# A survey of the foundations of four-manifold theory in the topological
category

Stefan Friedl  Fakultät für Mathematik
Universität Regensburg
Germany<EMAIL_ADDRESS>,  Matthias Nagel  Department of Mathematics, ETH
Zürich, Switzerland<EMAIL_ADDRESS>,  Patrick Orson  Mathematics
Department
Boston College
USA<EMAIL_ADDRESS>and  Mark Powell  Department of Mathematical
Sciences
Durham University
UK<EMAIL_ADDRESS>
###### Abstract.

The goal of this survey is to state some foundational theorems on 4-manifolds,
especially in the topological category, give precise references, and provide
indications of the strategies employed in the proofs. Where appropriate we
give statements for manifolds of all dimensions.

##  1\. Introduction

Here and throughout the paper “manifold” refers to what is often called a
“topological manifold”; see Section 2 for a precise definition. Here are some
of the statements discussed in this article.

  1. (1)

Existence and uniqueness of collar neighbourhoods (Theorem 2.5).

  2. (2)

The Isotopy Extension theorem (Theorem 2.10).

  3. (3)

Existence of CW structures (Theorem 4.5).

  4. (4)

Multiplicativity of the Euler characteristic under finite covers (Corollary
4.8).

  5. (5)

The Annulus theorem 5.1 and the Stable Homeomorphism Theorem 5.3.

  6. (6)

Connected sum of two oriented connected 4-manifolds is well-defined (Theorem
5.11).

  7. (7)

The intersection form of the connected sum of two 4-manifolds is the sum of
the intersection forms of the summands (Proposition 5.15).

  8. (8)

Existence and uniqueness of tubular neighbourhoods of submanifolds (Theorems
6.8 and 6.9).

  9. (9)

Noncompact connected 4-manifolds admit a smooth structure (Theorem 8.1).

  10. (10)

When the Kirby-Siebenmann invariant of a 4-manifold vanishes, both connected
sum with copies of $S^{2}\times S^{2}$ and taking the product with
$\mathbb{R}$ yield smoothable manifolds (Theorem 8.6).

  11. (11)

Transversality for submanifolds and for maps (Theorems 10.3 and 10.8).

  12. (12)

Codimension one and two homology classes can be represented by submanifolds
(Theorem 10.9).

  13. (13)

Classification of 4-manifolds up to homeomorphism with trivial and cyclic
fundamental groups (Section 11).

  14. (14)

Compact orientable manifolds that are homeomorphic are stably diffeomorphic
(Theorem 12.2 and Corollary 12.5).

  15. (15)

Multiplicativity of signatures under finite covers (Theorem 13.1).

  16. (16)

The definition of Reidemeister torsion for compact manifolds and some of its
key technical properties (Section 14.3).

  17. (17)

Obstructions to concordance of knots and links (Theorem 15.2).

  18. (18)

Poincaré duality for compact manifolds with twisted coefficients (Theorem A.15
and Theorem A.16).

Many of these results are essential tools for the geometric topologist. Our
hope is that with the statements from this note the “working topologist” will
be equipped to handle most situations. For many of the topics discussed in
this paper the corresponding statements for $4$-manifolds with a smooth atlas
are basic results in differential topology. However for general $4$-manifolds,
it can be difficult to find precise references. We aim to rectify this
situation to some extent. Throughout the paper we make absolutely no claims of
originality.

### Conventions.

  1. (1)

Given a subset $A$ of a topological space $X$ we denote the interior by
$\operatorname{Int}A$.

  2. (2)

For $n\in\mathbb{N}_{0}$ we write $D^{n}=\\{x\in\mathbb{R}^{n}\,|\,\|x\|\leq
1\\}$ for the closed unit ball in $\mathbb{R}^{n}$. We refer to
$\operatorname{Int}D^{n}=\\{x\in\mathbb{R}^{n}\,|\,\|x\|<1\\}$ as the open
$n$-ball.

  3. (3)

Unless indicated otherwise $I$ denotes the interval $I=[0,1]$.

  4. (4)

On several occasions we use cup and cap products and we cite several results
from [Dol95, Bre97, Hat02, Fri19]. Different books on algebraic topology often
have different sign conventions for cup and cap products, but in all
statements that we give, the sign conventions are irrelevant, so it is not a
problem to mix results from different sources.

### Acknowledgments.

Thanks to Steve Boyer, Anthony Conway, Jim Davis, Fabian Hebestreit, Jonathan
Hillman, Min Hoon Kim, Alexander Kupers, Markus Land, Tye Lidman, Erik
Pedersen, George Raptis, Arunima Ray, Eamonn Tweedy, and Chuck Livingston for
very helpful conversations. We are particularly grateful to Gerrit Herrmann
for providing us with detailed notes which form the basis of the Appendix A.
SF was supported by the SFB 1085 ‘Higher Invariants’ at the University of
Regensburg, funded by the Deutsche Forschungsgemeinschaft (DFG). MN gratefully
acknowledges support by the SNSF Grant 181199. MP was supported by an NSERC
Discovery Grant. SF wishes to thank the Université du Québec à Montréal and
Durham University for warm hospitality.

###### Contents

  1. 1 Introduction
  2. 2 Manifolds
    1. 2.1 Collar neighbourhoods
    2. 2.2 The isotopy extension theorem
  3. 3 Smooth 4-manifolds and their intersection forms
  4. 4 CW structures
  5. 5 The annulus theorem
    1. 5.1 The stable homeomorphism theorem
    2. 5.2 The connected sum operation
    3. 5.3 Further results on connected sums
    4. 5.4 The product structure theorem
  6. 6 Tubular neighbourhoods
    1. 6.1 Submanifolds
    2. 6.2 Tubular neighbourhoods: existence and uniqueness
    3. 6.3 Normal vector bundles
    4. 6.4 Tubular neighbourhoods: proofs
  7. 7 Background on bundle structures
  8. 8 Smoothing 4-manifolds
    1. 8.1 Smoothing non-compact 4-manifolds
    2. 8.2 The Kirby-Siebenmann invariant and stable smoothing of 4-manifolds
  9. 9 Tubing of surfaces
  10. 10 Topological transversality
    1. 10.1 Transversality for submanifolds
    2. 10.2 Transversality for maps
    3. 10.3 Representing homology classes by submanifolds
  11. 11 Classification results for 4-manifolds
    1. 11.1 Simply connected $4$-manifolds
    2. 11.2 Non simply connected $4$-manifolds
  12. 12 Stable smoothing of homeomorphisms
    1. 12.1 Kreck’s modified surgery
    2. 12.2 Stable diffeomorphism of homeomorphic orientable 4-manifolds
    3. 12.3 Non-orientable 4-manifolds and stable diffeomorphism
  13. 13 Twisted intersection forms and twisted signatures
  14. 14 Reidemeister torsion in the topological category
    1. 14.1 The simple homotopy type of a manifold
    2. 14.2 The cellular chain complex and Poincaré triads
    3. 14.3 Reidemeister torsion
  15. 15 Obstructions to being topologically slice
    1. 15.1 The Fox-Milnor Theorem
    2. 15.2 A proof of the Fox-Milnor Theorem
  16. A Poincaré Duality with twisted coefficients
    1. A.1 Twisted homology and cohomology groups
    2. A.2 Cup and cap products on twisted (co-) chain complexes
    3. A.3 The Poincaré duality theorem
    4. A.4 Preparations for the proof of the Twisted Poincaré Duality Theorem A.15
    5. A.5 The main technical theorem regarding Poincaré Duality
    6. A.6 Proof of the Twisted Poincaré Duality Theorem A.15

##  2\. Manifolds

In the literature the notion of a “manifold” gets defined differently,
depending on the preferences of the authors. Thus we state in the following
what we mean by a manifold.

######  Definition 2.1.

Let $X$ be a topological space.

  1. (1)

We say that $X$ is _second countable_ if there exists a countable basis for
the topology.

  2. (2)

An _$n$ /̄dimensional chart for $X$ at a point $x\in X$_ is a homeomorphism
$\Phi\colon U\to V$ where $U$ is an open neighbourhood of $x$ and

    1. (i)

$V$ is an open subset of $\mathbb{R}^{n}$ or

    2. (ii)

$V$ is an open subset of the half-space
$H_{n}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\,|\,x_{n}\geq 0\\}$ and
$\Phi(x)$ lies on
$E_{n-1}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\,|\,x_{n}=0\\}$.

In the former case we say that $\Phi$ is a chart of type (i) in the latter
case we say that $\Phi$ is a chart of type (ii).

