition, write $W:=R\setminus\nu\Sigma_{R}$, and let
$\sigma(W)\in\mathbb{Z}$ be its signature. Then Cappell-Shaneson defined

$\alpha(R,f):=2\mu(\Sigma_{R})-\sigma(W)\mod{32}.$

In [CS-new-four-mflds-annals]*Proposition 2.1 they showed that $\alpha$ is a
well-defined map
$\alpha\colon\mathcal{N}(S^{1}\mathbin{\widetilde{\times}}S^{3})\to\mathbb{Z}/32$
as claimed. This invariant will be used again in Section 5.12; see also
Kreck’s invariant in Section 5.5, which was inspired by the Cappell-Shaneson
invariant.

Cappell-Shaneson constructed $R$ as follows. Let

$A:=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ -1&1&0\end{pmatrix}.$

The matrix $A$ induces a diffeomorphism $\phi_{A}\colon T^{3}\to T^{3}$. Let
$M_{A}$ be the mapping torus. Then [CS-new-four-mflds-annals]*Propositions 2.1
and 2.2 show that there is a degree one normal map $f\colon M_{A}\to
S^{1}\mathbin{\widetilde{\times}}S^{3}$ and that $\alpha(M_{A},f)$ is nonzero.
As such, just as in Section 5.5, the exotic behaviour is due to Rochlin’s
theorem. Now consider the punctured 3-torus $T^{3}_{0}$ and the corresponding
mapping torus for $A$, denoted $M_{A,0}$. Take $\mathbb{RP}^{4}$ and remove a
neighbourhood $S^{1}\mathbin{\widetilde{\times}}D^{3}$ of an embedded circle
representing the generator of $\pi_{1}(\mathbb{RP}^{4})\cong\mathbb{Z}/2$.
Then

$R:=(\mathbb{RP}^{4}\setminus
S^{1}\mathbin{\widetilde{\times}}D^{3})\cup_{\partial}M_{A,0}.$

In [CS-new-four-mflds-annals]*Theorem 3.1, Cappell and Shaneson use the
nontriviality of $\alpha(M_{A},f)$ to show that $R$ is not diffeomorphic to
$\mathbb{RP}^{4}$. This theorem shows that there is a homotopy equivalence
$h\colon R\to\mathbb{RP}^{4}$ that is not homotopic to a diffeomorphism. Then
since every homotopy equivalence of $\mathbb{RP}^{4}$ is homotopic to the
identity (cf. Proposition 5.3), it follows that there is no diffeomorphism
between $R$ and $\mathbb{RP}^{4}$. It was shown in [gompf-killing] that the
double cover of $R$ is diffeomorphic to $S^{4}$.

  * •

The manifolds $R$ and $\mathbb{RP}^{4}$ are smooth and nonorientable. They
have nontrivial fundamental group isomorphic to $\mathbb{Z}/2$.

  * •

They have equal Euler characteristics $\chi(R)=1=\chi(\mathbb{RP}^{4})$.

  * •

Cappell-Shaneson [CS-new-four-mflds-annals] showed that $R$ and
$\mathbb{RP}^{4}$ are homotopy equivalent.

  * •

After Freedman’s work it was later shown [HKT-nonorientable] that these
manifolds are furthermore homeomorphic.

  * •

By combining with their earlier work on $4$-dimensional surgery [CS-
on-4-d-surgery]*Theorem 2.4, Cappell-Shaneson [CS-new-four-mflds-annals]*p. 61
showed that, in addition to not being diffeomorphic, $R$ and $\mathbb{RP}^{4}$
are not stably diffeomorphic. Alternatively, this can be seen using the
$\alpha$ invariant, which turns out to be a stable diffeomorphism invariant,
via the reduction of stable diffeomorphism to bordism over the normal 1-type.
We refer to Section 5.12, page • ‣ 5.12, where we explain this in more detail
in the case of Akbulut’s examples.

  * •

Since they are homeomorphic they are stably homeomorphic, simple homotopy
equivalent, topologically $h$\- and $s$-cobordant, $\mathbb{CP}^{2}$-stably
homeomorphic and $\mathbb{CP}^{2}$-stably diffeomorphic.

  * •

Since they are not stably diffeomorphic they are not diffeomorphic. They are
not smoothly $h$\- or $s$-cobordant because they are not stably diffeomorphic.

  * •

As mentioned above, the Fintushel-Stern 4-manifold $R_{FS}$ has exactly the
same properties.

The Cappell-Shaneson construction can be varied by making different choices
for the matrix $A$, giving rise to an exotic $\mathbb{RP}^{4}$ denoted
$R_{A}$. The precise conditions are that $A\in\operatorname{GL}(3,\mathbb{Z})$
with $\det(A)=-1$ and $\det(I-A^{2})=\pm 1$; call such a matrix a _Cappell-
Shaneson matrix_. It can be seen from the construction that if $A$ and
$A^{\prime}$ are similar Cappell-Shaneson matrices then $R_{A}$ and
$R_{A^{\prime}}$ are diffeomorphic. The universal cover of $R_{A}$ is a smooth
homotopy 4-sphere. Many of these have been shown to be diffeomorphic to
$S^{4}$ – we refer to the introduction of [Kim-Yamada] for a detailed survey.
The most general result in this vein is that the universal cover of $R_{A}$
arising from a Cappell-Shaneson matrix $A$ with trace $n$ where $-64\leq n\leq
69$ is diffeomorphic to $S^{4}$ [ibid.]. However, this corresponds to only
finitely many similarity classes, and so there are infinitely many Cappell-
Shaneson homotopy $4$-spheres which remain as potential counter-examples to
the smooth 4-dimensional Poincaré conjecture.

###  5.7. $L\times S^{1}$ and $L^{\prime}\times S^{1}$

Let $L$ and $L^{\prime}$ be 3-dimensional lens spaces with the same
fundamental group that are homotopy equivalent but not homeomorphic. These are
of the form $L_{p,q_{1}}$ and $L_{p,q_{2}}$ with $\gcd(p,q_{i})=1$, and $1\leq
q_{i}<p$ for $i=1,2$, such that for some $n$, $q_{1}q_{2}\equiv\pm
n^{2}\mod{p}$ (for homotopy equivalent), and $q_{i}\not\equiv\pm q_{j}^{\pm
1}\mod{p}$ (for non-homeomorphic) [reidemeister]. See also [brody, davis-kirk,
Cohen].

  * •

The manifolds $L\times S^{1}$ and $L^{\prime}\times S^{1}$ are smooth and
orientable. They have fundamental group isomorphic to
$\mathbb{Z}/p\times\mathbb{Z}$, for some $p$.

  * •

They are homotopy equivalent because $L$ and $L^{\prime}$ are, and since they
are smooth they are therefore $\mathbb{CP}^{2}$-stably homeomorphic and
$\mathbb{CP}^{2}$-stably diffeomorphic.

  * •

The formula for the Whitehead torsion of a product of homotopy equivalences
[kwun-szczarba]*Corollary 1.3[lueck-surgery-intro] implies that $L\times
S^{1}$ and $L^{\prime}\times S^{1}$ are simple homotopy equivalent. Indeed,
let $f\colon L\to L^{\prime}$ be a homotopy equivalence. Let
$i\colon\mathbb{Z}\to\mathbb{Z}/p\times\mathbb{Z}$ and let
$j\colon\mathbb{Z}/p\to\mathbb{Z}/p\times\mathbb{Z}$ be the standard
inclusions. Then

$\tau(f\times\operatorname{Id})=j_{*}(\tau(f))\cdot\chi(S^{1})+\chi(L)\cdot
i_{*}(\tau(\operatorname{Id}))=0,$ (5.2)

since $\chi(S^{1})=0=\chi(L)$, where $i_{*}$ and $j_{*}$ are the induced maps
on Whitehead groups.

  * •

In the late 1960s it was proven that $L\times S^{1}$ and $L^{\prime}\times
S^{1}$ are not diffeomorphic. If they were, then $L$ and $L^{\prime}$ would be
smoothly $h$-cobordant, which can be seen by embedding $L$ in the infinite
cyclic cover $L^{\prime}\times\mathbb{R}$. Atiyah-Bott [Atiyah-Bott]*Theorem
7.27 and Milnor [Milnor-whitehead-torsion]*Corollary 12.12 showed that
smoothly $h$-cobordant lens spaces are homeomorphic. Therefore $L\times S^{1}$
and $L^{\prime}\times S^{1}$ are not diffeomorphic.

  * •

In the late 1980s, Turaev [Turaev1988] showed that moreover $L\times S^{1}$
and $L^{\prime}\times S^{1}$ are not homeomorphic. He showed that for any
3-manifolds $M$ and $M^{\prime}$ that do not fibre over $S^{1}$ with periodic
monodromy, the product $M\times S^{1}$ and $M^{\prime}\times S^{1}$ are
homeomorphic if and only if $M$ and $M^{\prime}$ are homeomorphic
[Turaev1988]*Theorem 1.5. To show this he proved that such $M$ and
$M^{\prime}$ are topologically $h$-cobordant if and only if $M$ and
$M^{\prime}$ are homeomorphic. It follows that $L\times S^{1}$ and
$L^{\prime}\times S^{1}$ are not homeomorphic. Of course this also reproves
that they are not diffeomorphic.

  * •

The manifolds $L\times S^{1}$ and $L^{\prime}\times S^{1}$ are stably
diffeomorphic, and therefore stably homeomorphic. We prove this in Proposition
5.3 below, by adapting a proof of Cappell-Shaneson [Cappell-Shaneson-
nonlinear-similarity]. That this is possible was stated in [Weinberger-higher-
rho].

  * •

By the topological $s$-cobordism theorem (Theorem 3.5), $L\times S^{1}$ and
$L^{\prime}\times S^{1}$ are not topologically $s$-cobordant and are therefore
not smoothly $s$-cobordant. Here we use that $\mathbb{Z}/p\times\mathbb{Z}$ is
a good group.

  * •

They are not topologically $h$-cobordant, as we explain in Proposition 5.4
using [Kwasik-Schultz]. It follows that they are not smoothly $h$-cobordant.

  * •

By choosing a large enough value of $p$, one may use the classification of
lens spaces to find arbitrarily large, finite sets $\\{L_{i}\times
S^{1}\\}_{i}$, such that the elements pairwise satisfy the above properties.

######  Proposition 5.3.

Let $L$ and $L^{\prime}$ be 3-dimensional lens spaces that are homotopy
equivalent but not homeomorphic. Then the $4$-manifolds $L\times S^{1}$ and
$L^{\prime}\times S^{1}$ are stably diffeomorphic.

###### Proof.

The strategy is as follows. Let
$\pi_{1}(L)\cong\pi_{1}(L^{\prime})\cong\mathbb{Z}/p$ and let $h\colon L\to
L^{\prime}$ be a homotopy equivalence. Let $\mathcal{S}^{h}(L^{\prime})$ be
the homotopy structure set of $L^{\prime}$ and consider the map in the surgery
sequence
$\eta\colon\mathcal{S}^{h}(L^{\prime})\to\mathcal{N}(L^{\prime})\cong[L^{\prime},G/\operatorname{TOP}]$
with target the normal invariants of $L^{\prime}$. This map is defined for
3-manifolds, even though there is no analogue of the entire surgery sequence
for 3-manifolds (but see [Kirby-Taylor]*Theorem 4 for a version with a
homology structure set). We will show that the homotopy equivalences $h\colon
L\to L^{\prime}$ and $\operatorname{Id}\colon L^{\prime}\to L^{\prime}$
determine equal elements
$\eta(h)=\eta(\operatorname{Id})\in\mathcal{N}(L^{\prime})$ i.e. normally
bordant degree one normal maps. Crossing with $S^{1}$ we see that
$\eta(h\times\operatorname{Id})=\eta(\operatorname{Id}\times\operatorname{Id})\in\mathcal{N}(L^{\prime}\times
S^{1})$. Since $\mathbb{Z}/p\times\mathbb{Z}$ is a good group, the surgery
sequence is exact. Therefore $[h\times\operatorname{Id}\colon L\times S^{1}\to
L^{\prime}\times S^{1}]$ and $[\operatorname{Id}\times\operatorname{Id}\colon
L^{\prime}\times S^{1}\to L^{\prime}\times S^{1}]$ are in the same orbit of
the action of $L_{5}^{h}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])$ on
$\mathcal{S}^{h}(L^{\prime}\times S^{1})$. Then, by the definition of the Wall
realisation $L_{5}^{h}$ action, it follows that $L\times S^{1}$ and
$L^{\prime}\times S^{1}$ are stably homeomorphic. Then since these 4-manifolds
are smooth and orientable, they are in fact stably diffeomorphic by Theorem
3.8.

We therefore have to show that
$\eta(h)=\eta(\operatorname{Id})\in\mathcal{N}(L^{\prime})$. For this we adapt
the proof of [Cappell-Shaneson-nonlinear-similarity]*Proposition 2.1, where
the corresponding fact for lens spaces with even order fundamental group was
proven in all dimensions, under an additional hypothesis that the double
covers are homeomorphic. We will show that in dimension 3 the extra hypotheses
are not needed.

First recall that there is a 4-connected map $k\colon G/\operatorname{TOP}\to
K(\mathbb{Z}/2,2)$, corresponding to a universal cohomology class $k\in
H^{2}(G/\operatorname{TOP};\mathbb{Z}/2)$ [Madsen-Milgram][Kirby-Taylor].
Consider $\eta(h)$ and $\eta(\operatorname{Id})$ as elements of
$[L^{\prime},G/\operatorname{TOP}]$. Then $\eta(\operatorname{Id})^{*}(k)=0\in
H^{2}(L^{\prime};\mathbb{Z}/2)$, so we have to show that $\eta(h)^{*}(k)=0$,
and then we will have shown that both maps $L^{\prime}\to
G/\operatorname{TOP}$ are null-homotopic, and hence that the two lens spaces
are normally bordant. If $p$ is odd, then $H^{2}(L^{\prime};\mathbb{Z}/2)\cong
H_{1}(L^{\prime};\mathbb{Z}/2)=0$, so we are done. We therefore assume that
$p$ is even, in which case $H^{2}(L^{\prime};\mathbb{Z}/2)=\mathbb{Z}/2$, and
we have something to check. So let $p=2r$, for some $r\geq 1$.

Now we diverge from the proof of [Cappell-Shaneson-nonlinear-similarity]. Let
$f\colon\widetilde{L}\to L$ and $f^{\prime}\colon\widetilde{L^{\prime}}\to
L^{\prime}$ be the $r$-fold covers so that
$\pi_{1}(\widetilde{L})\cong\mathbb{Z}/2\cong\pi_{1}(\widetilde{L^{\prime}})$.
Note that $\widetilde{L}$ and $\widetilde{L^{\prime}}$ are again lens spaces,
so $\widetilde{L}\cong\widetilde{L^{\prime}}\cong\mathbb{RP}^{3}=L_{2,1}$. We
claim that $(f^{\prime})^{*}\colon H^{2}(L^{\prime};\mathbb{Z}/2)\to
H^{2}(\mathbb{RP}^{3};\mathbb{Z}/2)$ is an isomorphism. Note that
$f^{\prime}_{*}\colon H_{1}(\mathbb{RP}^{3};\mathbb{Z})\to
H_{1}(L^{\prime};\mathbb{Z})$ is given by multiplication with $r$. Hence on
the cellular $\mathbb{Z}$-chain complexes, $f^{\prime}_{1}$ is given by
multiplication with $r$ and thus $f^{\prime}_{2}$ is the identity as can be
seen by considering the following commutative diagram of the cellular chain
complexes over $\mathbb{Z}$.

${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}.}$$\scriptstyle{0}$$\scriptstyle{f^{\prime}_{3}}$$\scriptstyle{\cdot
2}$$\scriptstyle{\cdot
1}$$\scriptstyle{f^{\prime}_{2}}$$\scriptstyle{0}$$\scriptstyle{\cdot
r}$$\scriptstyle{f^{\prime}_{1}}$$\scriptstyle{\cdot
1}$$\scriptstyle{f^{\prime}_{0}}$$\scriptstyle{0}$$\scriptstyle{\cdot
2r}$$\scriptstyle{0}$

The claim that $(f^{\prime})^{*}\colon H^{2}(L^{\prime};\mathbb{Z}/2)\to
H^{2}(\mathbb{RP}^{3};\mathbb{Z}/2)$ is an isomorphism immediately follows
from this.

By the claim, we have that $(f^{\prime})^{*}\circ\eta(h)^{*}(k)=0$ if and only
if $\eta(h)^{*}(k)=0$. But
$(f^{\prime})^{*}\circ\eta(h)^{*}(k)=\eta(\widetilde{h})^{*}(k)\in
H^{2}(\mathbb{RP}^{3};\mathbb{Z}/2)$, the normal invariant of the homotopy
self-equivalence $\widetilde{h}\colon\mathbb{RP}^{3}\to\mathbb{RP}^{3}$
obtained from lifting $h\circ f\colon\mathbb{RP}^{3}\to L^{\prime}$ to
$\mathbb{RP}^{3}$ along $f^{\prime}$ as in the diagram

${\mathbb{RP}^{3}}$${\mathbb{RP}^{3}}$${L}$${L^{\prime}.}$$\scriptstyle{\widetilde{h}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{h}$

We assert that every orientation-preserving homotopy self-equivalence of
$\mathbb{RP}^{3}$, and so in particular $\widetilde{h}$, is homotopic to the
identity. It then follows that $\widetilde{h}$ is trivial in the structure set
of $\mathbb{RP}^{3}$, and so
$\eta(\widetilde{h})=\eta(\operatorname{Id}_{\mathbb{RP}^{3}})$, hence
$\eta(\widetilde{h})^{*}(k)=0$ as desired.

It remains to prove the assertion that every orientation-preserving homotopy
self-equivalence of $\mathbb{RP}^{3}$ is homotopic to the identity. This can
be proven via obstruction theory. The obstruction to iteratively defining a
homotopy on the $k$ cells of $\mathbb{RP}^{3}\times[0,1]$, relative to the
subspace $\mathbb{RP}^{3}\times\\{0,1\\}$ where the homotopy is already
defined, lies in

$H^{k}(\mathbb{RP}^{3}\times[0,1],\mathbb{RP}^{3}\times\\{0,1\\};\pi_{k-1}(\mathbb{RP}^{3})).$

For $k=2$, the obstruction vanishes because both maps induce the identity on
$\pi_{1}(\mathbb{RP}^{3})\cong\mathbb{Z}/2$. Since
$\pi_{2}(\mathbb{RP}^{3})=\pi_{2}(S^{3})=0$ the remaining obstruction lies in

$H^{4}(\mathbb{RP}^{3}\times[0,1],\mathbb{RP}^{3}\times\\{0,1\\};\pi_{3}(\mathbb{RP}^{3}))\cong
H_{0}(\mathbb{RP}^{3}\times[0,1];\mathbb{Z})\cong\mathbb{Z}.$

The obstruction measures the difference in the degrees of the two maps. Since
both are degree 1, the obstruction vanishes and the assertion is proved. ∎

######  Proposition 5.4.

Let $L$ and $L^{\prime}$ be homotopy equivalent lens spaces that are not
homeomorphic. The manifolds $L\times S^{1}$ and $L^{\prime}\times S^{1}$ are
not topologically $h$-cobordant.