  3. (3)

We say that $X$ is an _$n$ /̄dimensional manifold_ if $X$ is second countable
and Hausdorff, and if for every $x\in X$ there exists an $n$/̄dimensional
chart $\Phi\colon U\to V$ at $x$.

  4. (4)

We say that a point $x$ on a manifold is a _boundary point_ if $x$ admits a
chart of type (ii). (A point cannot admit charts of both types [Hat02, Theorem
2B.3].) We denote the set of all boundary points of $X$ by $\partial X$.

  5. (5)

An _atlas_ for a manifold $X$ consists of a family of charts such that the
domains cover all of $X$. An atlas is _smooth_ if all transition maps are
smooth. A _smooth manifold_ is a manifold together with a smooth atlas.
Usually one suppresses the choice of a smooth atlas from the notation.

To avoid misunderstandings we want to stress once again that what we call a
“manifold” is often referred to as a “topological manifold”.

######  Definition 2.2.

An _orientation_ of an $n$-manifold $M$ is a choice of generators
$\alpha_{x}\in H_{n}(M,M\setminus\\{x\\};\mathbb{Z})$ for each $x\in
M\setminus\partial M$ such that for every $x\in M\setminus\partial M$ there
exists an open neighborhood $U\subset M\setminus\partial M$ and a class
$\beta\in H_{n}(M,M\setminus U,\mathbb{Z})$ such that $\beta$ projects to
$\alpha_{y}$ for each $y\in U$.

Using the cross product one can show that the product of two oriented
manifolds admits a natural orientation. Furthermore, the boundary of an
oriented manifold also comes with a natural orientation. The proof of the
latter statement is slightly delicate; we refer to [GH81, Chapter 28] or to
[Fri19, Chapter 45.9] for details.

###  2.1. Collar neighbourhoods

We discuss the existence of a collar neighbourhood of the boundary. First we
recall the definition of a neighbourhood.

######  Definition 2.3 (Neighbourhood).

Let $X$ be a space. A _neighbourhood_ of a subset $A\subseteq X$ is a set
$U\subseteq X$ for which there is an open set $V$ satisfying $A\subseteq
V\subseteq U$.

Next we give our definition of a collar neighbourhood.

######  Definition 2.4 (Collar neighbourhood).

Let $M$ be a manifold and let $B$ be a compact submanifold of $\partial M$. A
_collar neighbourhood_ is a map $\Phi\colon B\times[0,r]\to M$ for some $r>0$
with the following three properties:

  1. (1)

$\Phi$ is a homeomorphism onto its image,

  2. (2)

for all $P\in B$ we have $\Phi(P,0)=P$,

  3. (3)

we have $\Phi^{-1}(B\times[0,r])\cap\partial M=B$.

Often, by a slight abuse of language, we identify $B\times[0,r]$ with its
image $\Phi(B\times[0,r])$ and we refer to $B\times[0,r]$ also as a collar
neighbourhood.

It is a consequence of the invariance of domain theorem that a collar
neighbourhood of $\partial M$ is a neighbourhood of $\partial M$. Now we can
state the collar neighbourhood theorem in the formulation of [Arm70, Theorem
1]. The existence of collars is originally due to Brown [Bro62].

######  Theorem 2.5 (Collar neighbourhood theorem).

Let $M$ be an $n$-manifold. Let $C$ be a compact $(n-1)$-dimensional
submanifold of $\partial M$. $($In most cases we take $C=\varnothing$.$)$
Given any collar neighbourhood $C\times[0,2]$, the restriction to
$C\times[0,1]$ can be extended to a collar neighbourhood $\partial
M\times[0,1]$.

To formulate a uniqueness result for collar neighbourhoods it helps to
introduce the following definition.

######  Definition 2.6.

Let $f,g\colon X\to Y$ be two maps between topological spaces and let $Z$ be a
subset of $Y$. We say $f$ and $g$ are _ambiently isotopic rel. $Z$_ if there
exists an isotopy $H=\\{H_{t}\\}_{t\in[0,1]}\colon Y\times[0,1]\to Y$ such
that $H_{0}=\operatorname{Id}$, $H_{t}|_{Z}=\operatorname{Id}_{Z}$ for all $t$
and such that $H_{1}\circ f=g$.

######  Theorem 2.7.

Let $M$ be a compact manifold. Given two collar neighbourhoods
$\Phi\colon\partial M\times[0,2]\to M$ and $\Psi\colon\partial M\times[0,2]\to
M$, their restrictions $\Phi|_{\partial M\times[0,1]}$ and $\Psi|_{\partial
M\times[0,1]}$ are ambiently isotopic rel. $\partial M\times\\{0\\}$.

###### Proof.

The theorem is due to [Arm70, Theorem 2], although Armstrong comments that the
theorem is not new, and the proof he gives was told to him by Lashof. See also
[KS77, Essay I, Theorem A.2]. ∎

The following corollary is a straightforward consequence of the Collar
neighbourhood theorem 2.5. The corollary often makes it possible to reduce
arguments about manifolds with boundary to the case of closed manifolds.

######  Corollary 2.8.

  1. (1)

Let $N$ be an $n$-manifold, possibly disconnected, and let $f\colon A\to B$ be
a homeomorphism between disjoint collections of boundary components of $N$.
Then the quotient $N/\sim$ under the relation $a\sim f(a)$ is an $n$-manifold
with boundary $\partial(N/\sim)=\partial N\setminus(A\cup B)$.

  2. (2)

Let $M$ be an $n$-manifold. Its double $DM:=M\cup_{\partial M=\partial M}M$ is
an $n$-manifold with empty boundary.

###  2.2. The isotopy extension theorem

######  Definition 2.9.

Let $X$ be a $k$-dimensional manifold and let $M$ be an $m$-dimensional
manifold. Let $h\colon X\times[0,1]\to M$ be a homotopy.

  1. (1)

We say $h$ is _locally flat_ if for every $(x,t)\in X\times[0,1]$ there exists
a neighbourhood $[t_{0},t_{1}]$ of $t$ and if there are level-preserving
embeddings $\alpha\colon D^{k}\times[t_{0},t_{1}]\to X\times[0,1]$ and
$\beta\colon D^{k}\times D^{m-k}\times[t_{0},t_{1}]\to M\times[0,1]$ to
neighbourhoods of $(x,t)$ and $(h_{t}(x),t)$ respectively, such that the
following diagram commutes:

$\textstyle{D^{k}\times\\{0\\}\times[t_{0},t_{1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{D^{k}\times
D^{m-k}\times[t_{0},t_{1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{X\times[0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(x,t)\mapsto(h_{t}(x),t)}$$\textstyle{M\times[0,1].}$

  2. (2)

We say $h$ is _proper_ if for every $t\in[0,1]$ we have $h_{t}(X)\cap\partial
M=h_{t}(\partial X)$.

This definition allows us to formulate the following useful theorem [EK71,
Corollary 1.4].

######  Theorem 2.10.

_(Isotopy Extension Theorem)_ Let $h\colon X\times[0,1]\to M$ be a locally
flat proper isotopy of a compact manifold $X$ into a manifold $M$. Then $h$
can be covered by an ambient isotopy of $M$, i.e. there exists an isotopy
$H\colon M\times[0,1]\to M$ such that $H_{0}=\operatorname{Id}$ and
$h_{t}=H_{t}\circ h_{0}$ for all $t\in[0,1]$.

As an application of the isotopy extension theorem we will prove the following
theorem. We encourage the reader to find a more direct proof.

######  Proposition 2.11.

Let $M$ be a connected $n$–manifold. Then for any two points $x$, $y$, there
exists a chart $\phi\colon U\to\mathbb{R}^{n}$ with $x,y\in U$. Especially,
the points $x$, $y$ are connected by a locally flat embedded arc.

###### Proof.

Since $M$ is path-connected, there exist points
$x=x_{0},x_{1},\ldots,x_{k+1}=y$ such that there are charts $(U_{i},\psi_{i})$
for $i=0,\ldots,k$ and both $x_{i}$, $x_{i+1}$ are contained in $U_{i}$.
Additionally, arrange that $y\notin U_{i}$ for $i=0,\ldots,k-1$.