###### Proof.

Assume that there is a topological $h$-cobordism $W$ from $L\times S^{1}$ to
$L^{\prime}\times S^{1}$. As in (5.2) above, taking a product with $S^{1}$
kills the Whitehead torsion, i.e. $W\times S^{1}$ is an $s$-cobordism from
$L\times S^{1}\times S^{1}$ to $L^{\prime}\times S^{1}\times S^{1}$. The high-
dimensional $s$-cobordism theorem then implies that $L\times S^{1}\times
S^{1}$ and $L^{\prime}\times S^{1}\times S^{1}$ are homeomorphic. But this
implies that $L$ and $L^{\prime}$ are homeomorphic by the toral stability
property for lens spaces [Kwasik-Schultz] which can be seen using higher
$\rho$-invariants [Weinberger-higher-rho]. This is a contradiction to our
assumption on $L$ and $L^{\prime}$. ∎

Indeed if $L$ and $L^{\prime}$ are not homeomorphic, then even
$L\times\mathbb{R}^{2}$ and $L^{\prime}\times\mathbb{R}^{2}$ are not
homeomorphic [Kwasik-Schultz]*Theorem 1.4.

###  5.8. Donaldson’s examples $E(1)$ and the Dolgachev surface $E(1)_{2,3}$

As mentioned before, Kreck (Section 5.5) and Cappell-Shaneson (Section 5.6)
constructed the first examples of exotic $4$-manifolds. These were
nonorientable, and the obstructions used arose from Rochlin’s theorem.

New examples of exotic pairs, including simply connected examples, were
provided by Donaldson [Donaldson:1987-1], and many others after him (see e.g.
[GompfStip, akbulut-book]). Donaldson’s first examples consisted of
$E(1)=\mathbb{CP}^{2}\\#\,\,{\scalebox{1.2}{\\#}^{9}}\overline{\mathbb{CP}}^{2}$
and the _Dolgachev surface_ $E(1)_{2,3}$, which is obtained from $E(1)$ via
two log transforms. Let us recall the construction. The 4-manifold $E(1)$
admits the structure of an elliptic fibration $f\colon E(1)\to S^{2}$. Let
$T^{2}\subseteq E(1)$ be a generic fibre. Its normal bundle is a copy of
$T^{2}\times D^{2}$ embedded in $E(1)$.

In general, a _log transform_ is a surgery operation on a 4-manifold $X$ with
a smoothly embedded torus $T^{2}\subseteq X$ with trivial normal bundle which
cuts out a neighbourhood $T^{2}\times D^{2}$ of $T^{2}$, and glues it back via
a diffeomorphism of $\partial(T^{2}\times D^{2})=T^{3}$. Let
$\\{\alpha,\beta,[\partial D^{2}]\\}$ be a basis for $H_{1}(T^{3})$. Then by
definition we reglue to form

$X^{\prime}:=\overline{X\setminus(T^{2}\times
D^{2})}\cup_{\varphi}(T^{2}\times D^{2})$

using a diffeomorphism $\varphi_{p}\colon T^{3}\to T^{3}$, for some
$p\in\mathbb{Z}$, corresponding to an element

$\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&-1&p\end{pmatrix}$

of $\operatorname{GL}(3,\mathbb{Z})$. Perform two of these log transform
operations on $E(1)$, on disjoint generic fibres of the elliptic fibration,
one with $p=2$ and one with $p=3$. The resulting 4-manifold is the Dolgachev
surface $E(1)_{2,3}$. One can also construct $E(1)_{2,3}$ by a single _knot
surgery_ operation [FS-ICM][FS-knot-surgery][GompfStip]*Section 10.3[akbulut-
book]*Section 6.5 on a generic fibre of $E(1)$, using a trefoil knot [park-
noncomplex]*p. 7[FS-lectures]*Lecture 6, Section 2. Akbulut used this in
[Akbuklut-kebab] to obtain a description of the Dolgachev surface without
$1$\- or $3$-handles.

  * •

$E(1)$ and $E(1)_{2,3}$ are smooth, closed, orientable, and simply connected.

  * •

They have isometric intersection forms $\langle+1\rangle\oplus
9\langle-1\rangle$ and so are homeomorphic [F, FQ], and therefore (simple)
homotopy equivalent, stably homeomorphic, topologically $h$\- and
$s$-cobordant, and $\mathbb{CP}^{2}$-stably homeomorphic.

  * •

They are homeomorphic and orientable and therefore stably diffeomorphic and
$\mathbb{CP}^{2}$-stably diffeomorphic (Theorem 3.8).

  * •

They are smoothly $h$-cobordant by [Wall-on-simply-conn-4mflds]*Theorem 2, and
are therefore smoothly $s$-cobordant since the Whitehead group of the trivial
group is trivial.

  * •

They are not diffeomorphic [Donaldson:1987-1], via tools of Yang-Mills gauge
theory.

While Rochlin’s theorem suffices to construct exotic pairs of nonorientable
$4$-manifolds (see Sections 5.5 and 5.6) it seems that one requires the full
force of gauge theory to detect orientable exotic pairs. After Donaldson’s
work, Seiberg-Witten theory provided an easier, but nonetheless still highly
nontrivial, way to distinguish manifolds such as $E(1)$ and $E(1)_{2,3}$ which
are related by log transforms or knot surgery. There is a large literature on
generalisations of Donaldson’s example, as described in [GompfStip, akbulut-
book]. As stated before, since this is not our focus we restrict ourselves to
recalling the first known example.

###  5.9. $\,\,{\scalebox{1.2}{\\#}^{3}}E_{8}$ and the Leech manifold

Next we present 4-manifolds that are stably homeomorphic but not homotopy
equivalent. Freedman [F] showed that every nonsingular, symmetric, integral
bilinear form can be realised as the intersection pairing of a closed, simply
connected, topological $4$-manifold. The forms $3E_{8}$ and the Leech lattice
are even, symmetric, positive definite bilinear forms of rank and signature
24, so they are realised by closed $4$-manifolds that we denote by
$\,\,{\scalebox{1.2}{\\#}^{3}}E_{8}$ and $Le$ respectively.

  * •

The manifolds $\,\,{\scalebox{1.2}{\\#}^{3}}E_{8}$ and $Le$ are simply
connected and orientable with
$\chi(\,\,{\scalebox{1.2}{\\#}^{3}}E_{8})=26=\chi(Le)$.

  * •

Since they are spin with signature $24$, Rochlin’s theorem implies that the
manifolds both have nontrivial Kirby-Siebenmann invariant, and are therefore
not smoothable.

  * •

The manifolds have inequivalent intersection pairings and are therefore not
homotopy equivalent. As a result they are neither simple homotopy equivalent,
homeomorphic, topologically $h$-cobordant, nor topologically $s$-cobordant.

  * •

Since they are spin and the Euler characteristics and the signatures coincide,
$\,\,{\scalebox{1.2}{\\#}^{3}}E_{8}$ and $Le$ are stably homeomorphic and
therefore also $\mathbb{CP}^{2}$-stably homeomorphic, as follows. The stable
classification of closed, simply connected, spin topological 4-manifolds is
essentially due to Wall [Wall-on-simply-conn-4mflds]*Theorems 2 and 3: two
such 4-manifolds are stably homeomorphic if and only if there are choices of
orientations with respect to which the manifolds are equivalent in the
topological spin bordism group
$\Omega_{4}^{\operatorname{TOPSpin}}\cong\mathbb{Z}$, with the isomorphism
given by $[M]\mapsto\sigma(M)/8$. Wall worked in the smooth category, but the
analogous topological category result is straightforward to deduce
[KPT]*Section 2.2.

  * •

The smooth questions are not applicable to this pair.

The downside of this example is that the manifolds are not smoothable. It
turns out that this is inevitable when considering simply connected
4-manifolds, as shown by the next proposition.

######  Proposition 5.5.

Closed, smooth, simply connected 4-manifolds $M$ and $N$ with equal Euler
characteristics are stably diffeomorphic if and only if they are homotopy
equivalent.

###### Proof.

Let $M$ and $N$ be closed, smooth, and simply connected $4$-manifolds. Assume
that $M$ and $N$ are homotopy equivalent. Then they have isometric
intersection forms, for some choice of orientation, so by [Wall-on-simply-
conn-4mflds]*Theorem 2 they are smoothly $h$-cobordant, and by [Wall-on-
simply-conn-4mflds]*Theorem 3 (see also Proposition 5.3) they are stably
diffeomorphic.

For the other direction, assume $M$ and $N$ are stably diffeomorphic and
$\chi(M)=\chi(N)$. Then modulo changing orientations, $\sigma(M)=\sigma(N)$.
Since $\chi(M)=\chi(N)$ and $\sigma(M)=\sigma(N)$, the intersection forms of
$M$ and $N$ are either both definite or both indefinite. In the definite case,
the intersection forms must be diagonal by Donaldson’s theorem [Donaldson],
and so the intersection forms are isometric, and therefore the manifolds are
homotopy equivalent [Whitehead-4-complexes, Milnor-simply-
connected-4-manifolds]. For the indefinite case, note that since the
hyperbolic form is even, and the intersection forms of $M$ and $N$ become
isometric after stabilising, they are either both odd or both even. Indefinite
forms are determined up to isometry by the rank, parity, and signature
[MH]*Theorem 5.3 and so again $M$ and $N$ are homotopy equivalent. ∎

###  5.10. Kreck-Schafer manifolds

Kreck and Schafer [Kreck-Schafer] constructed smooth 4-manifolds that are
stably diffeomorphic but not homotopy equivalent. As observed in Proposition
5.5, their examples are necessarily not simply connected.