Now we construct a locally flat isotopy $h$ moving $x$ to $x_{k}$. Pick a
function $\beta\colon[0,1]\to[0,1]$ with $\beta(t)=0$ for $t\in[0,1/5]$ and
$\beta(t)=1$ for $t\in[4/5,1]$. Connect $x_{i}$ to $x_{i+1}$ by the linear
path in the chart $U_{i}$, but pass along it with speed determined by $\beta$.
That is, define

$h_{i}(t):=\psi_{i}^{-1}\Big{(}(1-\beta(t))\cdot\psi_{i}(x_{i})+\beta(t)\cdot\psi_{i}(x_{i+1})\Big{)}.$

Consider the composition (of isotopies) $h=h_{0}*\cdots*h_{k}$, defined so
that $h|_{[i/{k},i+1/{k}]}$ corresponds to $h_{i}$. Each $h_{i}$ is locally
flat via the discs $D_{i}(t)=\psi_{i}^{-1}(B_{1}(\psi_{i}(h_{i}(t))))$, say.
Also, $h$ is locally flat at the times $i/k$, since it is the constant isotopy
of a point in a neighbourhood of time $i/k$. Since being locally flat is a
local condition, we deduce that $h$ is a locally flat isotopy from $x$ to
$x_{k}$. Since $y\notin U_{i}$ for $i=0,\ldots,k-1$, the image of $h$ is
disjoint from $y$. Consequently, we can upgrade $h$ to a locally flat isotopy
$S^{0}\times I\to M$ by declaring $(-1,t)\mapsto y$ and $(1,t)\mapsto h(t)$.
The Isotopy Extension Theorem 2.10 yields an ambient isotopy $H_{t}\colon M\to
M$ with $H_{t}(x)=h(t)$ and $H_{t}(y)=y$ for all $t\in I$.

Now we stretch the last chart $U_{k}$ all the way back to $x=x_{0}$. Define
$U=(H_{1})^{-1}(U_{k})$ and note that $U$ contains both $x$ and $y$. The
homeomorphism $\phi=\psi_{k}\circ H_{1}\colon U\to\mathbb{R}^{n}$ defines the
required chart on $M$.

To obtain a locally flat arc connecting $x$ and $y$, connect the points $x$
and $y$ by a straight line in the chart $(U,\phi)$. ∎

A related result [DV09, Proposition 4.5.1] shows that one can approximate a
map of a compact polyhedra into a manifold by a (non-locally flat) embedding.

##  3\. Smooth 4-manifolds and their intersection forms

In this chapter we consider some of the most famous 4-manifolds. Recall that
not all symmetric unimodular pairings over $\mathbb{Z}$ can be realised as the
intersection forms of closed, smooth 4-manifolds. We discuss some of these
limitations below.

We start out with the definition of the intersection form.

######  Definition 3.1.

  1. (1)

Given a finitely generated abelian group $H$ we write
$\operatorname{F}\\!H:=H/\mbox{torsion subgroup}$.

  2. (2)

Given an oriented compact 4-manifold $M$ we refer to the map

$\begin{array}[]{rcl}Q_{M}\colon\operatorname{F}\\!H_{2}(M;\mathbb{Z})\times\operatorname{F}\\!H_{2}(M;\mathbb{Z})&\to&\mathbb{Z}\\\
(a,b)&\mapsto&Q_{M}(a,b):=\langle\operatorname{PD}_{M}^{-1}(a)\acup\operatorname{PD}_{M}^{-1}(b),[M]\rangle\end{array}$

as the _intersection form_.

Let $E_{8}$ denote the even $8\times 8$ Cartan matrix of the eponymous
exceptional Lie algebra; that is,

$E_{8}=\begin{pmatrix}2&1&0&0&0&0&0&0\\\ 1&2&1&0&0&0&0&0\\\ 0&1&2&1&0&0&0&0\\\
0&0&1&2&1&0&0&0\\\ 0&0&0&1&2&1&0&1\\\ 0&0&0&0&1&2&1&0\\\ 0&0&0&0&0&1&2&0\\\
0&0&0&0&1&0&0&2\end{pmatrix}.$

Note that this is a symmetric integral matrix with determinant one.

######  Example 3.2.

Here are some important closed, smooth 4-manifolds.

  1. (1)

The 4-sphere $S^{4}$. This is simply connected and has
$H_{2}(S^{4};\mathbb{Z})=\\{0\\}$.

  2. (2)

The complex projective plane $\mathbb{C}\textup{P}^{2}$, which comes with a
canonical orientation. The same underlying manifold with the opposite
orientation is $\overline{\mathbb{C}\textup{P}^{2}}$. They are simply-
connected manifolds with
$H_{2}(\mathbb{C}\textup{P}^{2};\mathbb{Z})\cong\mathbb{Z}$. The intersection
form of $\mathbb{C}\textup{P}^{2}$ is $(1)$ and the intersection form of
$\overline{\mathbb{C}\textup{P}^{2}}$ is $(-1)$.

  3. (3)

The real projective space $\mathbb{RP}^{4}$, which is non-orientable and not
simply-connected.

  4. (4)

The products $S^{2}\times S^{2}$ and $S^{1}\times S^{3}$. The manifold
$S^{2}\times S^{2}$ is simply-connected and $H_{2}(S^{2}\times
S^{2};\mathbb{Z})\cong\mathbb{Z}\oplus\mathbb{Z}$. The intersection form of
$S^{2}\times S^{2}$ is represented by the standard hyperbolic form
$H:=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$.

  5. (5)

The $K3$ surface or Kummer surface

$K3\,\,:=\,\,\big{\\{}[z_{1}:z_{2}:z_{3}:z_{4}]\in\mathbb{C}\textup{P}^{3}\,\big{|}\,z_{1}^{4}+z_{2}^{4}+z_{3}^{4}+z_{4}^{4}=0\big{\\}}$

This is a simply connected, smooth, spin, closed 4-manifold with
$H_{2}(K3;\mathbb{Z})\cong\mathbb{Z}^{22}$. As is shown in [GS99, Theorem
1.3.8] or alternatively [MS17, p. 176], the intersection form of K3 is
isometric to $E_{8}\oplus E_{8}\oplus H\oplus H\oplus H$.

In Theorem 11.2 we will see that any unimodular symmetric form occurs as the
intersection form of a closed oriented 4-manifold. In the following we survey
results on intersection forms of closed oriented smooth 4-manifolds. As we
will see, the results in the smooth setting differ dramatically from the
results in the topological setting.

######  Proposition 3.3.

Let $M$ be a closed, oriented, spin 4-manifold. Then the signature
$\operatorname{sign}(M)$ is divisible by 8.

###### Proof.

Write $\operatorname{Sq}^{k}$ for the $k$th Steenrod square operation. The
$k$th Wu class $v_{k}\in H^{k}(M;\mathbb{Z}/2)$ satisfies
$\operatorname{Sq}^{k}(a)=v_{k}\cup a$, for every class $a\in
H^{4-k}(M;\mathbb{Z}/2)$. Hence if $a\in H^{2}(M;\mathbb{Z}/2)$ then
$v_{2}\cup a=\operatorname{Sq}^{2}(a)=a\cup a$. But the $n$th Stiefel-Whitney
class of $M$ is given by $w_{n}=\sum_{i}\operatorname{Sq}^{i}(v_{n-i})$ (see
[MS74, Theorem 11.14]). Since $M$ is oriented and spin, we have
$0=w_{1}=\operatorname{Sq}^{0}(v_{1})=v_{1}$, and
$0=w_{2}=\operatorname{Sq}^{0}(v_{2})+\operatorname{Sq}^{1}(v_{1})=v_{2}$. So
for any $a\in H^{2}(M;\mathbb{Z}/2)$, we have $a\cup a=0\cup
a=0\in\mathbb{Z}/2$. But this implies that for any $x\in FH_{2}(M;\mathbb{Z})$
we have that
$Q_{M}(x,x)=\langle\operatorname{PD}^{-1}(x)\cup\operatorname{PD}^{-1}(x),[M]\rangle\equiv
0\pmod{2}$. In other words $Q_{M}$ is an even form. It is then an algebraic
fact, see e.g. [MH73, Theorem 5.1], that for any symmetric nonsingular
bilinear even form $Q$ the signature is divisible by 8. ∎

Rochlin’s Theorem [Roh52] gives an extra restriction on the signatures of
intersection forms of spin 4-manifolds that admit a smooth structure.

######  Theorem 3.4 (Rochlin).

Let $M$ be a closed, oriented, spin, smooth 4-manifold. Then the signature
$\operatorname{sign}(M)$ is divisible by 16.