They used the following general construction. Start with a finite presentation
of a group. Form the corresponding presentation 2-complex $X$. Thicken it to a
5-dimensional manifold $N(X)$. Then consider the 4-manifold $\partial N(X)$.
One can use this to find a 4-manifold with any given finitely presented
fundamental group.

For any two finite 2-complexes $X$ and $X^{\prime}$ with the same fundamental
group, there are integers $m,n$ such that $X\vee^{m}S^{2}\simeq
X^{\prime}\vee^{n}S^{2}$ [lms197]*(40). It follows that the boundaries of the
5-dimensional thickenings $\partial N(X)$ and $\partial N(X^{\prime})$ are
stably diffeomorphic.

Kreck and Schafer used finite 2-complexes $X$ and $X^{\prime}$ with the same
fundamental group, as above, that have the same Euler characteristic but are
not homotopy equivalent. Finding examples of 2-complexes with this property is
rather difficult, but examples are known [Metzler, Sieradski, Lustig]. Kreck
and Schafer’s obstruction applies for many nontrivial fundamental groups, the
smallest of which is $\mathbb{Z}/5\times\mathbb{Z}/5\times\mathbb{Z}/5$. Kreck
and Schafer then showed that for their particular choices of $X$ and
$X^{\prime}$, the 4-manifolds $\partial N(X)$ and $\partial N(X^{\prime})$ are
not homotopy equivalent. These manifolds have the additional interesting
property that their intersection forms, and indeed their equivariant
intersection forms, are hyperbolic.

  * •

The manifolds $\partial N(X)$ and $\partial N(X^{\prime})$ are smooth, non-
simply-connected, oriented manifolds with the same Euler characteristic that
are stably diffeomorphic but not homotopy equivalent.

  * •

Since they are not homotopy equivalent, they are also not homeomorphic, nor
diffeomorphic, nor $h$\- or $s$-cobordant in either category.

  * •

Since they are stably diffeomorphic they are stably homeomorphic,
$\mathbb{CP}^{2}$-stably diffeomorphic, and $\mathbb{CP}^{2}$-stably
homeomorphic.

Kreck and Schafer found pairs of $4$-manifolds with the properties listed. Are
there stable diffeomorphism classes of smooth, oriented $4$-manifolds
containing infinitely many homotopy equivalence classes, all with the same
Euler characteristic? Or even arbitrarily many?

###  5.11. Teichner’s $E\\#E$ and $*E\\#\mathop{}\\!*E$

We begin by describing the star construction, and proving some basic
properties about it. A _star partner_ of a $4$-manifold $M$ is a manifold
$\mathop{}\\!*M$ such that there exists a homeomorphism
$M\\#\mathop{}\\!*\mathbb{CP}^{2}\cong\mathop{}\\!*M\\#\mathbb{CP}^{2}$
preserving the decomposition on $\pi_{2}$, where
$\mathop{}\\!*\mathbb{CP}^{2}$ is the Chern manifold whose construction we
recalled in Section 5.3.

######  Lemma 5.6.

The 4-manifolds $M$ and $\mathop{}\\!*M$ satisfying the above condition are
simple homotopy equivalent.

###### Proof.

We use an argument due to Stong [Stong-conn-sum]*Section 2. For a 4-manifold
$M$ and $\beta\in\pi_{2}(M)$, let $\operatorname{cap}(M,\beta)$ be the result
$M\cup_{\beta}D^{3}$ of adding a 3-cell to $M$ along $\beta$. Let
$h\colon\mathop{}\\!*M\\#\mathbb{CP}^{2}\to M\\#\mathop{}\\!*\mathbb{CP}^{2}$
be a homeomorphism preserving the decomposition of $\pi_{2}$. Let
$\alpha\in\pi_{2}(\mathbb{CP}^{2})\cong\mathbb{Z}$ be a generator. Then
$h_{*}(\alpha)$ generates $\pi_{2}(*\mathbb{CP}^{2})$. Further
$\operatorname{cap}(\mathbb{CP}^{2},\alpha)\simeq_{s}S^{4}$ and
$\operatorname{cap}(*\mathbb{CP}^{2},h_{*}(\alpha))\simeq_{s}S^{4}$. Therefore

$\displaystyle M$ $\displaystyle\cong
M\\#S^{4}\simeq_{s}M\\#\operatorname{cap}(\mathop{}\\!*\mathbb{CP}^{2},h_{*}(\alpha))\cong\operatorname{cap}(M\\#\mathop{}\\!*\mathbb{CP}^{2},h_{*}(\alpha))\cong\operatorname{cap}(\mathop{}\\!*M\\#\mathbb{CP}^{2},h_{*}^{-1}\circ
h_{*}(\alpha))$
$\displaystyle\cong\operatorname{cap}(\mathop{}\\!*M\\#\mathbb{CP}^{2},\alpha)\cong\mathop{}\\!*M\\#\operatorname{cap}(\mathbb{CP}^{2},\alpha)\simeq_{s}\mathop{}\\!*M\\#S^{4}\cong\mathop{}\\!*M$

as desired. ∎

Note that $\mathop{}\\!*M$ and $M$ have opposite Kirby–Siebenmann invariants,
by additivity of the Kirby-Siebenmann invariant under connected sum
[guide]*Theorem 8.2. Next we give a general criterion for when star partners
exist. The proof will use the following notion.

######  Definition 5.7.

Let $M$ be a $4$-manifold. An immersion of a $2$-sphere $\alpha\colon
S^{2}\looparrowright M$ is said to be _$\mathbb{RP}^{2}$ -characteristic_ if
for every immersion $R\colon\mathbb{RP}^{2}\looparrowright M$ such that
$R^{*}w_{1}(M)=0$, we have $\alpha\cdot R\equiv R\cdot R\in\mathbb{Z}/2$.

######  Proposition 5.8.

Let $M$ be a 4-manifold with good fundamental group and containing an
immersion $R\colon\mathbb{RP}^{2}\looparrowright M$ such that $R\cdot R\equiv
1\mod 2$ and $R^{*}w_{1}(M)=0$. Then a star partner $\mathop{}\\!*M$ exists.

###### Proof.

The manifold $\mathop{}\\!*M$ can be constructed as follows. Start with
$M\\#\mathop{}\\!*\mathbb{CP}^{2}$ and let $\alpha$ be an immersed sphere in
$\mathop{}\\!*\mathbb{CP}^{2}$ with trivial self-intersection number,
$\mu(\alpha)=0$, representing a generator of
$\pi_{2}(\mathop{}\\!*\mathbb{CP}^{2})$. In the construction of
$\mathop{}\\!*\mathbb{CP}^{2}$ from Section 5.3, one can find such an $\alpha$
by gluing together the track of a null-homotopy for the trefoil in $D^{4}$
with the core of the attached $2$-handle, and then adjusting the self-
intersection number by adding small cusps. Since the mod 2 intersection
numbers are such that $\alpha\cdot R=0\neq R\cdot R\in\mathbb{Z}/2$, $\alpha$
is not $\mathbb{RP}^{2}$-characteristic. Stong [Stong]*p. 1310 proved that in
this setting, where $\pi_{1}(M)$ is good, and $\alpha$ admits an algebraically
dual immersed sphere but is not $\mathbb{RP}^{2}$-characteristic, then
$\alpha$ is homotopic to an embedding $\alpha^{\prime}$. Since
$\lambda(\alpha^{\prime},\alpha^{\prime})=1$, it follows that
$\alpha^{\prime}$ has a regular neighbourhood with boundary $S^{3}$.
Consequently, $M\\#\mathop{}\\!*\mathbb{CP}^{2}\cong N\\#\mathbb{CP}^{2}$,
where $N$ is obtained from $M\\#\mathop{}\\!*\mathbb{CP}^{2}$ by replacing a
regular neighbourhood of $\alpha^{\prime}$ by $D^{4}$. By construction, $N$ is
a star partner for $M$, which we denote by $\mathop{}\\!*M$. ∎

$0$$1$ Figure 3. A Kirby diagram for the manifold $E$.

Let $E$ denote the unique fibre bundle over $\mathbb{RP}^{2}$ with fibre
$S^{2}$, that has orientable but not spin total space. We give a Kirby diagram
in Figure 3. This is a smooth, closed, orientable 4-manifold with fundamental
group $\mathbb{Z}/2$. By Propositions 5.8 and 5.6, $E$ has a star partner
$\mathop{}\\!*E$ which is simple homotopy equivalent to $E$ but has opposite
Kirby-Siebenmann invariant.

It follows from the surgery exact sequence that if
$\pi_{1}(M)\cong\mathbb{Z}/2$, then $\mathop{}\\!*M$ is unique up to
homeomorphism if it exists (see also [Teichner:1997-1]*Theorem 1). In
particular, this means that $\mathop{}\\!*E$ is the unique star partner for
$E$. If the universal cover $\widetilde{M}$ is non spin and the fundamental
group of $M$ is good, then $\mathop{}\\!*M$ is unique up to homeomorphism
[Stong-conn-sum]*Corollary 1.2. However, it was observed by Teichner
[Teichner:1997-1]*Proposition 3 that in general there can be more than one
star partner. Teichner’s example consists of the pair of manifolds $E\\#E$ and
$\mathop{}\\!*E\\#\mathop{}\\!*E$; they are both star partners for
$E\\#\mathop{}\\!*E$ by the following proposition. These manifolds are not
spin, but their universal covers are spin.

######  Proposition 5.9.

Let $M$ be a $4$-manifold with a star partner $\mathop{}\\!*M$. The relation
of being a star partner is symmetric, i.e. $M$ is a star partner of
$\mathop{}\\!*M$, whenever $\pi_{1}(M)$ is good.

###### Proof.

We must show that there is a homeomorphism
$M\\#\mathbb{CP}^{2}\cong\mathop{}\\!*M\\#\mathop{}\\!*\mathbb{CP}^{2}$,
preserving the decomposition on $\pi_{2}$. We will use the uniqueness of star
partners for manifolds with non-spin universal covers and good fundamental
group mentioned above, to show that both $M\\#\mathbb{CP}^{2}$ and
$\mathop{}\\!*M\\#\mathop{}\\!*\mathbb{CP}^{2}$ are star partners for
$M\\#\mathop{}\\!*\mathbb{CP}^{2}$. This will imply that there is a
homeomorphism
$M\\#\mathbb{CP}^{2}\xrightarrow{\cong}\mathop{}\\!*M\\#\mathop{}\\!*\mathbb{CP}^{2}$,
and we will see it preserves the decomposition on $\pi_{2}$ from the
construction.

By the classification of closed, simply connected 4-manifolds [F, FQ], there
is a homeomorphism

$\mathop{}\\!*\mathbb{CP}^{2}\\#\mathop{}\\!*\mathbb{CP}^{2}\cong\mathbb{CP}^{2}\\#\mathbb{CP}^{2}$

preserving the decomposition on $\pi_{2}$. Therefore,
$(M\\#\mathbb{CP}^{2})\\#\mathbb{CP}^{2}\cong(M\\#\mathop{}\\!*\mathbb{CP}^{2})\\#\mathop{}\\!*\mathbb{CP}^{2}$,
preserving the decomposition on $\pi_{2}$, as needed. This shows that
$M\\#\mathbb{CP}^{2}$ is a star partner for
$M\\#\mathop{}\\!*\mathbb{CP}^{2}$.

Since $\mathop{}\\!*M$ is a star partner of $M$, we know that
$\mathop{}\\!*M\\#\mathbb{CP}^{2}\cong M\\#\mathop{}\\!*\mathbb{CP}^{2}$,
preserving the decomposition on $\pi_{2}$. By taking a connected sum on both
sides with $\mathop{}\\!*\mathbb{CP}^{2}$, we see that
$\mathop{}\\!*M\\#\mathop{}\\!*\mathbb{CP}^{2}$ is a star partner of
$M\\#\mathop{}\\!*\mathbb{CP}^{2}$. ∎

Returning to the case of $E\\#\mathop{}\\!*E$, we have the homeomorphisms

$E\\#(E\\#\mathbb{CP}^{2})\cong
E\\#(\mathop{}\\!*E\\#\mathop{}\\!*\mathbb{CP}^{2})=\mathop{}\\!*E\\#(E\\#\mathop{}\\!*\mathbb{CP}^{2})\cong\mathop{}\\!*E\\#(\mathop{}\\!*E\\#\mathbb{CP}^{2})$

where we have used that
$E\\#\mathbb{CP}^{2}\cong\mathop{}\\!*E\\#\mathop{}\\!*\mathbb{CP}^{2}$ by
Proposition 5.9, and that
$E\\#\mathop{}\\!*\mathbb{CP}^{2}\cong\mathop{}\\!*E\\#\mathbb{CP}^{2}$, by
the definition of $\mathop{}\\!*E$.

  * •

The manifolds $E\\#E$ and $\mathop{}\\!*E\\#\mathop{}\\!*E$ are orientable,
with nontrivial fundamental group isomorphic to $\mathbb{Z}/2*\mathbb{Z}/2$.

  * •

The manifold $E$ is smooth by construction, and therefore so is $E\\#E$. The
manifold $\mathop{}\\!*E$ is not smoothable, but it is currently open whether
$\mathop{}\\!*E\\#\mathop{}\\!*E$ is smoothable. Indeed
$\mathop{}\\!*E\\#\mathop{}\\!*E$ has vanishing Kirby-Siebenmann invariant and
so is stably smoothable, i.e. there exists a $k$ such that
$\mathop{}\\!*E\\#\mathop{}\\!*E\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times
S^{2})$ is a smooth manifold.

  * •

Teichner [Teichner:1997-1]*Proposition 3 showed that $E\\#E$ and
$\mathop{}\\!*E\\#\mathop{}\\!*E$ are not stably homeomorphic, and therefore
they are not homeomorphic and not $s$\- or $h$\- cobordant.

  * •

They are simple homotopy equivalent by Lemma 5.6. Recall also that the
Whitehead group of the infinite dihedral group is trivial, and so it also
suffices to know that they are homotopy equivalent.

  * •

Since they are homotopy equivalent and have trivial Kirby-Siebenmann invariant
they are $\mathbb{CP}^{2}$-stably homeomorphic.

  * •

For smooth examples with the same properties, consider the pair
$E\\#E\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times S^{2})$ and
$\mathop{}\\!*E\\#\mathop{}\\!*E\\#\,\,{\scalebox{1.2}{\\#}^{k}}(S^{2}\times
S^{2})$. As mentioned above, there exists $k$ for which these are smooth since
$\operatorname{ks}(E\\#E)=\operatorname{ks}(\mathop{}\\!*{E}\\#\mathop{}\\!*{E})=0$.
These two manifolds are still simple homotopy equivalent,
$\mathbb{CP}^{2}$-stably homeomorphic and therefore $\mathbb{CP}^{2}$-stably
diffeomorphic, but not stably homeomorphic, and therefore not stably
diffeomorphic, neither smoothly $s$\- nor $h$-cobordant, and not
diffeomorphic.

Taking connected sums of $E\\#E$ and $\mathop{}\\!*E\\#\mathop{}\\!*E$, as
well as sufficiently many copies of $S^{2}\times S^{2}$, one can construct
homotopy equivalence classes containing arbitrarily many stable homeomorphism
classes of smooth, orientable $4$-manifolds using the techniques of
[Teichner:1997-1]. We omit the details. In [teichner-thesis]*Example 5.2.4,
Teichner also constructed similar examples for finite fundamental groups with
quaternionic 2-Sylow subgroup.

###  5.12. Akbulut’s fake
$(S^{1}\mathbin{\widetilde{\times}}S^{3})\\#(S^{2}\times S^{2})$

In [Akbulut-fake-4-manifold], Akbulut constructed a smooth, closed
$4$-manifold $P$ that is homotopy equivalent to
$Q:=(S^{1}\mathbin{\widetilde{\times}}S^{3})\\#(S^{2}\times S^{2})$, but not
diffeomorphic to $Q$. The manifold $P$ has a relatively simple handle
decomposition [Akbulut-fake-4-manifold]*Figure 4.6 consisting of one
$0$-handle, one $1$-handle, two $2$-handles, one $3$-handle, and one
$4$-handle. Since $P$ is nonorientable, the $1$-handle is necessarily
nonorientable. In other words, $P$ is obtained by attaching two $2$-handles to
$S^{1}\mathbin{\widetilde{\times}}D^{3}$ and then capping off with another
copy of $S^{1}\mathbin{\widetilde{\times}}D^{3}$. Akbulut showed using
explicit moves on the handle decompositions that

$P_{0}\cup_{\partial}(\mathbb{RP}^{2}\widetilde{\times}D^{2})\cong
R\\#(S^{2}\times S^{2}),$

where $P_{0}:=P\setminus\operatorname{Int}{S^{1}\widetilde{\times}D^{3}}$ and
$R$ is the Cappell-Shaneson fake $\mathbb{RP}^{4}$ from Section 5.6. The
double cover of $P$ is the standard $(S^{1}\times
S^{3})\\#\,\,{\scalebox{1.2}{\\#}^{2}}(S^{2}\times S^{2})$, as shown in
[torres-gluck]*Proposition 9, using the fact that the double cover of $R$ is
diffeomorphic to $S^{4}$ [gompf-killing]. Akbulut’s construction could be
modified to use other Cappell-Shaneson $\mathbb{RP}^{4}$s, some of which are
not known to have standard double covers. Two other constructions of $P$ were
given in [akbulut-fake-gluck, Akbulut-on-fake]. We sketch them next.

First we describe the construction in [akbulut-fake-gluck]*Section 3 (see also
[akbulut-book]*Section 9.5). Let $\Sigma$ denote the $(2,3,7)$-Brieskorn
sphere. Recall that the Rochlin invariant of $\Sigma$, denoted $\mu(\Sigma)$,
is by definition the signature mod 16 of a smooth, spin 4-manifold with
boundary $\Sigma$. In this case $\mu(\Sigma)\equiv 8\mod{16}$. Attach a
nonorientable $1$-handle to $\Sigma\times[0,1]$, joining the two boundary
components, followed by a pair of $2$-handles, as shown in Figure 4. One then
checks that the boundary of this new handlebody is
$S^{1}\mathbin{\widetilde{\times}}S^{2}$, so it can be capped off with
$S^{1}\mathbin{\widetilde{\times}}D^{3}$, yielding the desired manifold $P$.

(a)(b)$-1$$-1$$0$$0$ Figure 4. (a) A Kirby diagram for the product
$\Sigma\times[0,1]$, where $\Sigma$ denotes the $(2,3,7)$-Brieskorn sphere.
The handle decomposition has a $3$-handle that is not pictured. The dotted
circle indicates the complement in $D^{3}\times[0,1]$ of the product
concordance from the figure eight knot to itself. A $-1$-framed $2$-handle is
then attached along this concordance. The resulting handlebody has boundary
$\Sigma\\#-\Sigma$. The $3$-handle is attached along the connected sum sphere
to finish the construction. Here we have used that $\Sigma$ is the result of
$-1$-framed Dehn surgery on the figure eight knot. (b) A nonorientable
$1$-handle is attached, along with two $2$-handles. The attaching sphere of
the new $1$-handle is shown using the notation of [Akbulut-
fake-4-manifold][akbulut-book]*Section 1.5: they are identified via the
orientation preserving diffeomorphism $(x,y,z)\mapsto(x,-y,-z)$, with respect
to coordinates based at the centre. One of the new $2$-handles is shown in
grey to help the reader distinguish between the two $2$-handles. It is
required, but can be checked, that the $2$-handles are attached along curves
disjoint from the attaching $2$-sphere of the $3$-handle from (a). The
manifold $P$ is formed by gluing on $S^{1}\mathbin{\widetilde{\times}}D^{3}$.