######  Remark 3.5.

Let $M$ be a closed oriented 4-manifold with an even intersection form and
such that $H_{1}(M;\mathbb{Z})$ has no 2-torsion. This implies
$H^{2}(M;\mathbb{Z}/2)\cong\operatorname{Hom}(H_{2}(M;\mathbb{Z}),\mathbb{Z}/2)$
and that the mod 2 reduction of $Q_{M}$ is isomorphic to the pairing
$(a,b)=\langle a\cup b,[M]\rangle$ on $H^{2}(M;\mathbb{Z}/2)$. As $Q_{M}$ is
even, this implies that $(a,a)=0\in\mathbb{Z}/2$ for any $a\in
H^{2}(M;\mathbb{Z}/2)$. But we saw above that $a\cup a=v_{2}\cup a$, so we
must have that $v_{2}=0$ as this pairing is nondegenerate. We also saw above
that $v_{2}=w_{2}$ when $M$ is oriented, so in fact $w_{2}=0$ and $M$ admits a
spin structure.

It is not true that simply having an even intersection form implies $M$ is
spin. Indeed, it is possible to construct a closed oriented $4$-manifold $M$
that has $Q_{M}=0$ (which is in particular an even form), but has non-
vanishing $w_{2}$ [GS99, Exercise 5.7.7(a)]. In a similar spirit, by [Hab82,
FS84] there exists a closed oriented 4/̄dimensional smooth manifold $M$ with
an even intersection form $Q_{M}$ that satisfies $\operatorname{sign}(M)=8$.
Hence this must also fail to be spin, now by Rochlin’s theorem.

######  Corollary 3.6.

There exists a closed orientable 4-manifold that does not admit a smooth
structure.

###### Proof.

By Theorem 11.2 there exists a simply connected closed orientable 4-manifold
$M$ with $Q_{M}\cong E_{8}$. By Rochlin’s Theorem 3.4 this manifold does not
admit a smooth structure. ∎

In a remarkable twist, shortly after Freedman proved Theorem 11.2, Donaldson
[Don83, Theorem A] [Don87, Theorem 1], proved the following result regarding
intersection forms of smooth 4-manifolds.

######  Theorem 3.7 (Donaldson).

Let $M$ be a closed oriented smooth 4-manifold. If $Q_{M}$ is positive-
definite, then $Q_{M}$ can be represented by the identity matrix.

To understand the significance of Donaldson’s Theorem it is helpful consider
the following table from [MH73, p. 28], which basically says that there are
lots of isometry types of nonsingular positive definite forms.

$\begin{array}[]{rcccccc}\mbox{Dimension:}&\,\,\,\,8&\,\,16&\,\,24&\,\,32&\,\,40\\\
\hline\cr\begin{array}[]{r}\mbox{Number of isometry types of nonsingular}\\\
\mbox{positive definite even symmetric forms:}\end{array}&1&2&24&\geq
10^{7}&\geq 10^{51}\\\ \end{array}$

It follows from [MH73, Theorem II.5.3] that every nonsingular indefinite _odd_
symmetric form is isometric to $k\cdot(1)\oplus\ell\cdot(-1)$. These are
realised by
$k\cdot\mathbb{C}\textup{P}^{2}\\#\ell\cdot\overline{\mathbb{C}\textup{P}}^{2}$.
Therefore we only need to discuss the realisability of nonsingular indefinite
_even_ symmetric forms. Again by [MH73, Theorem II.5.3], every nonsingular
_even_ indefinite symmetric form is isometric to $n\cdot E_{8}\oplus m\cdot H$
for some $(m,n)\in\mathbb{N}_{0}\times\mathbb{Z}\setminus\\{(0,0)\\}$. The
following theorem, proved by Furuta [Fur01], gives some restrictions on the
possible values of $m$ and $n$.

######  Theorem 3.8 (Furuta’s 10/8 Theorem).

If $M$ is a closed oriented connected smooth 4-manifold with _indefinite_ even
intersection form, then

$b_{2}(M)\,\,\geq\,\,\mbox{\large$\frac{10}{8}$}\cdot|\operatorname{sign}(M)|+2.$

In particular $Q_{M}\cong n\cdot E_{8}\oplus m\cdot H$ for some $n\in
2\mathbb{Z}$ and $m\in\mathbb{N}$ with $m\geq|n|+1$.

Furuta’s 10/8 Theorem is not yet optimal since it does not quite close the gap
between the forms we can realise by smooth manifolds and the forms we can
exclude. More precisely, it follows from the calculation of the intersection
form of the K3-surface and of $S^{2}\times S^{2}$ that for any
$n=2p\in\mathbb{Z}$ and every $m\geq 3|p|$ there exists a closed oriented
simply connected 4/̄dimensional smooth manifold with intersection form
isometric to $n\cdot E_{8}\oplus m\cdot H$. In other words, we have

$\mbox{intersection form of $p\cdot\textup{K}3\,\\#\,(m-3|p|)\cdot(S^{2}\times
S^{2})$}\,\,\cong\,\,2p\cdot E_{8}\oplus m\cdot H.$

The following conjecture predicts that this result is optimal.

######  Conjecture 3.9 (11/8-Conjecture).

If $M$ is a closed oriented smooth 4-manifold with indefinite even
intersection form, then

$b_{2}(M)\,\,\geq\,\,\mbox{\large$\frac{11}{8}$}\cdot|\operatorname{sign}(M)|.$

Equivalently, if $Q_{M}\cong 2p\cdot E_{8}\oplus m\cdot H$ with $p\neq 0$,
then $m\geq 3|p|$.

######  Remark 3.10.

  1. (1)

A proof of the 11/8-Conjecture would imply, by Freedman’s Theorem 11.2, that
any closed oriented simply connected smooth 4-manifold is _homeomorphic_ to
either a connected sum of the form
$k\cdot\mathbb{C}\textup{P}^{2}\\#\ell\cdot\overline{\mathbb{C}\textup{P}}^{2}$
or to a connected sum of the form $n\cdot\textup{K}3\\#m\cdot(S^{2}\times
S^{2})$.

  2. (2)

Currently the best known result in the direction of the 11/8-Conjecture is
[HLSX18, Corollary 1.13], which says that if $M$ is a closed oriented simply-
connected 4-manifold that is not homeomorphic to $S^{4}$, $S^{2}\times S^{2}$
or the K3-surface and whose intersection form is indefinite and even, then
$b_{2}(M)\geq\frac{10}{8}\cdot|\operatorname{sign}(M)|+4$.

##  4\. CW structures

In this chapter we will discuss the existence of CW-structures on manifolds.

######  Definition 4.1 (CW complex).

A _CW complex_ is a topological space $X$ together with a filtration

$\varnothing=X_{-1}\subseteq X_{0}\subseteq X_{1}\subseteq
X_{2}\subseteq\cdots$

with
$X=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>X_{n}$, such that for each $n\geq 0$, the space $X_{n}$ arises as a pushout

$\textstyle{\coprod_{j\in\mathcal{J}_{n}}S^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod_{j\in\mathcal{J}_{n}}D^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n}}$

where $\mathcal{J}_{n}$ indexes the discs $D^{n}$. It is implicit in the
statement
$X=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>X_{n}$ that the topology of $X$ agrees with the weak topology induced from the
discs $D^{n}$. The interiors $\operatorname{Int}D^{n}$ of the discs are called
the _$n$ -cells_. For $n\geq 0$, a CW complex $X$ is said to be _$n$
-dimensional_ if $X_{n}\setminus X_{n-1}\neq\varnothing$ and $X_{i}=X_{i+1}$
for all $i\geq n$. A manifold $M$ _admits a CW-structure_ if $M$ is
homeomorphic to a CW complex.

First we discuss the case of smooth manifolds. In [Mun66, Theorem 10.6] and
[Whi57, Chapter IV.12] it is shown that every smooth manifold admits a
simplicial structure, in particular it admits a CW-structure. Alternatively,
it is shown in [Mil63, Section 3] and [Hir94, Section 6.4] that every compact
smooth manifold admits a handle decomposition which implies by [Mil63, Theorem
3.5] that every compact smooth manifold is homotopy equivalent to a compact
CW-complex.

It is natural to ask whether a similar result holds if we drop the smoothness
hypothesis. The next theorem summarises what seems to be the state of the art.

######  Theorem 4.2.

  1. (1)

For $n\leq 3$ every compact $n$-manifold admits the structure of a finite
$n$-dimensional CW complex.

  2. (2)

Let $n\geq 5$ and let $M$ be a compact $n$-manifold. Then $M$ is homeomorphic
to the mapping cylinder of some map $f\colon\partial M\to X$, where $X$ is a
finite CW complex.

  3. (3)

For $n\geq 5$, every _closed_ $n$-manifold admits the structure of a finite
$n$-dimensional CW complex.