Via explicit handle manipulation [akbulut-fake-gluck][akbulut-book]*Exercise
9.3, this construction can be seen to be diffeomorphic to the result of a
Gluck twist on an embedded $2$-sphere in $Q$, which is the third construction
of $P$ [Akbulut-on-fake]*Theorem 1[akbulut-fake-gluck]. Notably the pair $P$
and $Q$ comprise the first example where the Gluck twist operation on a
2-sphere changes the smooth structure of a $4$-manifold. Whether this is
possible in the orientable setting remains open. Other examples of the
operation changing the smooth structure on nonorientable $4$-manifolds are
given in [torres-gluck]. See also [akbulut-yasui] for a condition that implies
the Gluck twist operation does not change the diffeomorphism type.

In the above references it is also claimed that the construction of $P$ in the
first paragraph of this section [Akbulut-fake-4-manifold] yields the same
manifold as in the second paragraph [Akbulut-on-fake]*Theorem 1[akbulut-fake-
gluck], but a proof has so far not appeared.

Another construction of a manifold homeomorphic but not diffeomorphic to $Q$
was given by Fintushel-Stern in [FS-fake], using the technology of [Fintushel-
Stern-exotic-free]. By surgering an exceptional fibre of the
$(3,5,19)$-Brieskorn sphere, they constructed $K$, a homology $S^{2}\times
S^{1}$. They then formed $X$, the mapping cylinder of the quotient map $K\to
K/t$ where $t$ is the free involution contained in the $S^{1}$-action on $K$,
as in Sections 5.4 and 5.6. To finish the construction Fintushel and Stern
showed that $K$ is the boundary of a homotopy $(S^{1}\times
D^{3})\\#(S^{2}\times S^{2})$, whose union with $X$ is the desired manifold
$M$. Using the invariant defined in [Fintushel-Stern-exotic-free], they showed
that $M$ is not smoothly $s$-cobordant to $Q$. They also showed using the
handle decomposition, and the fact that $t$ is isotopic to the identity, that
the double cover of $M$ is diffeomorphic to the standard $(S^{1}\times
S^{3})\\#\,\,{\scalebox{1.2}{\\#}^{2}}(S^{2}\times S^{2})$. It is not known
whether Akbulut’s $P$ is diffeomorphic to $M$.

  * •

The manifolds $P$ and $Q$ are smooth, closed, and nonorientable. They have
nontrivial fundamental group $\mathbb{Z}$.

  * •

The Euler characteristics are equal: $\chi(P)=2=\chi(Q)$.

  * •

Wang’s classification [Wang-nonorientable] of nonorientable $4$-manifolds with
fundamental group $\mathbb{Z}$ up to homeomorphism shows that $P$ and $Q$ are
homeomorphic, since they have isomorphic equivariant intersection forms and
equal Kirby-Siebenmann invariant.

  * •

Since they are homeomorphic, $P$ and $Q$ are topologically $h$\- and
$s$-cobordant, and (simple) homotopy equivalent. Recall that the Whitehead
group of $\mathbb{Z}$ is trivial. Direct arguments that $P$ and $Q$ are
homotopy equivalent were also given in [Akbulut-fake-4-manifold, akbulut-fake-
gluck, Akbulut-on-fake].

  * •

They are not smoothly $h$-cobordant [Akbulut-on-fake], and are therefore not
smoothly $s$-cobordant, nor diffeomorphic. We sketch the proof. Compose an
arbitrary homotopy equivalence $P\to Q$ with the collapse map to construct a
degree one normal map $g\colon P\to S^{1}\mathbin{\widetilde{\times}}S^{3}$.
We consider $(P,g)$ as an element of the set of degree one normal maps
$\mathcal{N}(Q)$, considered modulo smooth normal bordism. Next we use the
Cappell-Shaneson $\alpha$-invariant
$\alpha\colon\mathcal{N}(S^{1}\mathbin{\widetilde{\times}}S^{3})\to\mathbb{Z}/32$
described in Section 5.6, and compute

$\alpha(P,g)=2\mu(\Sigma)-\sigma(S^{3}\times[0,1]\\#(S^{2}\times S^{2}))\equiv
16\mod{32}.$

Now suppose that $P$ were $h$-cobordant to $Q$ via some smooth $h$-cobordism
$V$. Then we can extend $f$ over the cobordism obtained from $V$ by attaching
a $3$-handle to one of the $S^{2}$-factors in the $S^{2}\times S^{2}$ summand
of $Q$. This gives a smooth normal bordism between $(P,g)$ and
$(S^{1}\mathbin{\widetilde{\times}}S^{3},h)$, for a homotopy equivalence
$h\colon S^{1}\mathbin{\widetilde{\times}}S^{3}\to
S^{1}\mathbin{\widetilde{\times}}S^{3}$. But
$\alpha(S^{1}\mathbin{\widetilde{\times}}S^{3},h)=0$, which is a
contradiction.

In [Akbulut-fake-4-manifold] a different proof was given to show that $P$ is
not diffeomorphic to $Q$, for the construction in that paper. This consisted
of explicit moves on the handle decompositions reducing the problem to the
fact that the Cappell-Shaneson fake $\mathbb{RP}^{4}$ from Section 5.6 is not
diffeomorphic to $\mathbb{RP}^{4}$.

  * •

Since they have differing $\alpha$ invariants, the manifolds $P$ and $Q$ are
not stably diffeomorphic. In more detail, by the same method as in Theorem
5.2, $P$ and $Q$ are stably diffeomorphic if and only if they induce
equivalent elements in $\Omega_{4}(B,\xi)$, where
$B:=S^{1}\times\operatorname{BSpin}\xrightarrow{\xi}BO$ is a fibration similar
to that in Section 5.5 but with $B\mathbb{Z}/2$ replaced by
$B\mathbb{Z}=S^{1}$. In this case the group is
$\Omega_{4}(B,\xi)\cong\mathbb{Z}/2$, which can be computed by analysing the
appropriate Atiyah-Hirzebruch spectral sequence. The $\alpha$ invariant
defines a map $\alpha\colon\Omega_{4}(B,\xi)\to\mathbb{Z}/32$ with image
$16\mathbb{Z}/32\mathbb{Z}\cong\mathbb{Z}/2$. Since $\alpha(P)\equiv 16$, $P$
represents the nontrivial element in the bordism group. On the other hand $Q$
is null-bordant, which implies that $P$ and $Q$ are not stably diffeomorphic.

  * •

They are $\mathbb{CP}^{2}$-stably diffeomorphic and $\mathbb{CP}^{2}$-stably
homeomorphic, because they are homotopy equivalent and both have vanishing
Kirby-Siebenmann invariant. Indeed since they are related by a Gluck twist, we
have that $P\\#\mathbb{CP}^{2}\cong Q\\#\mathbb{CP}^{2}$ [Akbulut-on-
fake]*Corollary 2.

###  5.13. Kwasik-Schultz manifolds homotopy equivalent to $L\times S^{1}$

In this section we prove Theorem 1.2 from the introduction, which we restate
now. Recall this was first proven in [Kwasik-Schultz]*Theorem 1.2, but we give
a new proof.

######  Theorem 1.2.

Let $M:=L\times S^{1}$, where $L$ is a lens space $L_{p,q}$ with $p\geq 2$,
$1\leq q<p$, and $(p,q)=1$. Then there is an infinite collection of closed,
orientable, topological $4$-manifolds $\\{M_{i}\\}_{i=1}^{\infty}$, that are
all simple homotopy equivalent to $M$ but pairwise not homeomorphic.

###### Proof.

The proof will use the simple surgery exact sequence. The simple $L$-group
satisfies

$L_{5}^{s}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])\cong\mathbb{Z}^{r}\oplus(\text{torsion}),$

where

$r=\begin{cases}(p+1)/2&p\text{ odd}\\\ (p+2)/2&p\text{ even}.\end{cases}$

In both cases $r>1$ since $p\geq 2$. To compute these $L$-groups, first use
Shaneson splitting [Shaneson-splitting] to obtain

$L_{5}^{s}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])\cong
L_{5}^{s}(\mathbb{Z}[\mathbb{Z}/p])\oplus
L_{4}^{h}(\mathbb{Z}[\mathbb{Z}/p]).$

Then $L_{4}^{h}(\mathbb{Z}[\mathbb{Z}/p])\cong\mathbb{Z}^{r}\oplus T$, where
$T$ is a torsion group and the free part is detected by a multi-signature
invariant: see [Bak-even-L-groups-of-odd-order-groups, Bak-BAMS] for $p$ odd,
[Bak-computation-L-groups]*Theorem 2 for $p=2^{k}$, and [Hambleton-Taylor-
guide]*p. 227 and Proposition 12.1 for the deduction of the general case. On
the other hand $L_{5}^{s}(\mathbb{Z}[\mathbb{Z}/p])=0$, as shown in [Bak-odd-
L-groups-of-odd-order-groups, Bak-BAMS] and [Hambleton-Taylor-guide]*Theorem
10.1 for $p$ odd, [Bak-computation-L-groups]*Theorem 7 for $p=2^{k}$, and
again [Hambleton-Taylor-guide]*p. 227 and Proposition 12.1 for general $p$.

Since $\mathbb{Z}/p\times\mathbb{Z}$ is a good group, the simple surgery
sequence is exact. The normal maps
$\mathcal{N}(M\times[0,1],M\times\\{0,1\\})$ fit into an extension of
$H^{2}(M\times[0,1],M\times\\{0,1\\};\mathbb{Z}/2)$ and

$H^{4}(M\times[0,1],M\times\\{0,1\\};\mathbb{Z})\cong
H_{1}(M\times[0,1];\mathbb{Z})\cong\mathbb{Z}\oplus\mathbb{Z}/p.$

In particular, the normal maps have rank $1$. Hence the quotient

$L_{5}^{s}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])/\sigma(\mathcal{N}(M\times[0,1],M\times\\{0,1\\}))$

is infinite. By exactness this quotient acts freely on the structure set
$\mathcal{S}^{s}(M)$, and so the structure set of $M$ is also infinite. In
order to complete the proof, we need to consider the manifold set:

$\mathcal{M}(M):=\\{N\text{ a closed 4-manifold}\mid
N\simeq_{s}M\\}/\text{homeomorphism.}$

This set is isomorphic to the simple structure set $\mathcal{S}^{s}(M)$ modulo
the action of the simple homotopy self-equivalences of $M$. We will show that
the group $\operatorname{hAut}(M)$ of homotopy classes of homotopy self-
equivalences of $M$ is finite in Lemma 5.10 below. It follows that the group
of simple homotopy self-equivalences $\operatorname{hAut}^{s}(M)$ is also
finite. Then $\mathcal{M}(M)$ is the quotient of an infinite set
$\mathcal{S}^{s}(M)$ by a finite group, so is again infinite. The elements of
$\mathcal{M}(M)$ comprise the manifolds $\\{M_{i}\\}_{i=1}^{\infty}$ in the
theorem statement. ∎

  * •

The elements of $\mathcal{M}(M)$ are orientable and have nontrivial
fundamental group isomorphic to $\mathbb{Z}/p\times\mathbb{Z}$.

  * •

They are in general not known to be smoothable, but their Kirby-Siebenmann
invariants vanish because they are all bordant to the smooth manifold $M$. For
$L^{\prime}$ homotopy equivalent, but not homeomorphic, to $L$, the smooth
4-manifold $L^{\prime}\times S^{1}$ lies in $\mathcal{M}(M)$. In particular,
we know that $L^{\prime}\times S^{1}$ is simple homotopy equivalent to $M$ by
(5.2). However, such examples account for at most finitely many of the
elements of $\mathcal{M}(M)$. Therefore the smooth equivalence relations are
not applicable in general.

  * •

As they lie in the same simple structure set, they are all homotopy equivalent
and simple homotopy equivalent to one another, and therefore in particular all
have equal (vanishing) Euler characteristics.

  * •

Since they are obtained from the action of
$L_{5}^{s}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])$, the elements of
$\mathcal{M}(M)$ are stably homeomorphic and $\mathbb{CP}^{2}$-stably
homeomorphic.

  * •

The elements of $\mathcal{M}(M)$ are by definition pairwise non-homeomorphic
[Kwasik-Schultz]*Theorem 1.2. As a result, since
$\mathbb{Z}/p\times\mathbb{Z}$ is a good group, they are also not
topologically $s$-cobordant, by the $s$-cobordism theorem (Theorem 3.5).

  * •

An infinite subset of the manifolds in $\mathcal{M}(M)$ are in addition not
topologically $h$-cobordant to one another. To see this we argue as follows.
Infinitely many of the elements of $L_{5}^{s}$ that we used, namely those in
the $\mathbb{Z}^{r}$ summand detected by multisignatures, are nontrivial under
the forgetful map $L_{5}^{s}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])\to
L_{5}^{h}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])\cong\oplus^{r}\mathbb{Z}$.
This is because multisignatures are decoration independent. It follows that
the quotient of these elements by the image of the $\mathbb{Z}$ factor in the
normal invariants,

$L_{5}^{h}(\mathbb{Z}[\mathbb{Z}/p\times\mathbb{Z}])/\sigma(\mathcal{N}(M\times[0,1],M\times\\{0,1\\}))\cong\oplus^{r-1}\mathbb{Z},$

also act nontrivially on the homotopy structure set $\mathcal{S}^{h}(M)$. As
above $r\geq 2$ so this is infinite. Recall that the equivalence relation
defining this set is topological $h$-cobordism over $M$. The quotient of
$\mathcal{S}^{h}(M)$ by the group of homotopy self-equivalences of $M$ is the
manifold $h$-cobordism set:

$\mathcal{M}^{h}(M):=\\{N\text{ a closed 4-manifold}\mid N\simeq
M\\}/h\text{-cobordism}.$

As before, the homotopy self-equivalences form a finite group, so can only
identify finitely many of the manifolds. It follows that there is an infinite
subset of $\mathcal{M}(M)$ represented by manifolds that determine distinct
elements of $\mathcal{M}^{h}(M)$, and are therefore pairwise not topologically
$h$-cobordant.

These examples contrast with Teichner’s examples in Section 5.4 in that we
have infinitely many, and the stable homeomorphism statuses are different. A
similar phenomenon to the manifolds in $\mathcal{M}(M)$ arises for manifolds
homotopy equivalent to $\mathbb{RP}^{4}\\#\mathbb{RP}^{4}$ [BDK-07], except
that these manifolds are of course nonorientable.

######  Lemma 5.10.

The group of homotopy self-equivalences $\operatorname{hAut}(M)$ of $M$ is a
finite group.

###### Sketch of proof.

For this we will use the braid of exact sequences from Hambleton-Kreck
[Hambleton-Kreck-HSEs]*p. 148, which applies since $M$ is spin. This braid in
particular fits the group of homotopy self-equivalences
$\operatorname{hAut}(M)$ into an exact sequence [Hambleton-Kreck-
HSEs]*Corollary 2.13,

${\widehat{\Omega}_{5}^{\operatorname{Spin}}(B,M)}$${\operatorname{hAut}(M)}$${\operatorname{hAut}(B)}$

sandwiched between the homotopy automorphisms $\operatorname{hAut}(B)$ of the
Postnikov 2-type $B$, and a spin bordism group
$\widehat{\Omega}_{5}^{\operatorname{Spin}}(B,M)$ that we shall define below.

Since $\pi_{2}(M)=0$, the Postnikov 2-type of $M$ is
$B:=B(\mathbb{Z}/p\times\mathbb{Z})$. The homotopy classes of homotopy
equivalences of $B$ are therefore isomorphic to the automorphisms of the group
$\mathbb{Z}/p\times\mathbb{Z}$. This group of automorphisms is a finite group.

Let
$\widehat{\Omega}_{4}^{\operatorname{Spin}}(M)\subseteq\Omega_{4}^{\operatorname{Spin}}(M)$
denote the subset of bordism classes $(X,f)$ where the reference map $f\colon
X\to M$ has degree $0$, let
$\partial\colon\Omega_{5}^{\operatorname{Spin}}(B,M)\to\Omega_{4}^{\operatorname{Spin}}(M)$
be the boundary map in the long exact sequence of the pair, and let
$\widehat{\Omega}_{5}^{\operatorname{Spin}}(B,M)\subseteq\Omega_{5}^{\operatorname{Spin}}(B,M)$
be $\partial^{-1}(\widehat{\Omega}_{4}^{\operatorname{Spin}}(M))$. By
[Hambleton-Kreck-HSEs]*Lemma 2.2, there is a long exact sequence

${\Omega_{5}^{\operatorname{Spin}}(M)}$${\Omega_{5}^{\operatorname{Spin}}(B)}$${\widehat{\Omega}_{5}^{\operatorname{Spin}}(B,M)}$${\widehat{\Omega}_{4}^{\operatorname{Spin}}(M)}$${\Omega_{4}^{\operatorname{Spin}}(B).}$

The four non-relative spin bordism groups, and the first and last maps in the
sequence, can be computed using the (natural) Atiyah-Hirzebruch spectral
sequence for the generalised homology theory of spin bordism. We omit the
details, because very similar details will appear below in the proof of Lemma
5.11. There we will restrict to $p$ a power of $2$, but the computation that
both the cokernel of the map
$\Omega_{5}^{\operatorname{Spin}}(M)\to\Omega_{5}^{\operatorname{Spin}}(B)$
and the kernel of the map
$\widehat{\Omega}_{4}^{\operatorname{Spin}}(M)\to\Omega_{4}^{\operatorname{Spin}}(B)$
are finite groups is similar for all $p\geq 2$. To avoid essentially repeating
ourselves, we only give the details in the proof in Section 5.14 below, since
in that case the result is new, and more precise upper bounds are required. It
follows that $\widehat{\Omega}_{5}^{\operatorname{Spin}}(B,M)$ is finite. The
braid from [Hambleton-Kreck-HSEs]*p. 148 therefore shows that
$\operatorname{hAut}(M)$ is finite, as desired. ∎

###  5.14. Simple homotopy equivalent, $h$-cobordant $4$-manifolds that are
not $s$-cobordant

We construct arbitrarily large collections of closed, orientable, topological
4-manifolds that are simple homotopy equivalent and $h$-cobordant but not
topologically $s$-cobordant. This will prove Theorem 1.1, which we restate
below. We will employ the same scheme as in the previous subsection. The
manifolds in each collection will be simple homotopy equivalent and
$h$-cobordant to a fixed 4-manifold $L_{2^{r},1}\times S^{1}$, for some $r$.
We will show that by making $r$ large enough we can obtain a collection of
4-manifolds of any given size, with the following properties.

  * •

They are orientable and have nontrivial fundamental group isomorphic to
$\mathbb{Z}/{2^{r}}\times\mathbb{Z}$ for some $r\geq 8$.