###### Proof.

Radó [Rad26] showed in 1926 that every compact 2-manifold has a simplicial
structure and so in particular has a CW structure. Hatcher’s exposition
[Hat13] is well worth reading. Moise [Moi52, Moi77] proved the analogous
result for 3-manifolds. See also [Ham76]. Since a CW complex is finite if and
only if it is compact we have shown (1).

For $M$ an $n$-manifold with boundary, Kirby-Siebenmann [KS77, Essay III.2,
Theorem 2.1] showed for $n\geq 6$ that $M$ has a topological handlebody
structure rel. $\partial M\times I$. Quinn [Qui82, Theorem 2.3.1] extended
this result to $n=5$. Kirby-Siebenmann [KS77, Essay III.2, Theorem 2.2] then
says that $M$ is homeomorphic to the mapping cylinder of some map
$f\colon\partial M\to X$, where $X$ is a finite CW complex. Thus (2) holds.

In particular, if $M$ is closed, it admits the structure of a finite
$n$-dimensional CW complex, which shows (3). ∎

It is not clear to us whether Theorem 4.2 suffices to show that every compact
high-dimensional manifold admits a CW structure. Put differently, to the best
of our knowledge the following question is still open for manifolds with
nonempty boundary.

######  Question 4.3.

Let $n\geq 5$. Does every compact $n$-manifold have a CW structure?

Casson [AM90, p. xvi] showed in the 1980s that there exist closed 4-manifolds
that do not have a simplicial structure. It is now known that in every
dimension $n\geq 5$, there exists a closed $n$-manifold that does not admit a
simplicial structure. This question was reduced to a problem about homology
$3$-spheres [Mat78, GS80], which was then solved by Manolescu [Man16]. Note
that a simplicial structure is not necessarily a $\operatorname{PL}$
structure; an $n$-dimensional $\operatorname{PL}$ structure satisfies the
additional condition that the link of an $m$-simplex be homeomorphic to an
$(n-m-1)$-dimensional sphere. To the best of our knowledge the following
question is still open.

######  Question 4.4.

Does every compact $4$-manifold have a CW structure?

We have the following theorem regarding CW structures and manifolds. Given
Theorem 4.2, the first statement is only of interest in dimension 4. The
second statement appears in [Wal67, Theorem 2.2] for $n\geq 4$.

######  Theorem 4.5.

  1. (1)

Every compact $n$-manifold is homotopy equivalent to an $n$-dimensional finite
CW complex.

  2. (2)

Every connected compact $n$-manifold with nonempty boundary is homotopy
equivalent to an $(n-1)$-dimensional finite CW complex.

In the proof of Theorem 4.5 we will use the following theorem proved by Wall
[Wal66, Corollary 5.1].

######  Theorem 4.6.

Let $X$ be a finite connected CW complex. Suppose that there is an integer
$n\geq 3$ such that $H^{i}(X;\mathbb{Z}[\pi])=0$ for all $i>n$. Then $X$ is
homotopy equivalent to an $n$-dimensional finite CW complex.

The proof of Theorem 4.5 also makes use of the following definition. Recall
that a space is metrisable if it admits the structure of a metric space
inducing the given topology.

######  Definition 4.7 (Absolute Neighbourhood Retract (ANR)).

A space $X$ is called an _absolute neighbourhood retract_ if $X$ is metrisable
and if whenever $X\subseteq Y$ is a closed subset of a metrisable space $Y$,
then $X$ is a neighbourhood retract of $Y$. That is, there is an open
neighbourhood $U\subseteq Y$ containing $X$, with a map $r\colon U\to X$ such
that the composition $X\to U\xrightarrow{r}X$ is equal to the identity on $X$.

###### Proof of Theorem 4.5.

Hanner [Han51, Theorem 3.3] showed that every manifold is an ANR, and West
[Wes77] showed that every compact ANR is homotopy equivalent to a finite CW
complex. (An alternative proof that every compact manifold has the homotopy
type of a finite CW complex is given in Kirby-Siebenmann [KS69, Section 1
(III)].)

Let $M$ be a compact $n$-manifold. By the results above, $M$ is homotopy
equivalent to a finite CW complex $X$. By Theorem 4.2 we need to complete the
proof of (1) only in the case $n=4$. But since the subsequent argument works
for all $n\geq 4$ we also state it for all $n\geq 4$. Since $M$ is
$n$-dimensional it follows from Universal Poincaré duality (Theorem A.15) that
for any $k>n$ and any $\mathbb{Z}[\pi_{1}(X)]$-module $\Lambda$ we have

$H^{k}(X;\Lambda)\,\,\cong\,\,H_{n-k}(X;\Lambda)\,\,=\,\,0.$

By Theorem 4.6, $X$ is homotopy equivalent to an $n$-dimensional finite CW
complex. Note that to apply Theorem 4.6 we have used that $n\geq 3$.

Now we turn to the proof of (2). We start with $n=1,2$ or $3$. We saw in the
proof of Theorem 4.2 that a compact connected $n$-manifold admits a simplicial
structure. It is well known that a compact connected $n$-manifold with
nonempty boundary and a simplicial structure is homotopy equivalent to an
$(n-1)$-dimensional simplicial complex: collapse top dimensional simplices
starting with those that have a face on the boundary. In particular such a
manifold is homotopy equivalent to an $(n-1)$-dimensional finite CW complex.

Now suppose that $n\geq 4$. By (1) we know that if $M$ is a connected
$n$-manifold with nonempty boundary, then it admits the structure of a finite
CW complex. Let $k\geq n$ and let $\Lambda$ be a
$\mathbb{Z}[\pi_{1}(X)]$-module. By Poincaré-Lefschetz duality (Section A) we
have

$H^{k}(X;\Lambda)\,\,\cong\,\,H_{n-k}(X,\partial X;\Lambda)\,\,=\,\,0.$

Here the last conclusion is obvious for $k>n$. For $k=n$ the conclusion
follows from the fact that $\partial X\neq\varnothing$, that $X$ is connected
and the explicit calculation of 0-th twisted homology groups as given in
[HS97, Chapter VI.3]. It follows from Theorem 4.6 that $X$ is homotopy
equivalent to an $(n-1)$-dimensional finite CW complex. Note here that to
apply Theorem 4.6 we used $n\geq 4$. ∎

Theorem 4.5 is strong enough to recover many familiar statements.

######  Corollary 4.8.

Let $M$ be a compact connected manifold.

  1. (1)

The group $\pi_{1}(M)$ is finitely presented.

  2. (2)

All homology groups $H_{k}(M;\mathbb{Z})$ are finitely generated abelian
groups, in particular it makes sense to define the Euler characteristic

$\chi(M)\,\,:=\,\,\mbox{$\textstyle\sum\limits_{n}$}(-1)^{n}\cdot b_{n}(M).$

  3. (3)

Let $p\colon\widetilde{M}\to M$ be a finite covering. Then

$\chi(\widetilde{M})\,\,=\,\,[\widetilde{M}:M]\cdot\chi(M).$

######  Remark 4.9.

Let $M$ be a compact, connected manifold. Borsuk’s theorem that $M$ is a
Euclidean Neighbourhood Retract shows that $M$ is a retract of a finite CW
complex; see [Hat02, Appendix A, Corollary A.9], [Bre97, Appendix E], and
[Fri19, Proposition 65.22]. This fact is nontrivial but it is much easier to
prove than Theorem 4.5. Borsuk’s Theorem implies immediately that the homology
groups of $M$ are finitely generated and that the fundamental group of $M$ is
finitely generated. In fact using a group theoretic lemma as in [Wal65, Lemma
1.3] or alternatively [FR01, Theorem 3.1], one actually obtains that
$\pi_{1}(M)$ is finitely presented. But it is not clear how Borsuk’s Theorem
can be used to prove Corollary 4.8 (3).

###### Proof.

The first two statements in the corollary are an immediate consequence of
Theorem 4.5 and standard results on fundamental groups and homology groups of
finite CW complexes. We turn to the final statement. Let $X$ be a finite CW
complex homotopy equivalent to $M$. Use the fact that the Euler characteristic
is multiplicative for finite covers of finite CW complexes and use that a
$k$-fold cover $\widetilde{M}$ of $M$ induces a $k$-fold cover $\widetilde{X}$
of $X$ such that $\widetilde{M}$ and $\widetilde{X}$ are homotopy equivalent,
to deduce the result. ∎

######  Remark 4.10.

As pointed out above, every compact smooth manifold admits the structure of a
finite CW complex. One can combine this fact with Theorem 8.6 below to obtain
an alternative proof of Corollary 4.8. Theorem 8.6 says that for any
4-manifold $M$ there is a $4$-manifold $N$ such that the connected sum $M\\#N$
admits a smooth structure.

##  5\. The annulus theorem

The annulus theorem is a fundamental result in the development of the theory
of manifolds. In high dimensions, it underpins the product structure theorem
[KS77, Essay I, Theorem 5.1], which itself underpins all the results of
[KS77]. We state the product structure theorem in Section 5.4. In dimension
four, the annulus theorem is needed for the proofs of smoothing theorems
(Section 8), existence and uniqueness of normal bundles (Section 6), and
transversality (Section 10). We discuss these developments in later sections.
Later in this section (Section 5.2), we will discuss a more immediate
application: showing that connected sum is a well-defined operation on
connected topological manifolds. Here is the annulus theorem.

######  Theorem 5.1 (Annulus theorem).

Let $n\in\mathbb{N}$ and let $f,g\colon D^{n}\to\mathbb{R}^{n}$ be two
orientation-preserving locally flat embeddings. If
$f(D^{n})\subset\operatorname{Int}(g(D^{n}))$, then
$g(D^{n})\setminus\operatorname{Int}(f(D^{n}))$ is homeomorphic to
$S^{n-1}\times[0,1]$.

For $n=2,3$ the annulus theorem follows from the work of Radó [Rad26] and
Moise [Moi52, Moi77] (see also [Edw84, p. 247]). The annulus theorem was
proved for dimensions $\neq 4$ by Kirby [Kir69] and in dimension $4$ by Quinn
[Qui82, p. 506]; see also [Edw84, p. 247].

The known proofs of the annulus theorem deduce it from the stable
homeomorphism theorem. In the next section we will state the stable
homeomorphism theorem 5.3 and we will show how the annulus theorem can be
deduced from that theorem.

###  5.1. The stable homeomorphism theorem

We reduce the annulus theorem to the stable homeomorphisms theorem stated in
Theorem 5.3. This follows from work of Brown and Gluck [BG64b], but since it
requires some work to find this deduction in [BG64b], we give the details
here.

######  Definition 5.2.

A homeomorphism $f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is said to be
_stable_ if there is a sequence of homeomorphisms
$f_{1},\dots,f_{n}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that
$f_{n}\circ\cdots\circ f_{1}=f$ and such that for each $i$, the homeomorphism
$f_{i}$ is _somewhere the identity_ , which means that there is an open
nonempty set $U\subseteq\mathbb{R}^{n}$ such that $f_{i}|_{U}$ is the identity
on $U$.

The key ingredient to the subsequent discussion is the following theorem.

######  Theorem 5.3 (Stable homeomorphism theorem).

Let $n\in\mathbb{N}$. Every orientation preserving homeomorphism from
$\mathbb{R}^{n}$ to itself is stable.

For $n\geq 5$ this was proved by Kirby [Kir69, p. 575]. Slightly more
precisely, Kirby [Kir69] showed that the stable homeomorphism theorem in
dimensions at least five is a consequence of the surgery theoretic
classification of $\operatorname{PL}$ homotopy tori which had been worked out
by Wall [Wal70],[Wal99, Section 15A] and independently by Hsiang-Shaneson
[HS69, p. 688], both proofs building on the work of Browder, Novikov and Wall.
For $n=4$ the stable homeomorphism theorem was proven by Quinn, see also
[Edw84, p. 247].

Before we discuss consequences of the Stable homeomorphism theorem 5.3 we
recall the two versions of the Alexander trick.

######  Lemma 5.4 (Alexander trick).

  1. (1)

Every homeomorphism of $S^{n-1}$ can be extended radially to a homeomorphism
of $D^{n}$ that sends 0 to 0.

  2. (2)

Let $f$ and $g$ be two homeomorphisms of $D^{n}$. If the restrictions of $f$
and $g$ to $S^{n-1}$ are isotopic, then $f$ and $g$ are isotopic
homeomorphisms of $D^{n}$.

The extension in the first statement can be obtained by coning: $f(t\cdot
x)=t\cdot f(x)$. The second one is an amusing exercise; see [Han89, Lemma 5.6]
for a proof. The same idea extends to show that the topological group
$\operatorname{Homeo}_{\partial}(D^{n})$ of homeomorphisms of $D^{n}$ fixing
the boundary pointwise is contractible.

We can now prove the following almost immediate consequence of the Stable
homeomorphism theorem 5.3.

######  Corollary 5.5.

Every orientation preserving self-homeomorphism of $S^{n}$ is isotopic to the
identity.

###### Proof.

We identify $S^{n}$ with $\mathbb{R}^{n}\cup\\{\infty\\}$. Let $h$ be a self-
homeomorphism of $S^{n}=\mathbb{R}^{n}\cup\\{\infty\\}$. After an isotopy
(using Theorem 2.10) we can assume that $h(\infty)=\infty$. By the Stable
homeomorphism theorem 5.3 we know that $h$ is stable. Thus we only have to
consider the case that $h$ fixes an open subset of
$\mathbb{R}^{n}\cup\\{\infty\\}=S^{n}$. After an isotopy we can assume that
$h$ fixes an open neighbourhood of $\infty$, so in particular there exists
$C>0$ such that $h$ is the identity on $\\{x\in\mathbb{R}^{n}\mid\|x\|\geq
C\\}$. It follows from Lemma 5.4 (2) that $h$ is isotopic to the identity. ∎

Denote the set of locally flat embeddings of $D^{n}$ into $\mathbb{R}^{n}$ by
$\operatorname{Emb}(D^{n},\mathbb{R}^{n})$; see Definition 6.2 for the
definition of locally flat.

######  Definition 5.6.

We say that two elements
$f_{0},f_{1}\in\operatorname{Emb}(D^{n},\mathbb{R}^{n})$ are _intertwined_ if
there exists an $h\in\operatorname{Homeo}(\mathbb{R}^{n},\mathbb{R}^{n})$ with
$h\circ f_{0}=f_{1}$.

We will need the following straightforward technical lemma.

######  Lemma 5.7.

Let $M$ be an $n$-dimensional manifold and let $f\colon D^{n}\to M$ be a
locally flat embedding into $\operatorname{Int}M=M\setminus\partial M$. Then
there exists a locally flat embedding $F\colon\mathbb{R}^{n}\to M$ such that
the restriction of $F$ to $D^{n}$ equals $f$.

###### Proof.

Let $f\colon D^{n}\to M$ be a locally flat embedding. By definition $f(D^{n})$
is a submanifold of $M$. It is straightforward to see that $W:=M\setminus
f(\operatorname{Int}D^{n})$ is also a submanifold of $M$. By the collar
neighbourhood theorem 2.5 there exists a collar $f(S^{n-1})\times[0,1]$. The
map

$\begin{array}[]{rcl}F\colon\mathbb{R}^{n}&\to&M\\\
x&\mapsto&\left\\{\begin{array}[]{ll}f(x),&\mbox{ if }x\in D^{n},\\\
\big{(}f(y),\frac{2}{\pi}\arctan(t-1)\big{)}&\mbox{ if $x=t\cdot y$ with
$t\in[1,\infty)$ and $y\in S^{n-1}$},\end{array}\right.\end{array}$

is easily seen to be a locally flat embedding. ∎

######  Lemma 5.8.

Any two elements $f_{0},f_{1}\in\operatorname{Emb}(D^{n},\mathbb{R}^{n})$ are
intertwined.

###### Proof.

It suffices to show that any $f\in\operatorname{Emb}(D^{n},\mathbb{R}^{n})$ is
intertwined with the standard embedding $D^{n}\subset\mathbb{R}^{n}$. So let
$f\in\operatorname{Emb}(D^{n},\mathbb{R}^{n})$. Apply Lemma 5.7 to extend $f$
to a locally flat embedding $F\colon D^{n}(\frac{3}{2})\to\mathbb{R}^{n}$.
Note that $F$ restricts to a locally flat embedding of
$S^{n-1}\times[\frac{1}{2},\frac{3}{2}]$ into
$S^{n}=\mathbb{R}^{n}\cup\\{\infty\\}$. Let $\widetilde{D}^{n}$ be another
copy of $D^{n}$. By the generalised Schoenflies theorem [Bro60, Theorem 5]
there exists a homeomorphism $g\colon\widetilde{D}^{n}\to S^{n}\setminus
f(\operatorname{Int}\widetilde{D}^{n})$. Since the homeomorphisms of $D^{n}$
act transitively on the interior of $\widetilde{D}^{n}$, arrange that
$g(0)=\infty$.