  * •

They are all simple homotopy equivalent and topologically $h$-cobordant to one
another. As a result, they are homotopy equivalent, stably homeomorphic, and
$\mathbb{CP}^{2}$-stably homeomorphic. They are are pairwise not topologically
$s$-cobordant and therefore not homeomorphic.

  * •

Since they are homotopy equivalent they all have vanishing Euler
characteristic, the same as $L_{2^{r},1}\times S^{1}$.

  * •

They are all stably homeomorphic and $\mathbb{CP}^{2}$-stably homeomorphic.

  * •

We do not know whether they are smoothable, and therefore the smooth questions
are not applicable.

######  Theorem 1.1.

For every $n\geq 1$, there is a collection $\\{N_{i}\\}_{i=1}^{n}$ of closed,
orientable, topological 4-manifolds, that are all simple homotopy equivalent
and $h$-cobordant to one another, but which are pairwise not $s$-cobordant.

###### Proof.

Let $M_{r}:=L_{2^{r},1}\times S^{1}$, for $r\geq 1$. Let
$\pi_{r}:=\mathbb{Z}/2^{r}\times\mathbb{Z}$, and let
$G_{r}:=\mathbb{Z}/2^{r}$. The proof will again use the surgery sequence, both
with the $h$ and $s$ decorations. We begin by investigating the individual
terms. Shaneson splitting [Shaneson-splitting] shows that

$L_{5}^{s}(\mathbb{Z}\pi_{r})\cong L_{5}^{s}(\mathbb{Z}G_{r})\oplus
L_{4}^{h}(\mathbb{Z}G_{r})\text{ and }L_{5}^{h}(\mathbb{Z}\pi_{r})\cong
L_{5}^{h}(\mathbb{Z}G_{r})\oplus L_{4}^{p}(\mathbb{Z}G_{r}).$

Then by [Bak-computation-L-groups]*Theorem 7, we know

$L_{5}^{s}(\mathbb{Z}G_{r})=0=L_{5}^{h}(\mathbb{Z}G_{r}).$

Here we use that $s=0$, in the notation of that theorem (this is a different
$s$ to the $s$-decoration of the $L$-groups). By [Bak-computation-L-
groups]*Theorem 1 we have that

$L_{4}^{p}(\mathbb{Z}G_{r})=\mathbb{Z}^{m(r)}.$

where $m(r):=2^{r-1}+1$. By [Bak-computation-L-groups]*Theorem 2 and [CS-
torsion-in-L-groups]*Theorem B:

$L_{4}^{h}(\mathbb{Z}G_{r})=\mathbb{Z}^{m(r)}\oplus(\mathbb{Z}/2)^{n(r)}$

where

$n(r):=\lfloor 2(2^{r-2}+2)/3\rfloor-\lfloor r/2\rfloor-1.$

Putting this all together we have:

$L_{5}^{s}(\mathbb{Z}\pi_{r})\cong\mathbb{Z}^{m(r)}\oplus(\mathbb{Z}/2)^{n(r)}\text{
and }L_{5}^{h}(\mathbb{Z}\pi_{r})\cong\mathbb{Z}^{m(r)}.$

The kernel of the forgetful map $L_{5}^{s}(\mathbb{Z}\pi_{r})\to
L_{5}^{h}(\mathbb{Z}\pi_{r})$ is the torsion summand

$K:=\ker(L_{5}^{s}(\mathbb{Z}\pi_{r})\to
L_{5}^{h}(\mathbb{Z}\pi_{r}))\cong(\mathbb{Z}/2)^{n(r)},$

since the summand $\mathbb{Z}^{m(r)}$ detected by the multisignature is
decoration independent. The elements in $K$ act on the simple structure set of
$M_{r}$, producing topological manifolds that are stably homeomorphic and
simple homotopy equivalent to $M_{r}$.

We can compute the normal maps as

$\displaystyle\mathcal{N}(M_{r}\times[0,1],M_{r}\times\\{0,1\\})$
$\displaystyle\cong
H^{4}(M_{r}\times[0,1],M_{r}\times\\{0,1\\};\mathbb{Z})\oplus
H^{2}(M_{r}\times[0,1],M_{r}\times\\{0,1\\};\mathbb{Z}/2)$
$\displaystyle\cong\mathbb{Z}\oplus\mathbb{Z}/2^{r}\oplus\mathbb{Z}/2.$

Here the $\mathbb{Z}$ summand maps to one of the multisignature summands in
$\mathbb{Z}^{m(r)}\subseteq L_{5}^{s}(\mathbb{Z}\pi_{r})$, under the surgery
obstruction map. The torsion summand $\mathbb{Z}/2^{r}\oplus\mathbb{Z}/2$
could map to the torsion elements in $L_{5}^{s}(\mathbb{Z}\pi_{r})$. But at
least a summand of $L_{5}^{s}(\mathbb{Z}\pi_{r})$, one isomorphic to
$(\mathbb{Z}/2)^{n(r)-2}$, acts nontrivially on the simple structure set. Note
that $|(\mathbb{Z}/2)^{n(r)-2}|=2^{n(r)-2}$.

Still, it might be the case that some of the elements of the simple structure
set $\mathcal{S}^{s}(M_{r})$ obtained by this action of $L_{5}^{s}$ are
identified by the action of the simple homotopy self-equivalences
$\operatorname{hAut}^{s}(M_{r})$. Note that
$|\operatorname{hAut}^{s}(M_{r})|\leq|\operatorname{hAut}(M_{r})|$. We will
show the following lemma.

######  Lemma 5.11.

Let $r\geq 8$. Then

$2^{n(r)-2}/|\operatorname{hAut}(M_{r})|>1.$

Moreover, for any $k$ there exists an $r$ with
$2^{n(r)-2}/|\operatorname{hAut}(M_{r})|>k.$

Every element $M_{r}(\kappa):=\kappa\cdot[\operatorname{Id}\colon M_{r}\to
M_{r}]\in\mathcal{S}^{s}(M_{r})$ arising from the action of an element
$\kappa\in K$ maps trivially to the homotopy structure set
$\mathcal{S}^{h}(M_{r})$, by the diagram

${L_{5}^{s}(\mathbb{Z}\pi_{r})}$${\mathcal{S}^{s}(M_{r})}$${L_{5}^{h}(\mathbb{Z}\pi_{r})}$${\mathcal{S}^{h}(M_{r}).}$$\scriptstyle{W}$$\scriptstyle{W}$

The $M_{r}(\kappa)$ are all therefore $h$-cobordant to $M_{r}$. Since they
arise from the action of the simple $L$-group $L_{5}^{s}(\mathbb{Z}\pi_{r})$,
they are all simple homotopy equivalent and stably homeomorphic to one
another. By Lemma 5.11 there is more than one orbit of
$\\{M_{r}(\kappa)\\}_{\kappa\in K}/\operatorname{hAut}^{s}(M_{r})$, and these
are not $s$-cobordant and therefore not homeomorphic manifolds. Moreover, for
a given $k$ we can choose $r$ so that there are at least $k$ orbits, and
therefore we find arbitrarily large collections. ∎

Now we prove Lemma 5.11.

###### Proof of Lemma 5.11.

As in Section 5.13, we use the braid from [Hambleton-Kreck-HSEs]. There we
claimed that $\operatorname{hAut}(L_{p,q}\times S^{1})$ is finite for any lens
space $L_{p,q}$ with $p$ odd. Now we claim the same when $p=2^{r}$, and
moreover in this case we compute an explicit upper bound, in terms of $r$, for
the order of $\operatorname{hAut}(L_{p,q}\times S^{1})$. The braid includes
the exact sequence [Hambleton-Kreck-HSEs]*Corollary 2.13

$\widehat{\Omega}_{5}^{\operatorname{Spin}}(B_{r},M_{r})\to\operatorname{hAut}(M_{r})\to\operatorname{hAut}(B_{r}),$
(5.3)

so we need upper bounds for the cardinalities of
$\widehat{\Omega}_{5}^{\operatorname{Spin}}(B_{r},M_{r})$ and
$\operatorname{hAut}(B_{r})$, where $\operatorname{hAut}(B_{r})$ denotes the
set of homotopy self-equivalences of the Postnikov 2-type $B_{r}$ up to
homotopy. The spin bordism group
$\widehat{\Omega}_{5}^{\operatorname{Spin}}(B_{r},M_{r})$ also appeared in the
proof of Lemma 5.10, and we refer the reader there for the definition.

First we compute $\operatorname{hAut}(B_{r})$. Since $\pi_{2}(M_{r})=0$, the
Postnikov 2-type $B_{r}$ is given by $B\pi_{r}$. Therefore,
$\operatorname{hAut}(B_{r})\cong\operatorname{Aut}(\mathbb{Z}/2^{r}\times\mathbb{Z})$.
There are two automorphisms of $\mathbb{Z}$, and an automorphism of
$\mathbb{Z}/2^{r}$ for each odd number $1\leq k\leq 2^{r}-1$. There are
therefore $2^{r-1}$ such automorphisms. For each of the $2^{r}$ pairs
$(f,g)\in(\operatorname{Aut}\mathbb{Z})\times(\operatorname{Aut}\mathbb{Z}/2^{r})$
there are also $2^{r}$ possible maps $\mathbb{Z}\to\mathbb{Z}/2^{r}$ that
describe the image of $(1,0)$ in $\mathbb{Z}/2^{r}$ under an automorphism of
$\mathbb{Z}/2^{4}\times\mathbb{Z}$. Therefore

$|\operatorname{hAut}(B_{r})|=2^{2r}.$

To find an upper bound for
$|\widehat{\Omega}_{5}^{\operatorname{Spin}}(B_{r},M_{r})|$ we use the exact
sequence [Hambleton-Kreck-HSEs]*Lemma 2.2