Note that $g^{-1}\circ f\colon S^{n-1}\to S^{n-1}$ is a homeomorphism. By
Lemma 5.4 (1) this homeomorphism extends to a homeomorphism $\phi$ of $D^{n}$.
Replace $g$ by $g\circ\phi$ if necessary to obtain that $f=g\colon S^{n-1}\to
f(S^{n-1})$. Identify
$S^{n}=\mathbb{R}^{n}\cup\\{\infty\\}=D^{n}\cup\widetilde{D}^{n}$ in such a
way that $0\in\widetilde{D}^{n}$ corresponds precisely to $\infty$. Consider
the map

$\begin{array}[]{rcl}F\colon S^{n}=D^{n}\cup\widetilde{D}^{n}&\to&S^{n}\\\
x&\mapsto&\begin{cases}f(x)&x\in D^{n}\\\
g(x)&x\in\widetilde{D}^{n}.\end{cases}\end{array}$

The maps $f$ and $g$ agree on the overlap, so the map is well-defined and is a
homeomorphism. Note that $F$ restricts to a homeomorphism of $\mathbb{R}^{n}$
which has the property that the restriction to $D^{n}$ equals $f$. This shows
that $F\circ\operatorname{Id}=f$ so $\operatorname{Id}$ and $f$ are
intertwined. ∎

######  Definition 5.9.

Let $f_{0},f_{1}\in\operatorname{Emb}(D^{n},\mathbb{R}^{n})$ with
$f_{0}(D^{n})\subset\operatorname{Int}f_{1}(D^{n})$. We say $f_{0}$ and
$f_{1}$ are _strictly annularly equivalent_ if there exists a map $F\colon
S^{n-1}\times I\to M$ that is a homeomorphism onto its image such that
$F(x,0)=f_{0}(x)$ and $F(x,1)=f_{1}(x)$ for all $x\in S^{n-1}$.

######  Theorem 5.10.

Let $f_{0},f_{1}\in\operatorname{Emb}(D^{n},\mathbb{R}^{n})$ with
$f_{0}(D^{n})\subset\operatorname{Int}f_{1}(D^{n})$. If $f_{0}$ and $f_{1}$
are orientation preserving, then they are strictly annularly equivalent if and
only if they are intertwined.

###### Proof.

If two such elements are strictly annularly equivalent, then they are
intertwined by [BG64b, Theorem 5.2].

Now suppose that $f_{0}$ and $f_{1}$ are intertwined, that is there exists an
$h\in\operatorname{Homeo}(\mathbb{R}^{n},\mathbb{R}^{n})$ with $h\circ
f_{0}=f_{1}$. By the Stable homeomorphism theorem 5.3 we know that $h$ is
stable. Thus we know from [BG64b, Theorem 5.4] that the embeddings are
annularly equivalent, i.e. there exist
$h_{0},\dots,h_{k}\in\operatorname{Emb}(D^{n},\mathbb{R}^{n})$ such that
$h_{0}=f_{0},h_{k}=f_{1}$ and for each $i$ the maps $h_{i}$ and $h_{i+1}$ are
strictly annularly equivalent. Since
$f_{0}(D^{n})\subset\operatorname{Int}f_{1}(D^{n})$, the embeddings of the
boundary spheres $f_{0}(\partial D^{n})$ and $f_{1}(\partial D^{n})$ are
disjoint. Therefore it follows from [BG64a, Theorem 3.5] that $f_{0}$ and
$f_{1}$ are not only annularly equivalent, but are moreover strictly annularly
equivalent. ∎

Now we can easily prove the Annulus theorem 5.1.

###### Proof of the Annulus theorem 5.1.

Let $f_{0},f_{1}\colon D^{n}\to\mathbb{R}^{n}$ be two orientation-preserving
locally flat embeddings with
$f_{0}(D^{n})\subset\operatorname{Int}(f_{1}(D^{n}))$. By Lemma 5.8 and
Theorem 5.10 the two maps $f_{0}$ and $f_{1}$ are strictly annularly
equivalent. But this immediately implies that
$f_{1}(D^{n})\setminus\operatorname{Int}(f_{0}(D^{n}))$ is homeomorphic to
$S^{n-1}\times[0,1]$. ∎

###  5.2. The connected sum operation

We recall the construction of the connected sum of two connected oriented
$n$-manifolds $M$ and $N$. Pick two orientation preserving locally flat
embeddings of $n$-balls $\Phi_{M}\colon D^{n}\to M$ and $\Phi_{N}\colon
D^{n}\to N$. Define the _connected sum_ $M\\#N$ of $M$ and $N$ by

$M\\#N:=(M\setminus\Phi_{M}(\operatorname{Int}(D^{n})))\cup_{\Phi_{M}(S^{n-1})=\Phi_{N}(S^{n-1})}(N\setminus\Phi_{N}(\operatorname{Int}(D^{n})))$

where we glue the left hand side to the right hand side via the map

$\Phi_{N}\circ\Phi_{M}^{-1}\colon\Phi_{M}(S^{n-1})\,\,\xrightarrow{\cong}\,\,\Phi_{N}(S^{n-1}).$

It follows from the Collar neighbourhood theorem 2.5 that the topological
space $M\\#N$ inherits the structure of an $n$-manifold; see [Lee11,
Proposition 6.6] for details. Furthermore $M\\#N$ can be oriented in such a
way that $M$ and $N$ are oriented submanifolds.

######  Theorem 5.11.

The connected sum $M\\#N$ of two connected oriented $n$-manifolds $M$ and $N$
is independent of the choice of embeddings of the $n$-balls.

######  Remark 5.12.

The manifolds $\mathbb{C}\text{P}^{2}\\#\overline{\mathbb{C}\text{P}^{2}}$ and
$\mathbb{C}\text{P}^{2}\\#\mathbb{C}\text{P}^{2}$ have different intersection
forms, so they are not homeomorphic; see Proposition 5.15. Thus connected sum
is not well-defined on orientable $4$-manifolds, rather it depends on the
choice of orientation. On the other hand for nonorientable manifolds, the
connected sum is well-defined. As discussed in [BCF+19], in contrast to the
case of orientable 3-dimensional manifolds, orientable 4-dimensional
topological manifolds do not admit a unique decomposition as a connected sum
of irreducible 4-manifolds.

The proof of Theorem 5.11 relies on the following two lemmas. The elementary
proof of the first lemma is left to the reader.

######  Lemma 5.13.

Let $D_{r}^{n}(x)$ and $D_{s}^{n}(y)$ be two Euclidean balls in
$\mathbb{R}^{n}$. There exists a homeomorphism
$f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $f(D_{r}^{n}(x))=D_{s}^{n}(y)$
such that $f$ is the identity outside of some compact set.

The next lemma is a consequence of the annulus theorem 5.1.

######  Lemma 5.14.

Let $\varphi,\psi\colon D^{n}\to\mathbb{R}^{n}$ be two orientation-preserving
locally flat embeddings. If
$\varphi(D^{n})\subset\operatorname{Int}(\psi(D^{n}))$, then there exists a
homeomorphism $f$ of $\mathbb{R}^{n}$ with $f(\varphi(D^{n}))=\psi(D^{n})$
such that $f$ is the identity outside of some compact set.

###### Proof.

By the annulus theorem 5.1 and the collar neighbourhood theorem 2.5 we can
find a locally flat embedding $\theta\colon S^{n-1}\times[-1,2]$ such that
$\theta(S^{n-1}\times[-1,0])$ is a collar for $\varphi(D^{n})$, such that
$\theta(S^{n-1}\times[0,1])=\psi(D^{n})\setminus\varphi(\operatorname{Int}D^{n})$
and such that $\theta(S^{n-1}\times[1,2])$ is a collar for
$\mathbb{R}^{n}\setminus\psi(\operatorname{Int}D^{n})$. It is now obvious that
we can find a homeomorphism $f$ with $f(\varphi(D^{n}))=\psi(D^{n})$ which is
the identity outside of $\theta(S^{n-1}\times[-1,2])$. ∎

The subsequent proof is partly based on the sketch given in [Rol90, p. 42].

###### Proof of Theorem 5.11.

We have to show that the connected sum is independent of the choice of
$\Phi_{M}\colon D^{n}\to M$ and $\Psi_{N}\colon D^{n}\to N$. By symmetry and
transitivity it suffices to show that the connected sum is independent of the
choice of $\Phi_{M}$. So suppose we are given two orientation-preserving
embeddings $\Phi_{1}\colon D^{n}\to M$ and $\Phi_{2}\colon D^{n}\to M$ and
suppose we are given an orientation-preserving embedding $\Psi\colon D^{n}\to
N$. We introduce the following notation.

  1. (1)

Write $D_{i}:=\Phi_{i}(D^{n})$.

  2. (2)

Let $X_{i}:=M\setminus\Phi_{i}(\operatorname{Int}D^{n})$ and let
$Y:=N\setminus\Psi(\operatorname{Int}D^{n})$,

  3. (3)

Denote the restriction of $\Phi_{i}$ to $S^{n-1}$ by $\varphi_{i}$ and denote
the restriction of $\Psi$ to $S^{n-1}$ by $\psi$.

Figure 1 hopefully makes it easier for the reader to internalise the notation.
We have to show that there exists a homeomorphism

$(X_{1}\cup Y)/\varphi_{1}(x)\sim\psi(x)\,\,\to\,\,(X_{2}\cup
Y)/\varphi_{2}(x)\sim\psi(x)$

where the gluing on both sides is given by taking $x\in S^{n-1}$.

Figure 1.

######  Claim.

There exists a homeomorphism $h$ of $M$ so that $h(D_{1})=D_{2}$.

First note that it follows from Lemma 5.7 and Lemma 5.13, together with our
hypothesis that $M$ is path connected, that there exists a homeomorphism $\mu$
of $M$ such that $\mu(D_{1})\subset\operatorname{Int}D_{2}$. Then apply Lemma
5.7 and Lemma 5.14 to find a homeomorphism $\nu$ of $M$ such that
$\nu(\mu(D_{1}))=D_{2}$. This concludes the proof of the claim.

It follows from the claim that $\varphi_{2}^{-1}\circ\varphi_{1}$ is a
homeomorphism of $S^{n-1}$. By Corollary 5.5 we know that there exists an
isotopy $H\colon S^{n-1}\times[0,1]\to S^{n-1}$ from
$\varphi_{2}^{-1}\circ\varphi_{1}$ to the identity.

We write $C:=\Phi(S^{n-1})$. By the collar neighbourhood theorem 2.5 we can
pick a collar $C\times[0,1]$ for $Y$. It is straightforward to verify that

$\begin{array}[]{rcl}(X_{1}\cup Y)/\varphi_{1}(x)\sim\psi(x)&\to&(X_{2}\cup
Y)/\varphi_{2}(x)\sim\psi(x)\\\
P&\mapsto&\left\\{\begin{array}[]{ll}h(P),&\mbox{ if }P\in X_{1},\\\
\psi(H(\psi^{-1}(Q),t))&\mbox{ if }P=(Q,t)\in C\times[0,1]\\\ P,&\mbox{ if
}Y\setminus P\in C\times[0,1]\end{array}\right.\end{array}$

is a well-defined map and is a homeomorphism. This shows that the connected
sums defined using $\Phi_{1}$ and $\Phi_{2}$ give rise to homeomorphic
manifolds. ∎

###  5.3. Further results on connected sums

The definition of the intersection form was given in Definition 3.1. The next
proposition shows that the intersection form is well behaved under the
connected sum operation.

######  Proposition 5.15.

Let $M$ and $N$ be two oriented compact $4$-manifolds. Then $Q_{M\\#N}$ is
isometric to $Q_{M}\oplus Q_{N}$.

###### Proof.

In the smooth case this statement follows immediately from the fact that any
class in second homology can be represented by an embedded oriented
submanifold [GS99, Proposition 1.2.3] and the fact that one can calculate the
intersection form in terms of algebraic intersection numbers of embedded
oriented surfaces [Bre97, Theorem VI.11.9]. To apply this approach to general
manifolds, one needs to use topological transversality, which holds, as
discussed in Section 10.

On the other hand the statement for general manifolds (and thus also for
smooth manifolds) can be proved directly with the usual tools of algebraic
topology, namely functoriality of the cup and cap products [Bre97, Theorem
VI.5.2.(4)] for maps between pairs of topological spaces, a Mayer-Vietoris
argument and the excision theorem. Full details are provided in [Fri19,
Proposition 72.11]. ∎

###  5.4. The product structure theorem

The product structure theorem [KS77, Essay I, Theorem 5.1], is a key result
for the development of topological manifold theory in high dimensions. It is a
consequence of the stable homeomorphism theorem 5.3, and in turn is used in
[KS77] to deduce handle structures, transversality, smoothing theory, and the
existence of a canonical simple homotopy type, for high dimensional $(n\geq
6)$ manifolds. We will give some examples of the use of the product structure
theorem below. Even though it is only for high dimensional manifolds, it still
appears in the development of the theory of 4-manifolds.

The product structure theorem will be stated for upgrading to either a smooth
or PL structure. A _concordance_ of (smooth, PL) structures
$\Sigma,\Sigma^{\prime}$ on a manifold $N$ is a (smooth, PL) structure
$\Omega$ on $N\times I$ that restricts to $\Sigma$ on $N\times\\{0\\}$ and
restricts to $\Sigma^{\prime}$ on $N\times\\{1\\}$.

######  Theorem 5.16 (Product structure theorem).

Let $M$ be a manifold of dimension $n\geq 5$. Let $\Sigma$ be a $($smooth,
PL$)$ structure on $M\times\mathbb{R}^{s}$, with $s\geq 1$. Let $U$ be an open
subset of $M$ with a $($smooth, PL$)$ structure $\rho$ on $U$ such that
$\rho\times\mathbb{R}^{s}=\Sigma|_{U\times\mathbb{R}^{s}}$. If $n=5$ then
suppose that $\partial M\subset U$.

Then there is a $($smooth, PL$)$ structure $\sigma$ on $M$ extending $\rho$,
together with a concordance of (smooth,PL) structures from $\Sigma$ to
$\sigma\times\mathbb{R}^{s}$, that is a product concordance in some
neighbourhood of $U\times\mathbb{R}^{s}$ and that is a product near
$M\times\mathbb{R}^{s}\times\\{i\\}$ for $i=0,1$.

######  Remark 5.17.

The statement of the product structure theorem was modelled on the Cairns-
Hirsch theorem [KS77, Essay I, Theorem 5.3], which was proven in the early
1960s, and provided the analogous upgrade from PL structures to smooth
structures. See [HM74] for a comprehensive treatment of smoothing theory for
PL manifolds. The Cairns-Hirsch theorem tells us that if $M$ already has a PL
structure $\varpi$, such that $\varpi\times\mathbb{R}^{s}$ is _Whitehead
compatible_ (see the discussion below [KS77, Essay I, Theorem 5.3] for
details) with a smooth structure $\Sigma$ on $M\times\mathbb{R}^{s}$, then the
smooth structure $\sigma$ on $M$ produced by Theorem 5.16 is Whitehead
compatible with $\varpi$.

In Section 14.1 on the simple homotopy type of a manifold we make use of the
following stronger local version.

######  Theorem 5.18 (Local product structure theorem).

Let $M$ be a manifold of dimension $n\geq 5$.

  1. (i)

Let $W$ be an open neighbourhood of $M\times\\{0\\}$ in
$M\times\mathbb{R}^{s}$, for some $s\geq 1$.

  2. (ii)

Let $\Sigma$ be a $($smooth, PL$)$ structure on $W$.

  3. (iii)

Let $C\subseteq M\times\\{0\\}$ be a closed subset such that there is a
neighbourhood $N(C)$ of $C$ on which the $($smooth,PL$)$ structure $\Sigma$ is
a product $\Sigma|_{N(C)}=\sigma\times\mathbb{R}^{s}$ for some $($smooth,PL$)$
structure $\sigma$ on $M$. If $n=5$ then suppose that $\partial M\subseteq C$.

  4. (iv)

Let $D\subset M\times\\{0\\}$ be another closed subset.

  5. (v)

Let $V\subseteq W$ be an open neighbourhood of $D\setminus C$.

Then we have the following.

  1. (1)

A $($smooth, PL$)$ structure $\Sigma^{\prime}$ on $W$ that equals $\Sigma$ on
$(W\setminus V)\cup((C\times\mathbb{R}^{s})\cap W)$ and is a product
$($smooth,PL$)$ structure $\rho\times\mathbb{R}^{s}$ on
$(N(D)\times\mathbb{R}^{s})\cap W$ for some neighbourhood $N(D)$ of $D$ and
for some $($smooth, PL$)$ structure $\rho$ on $N(D)$.

  2. (2)

A concordance of $($smooth, PL$)$ structures from $\Sigma$ to
$\Sigma^{\prime}$, that is a product concordance on some neighbourhood of
$(W\setminus V)\cup((C\times\mathbb{R}^{s})\cap W)$ and that is a product near
$W\times\\{i\\}$ for $i=0,1$.

Note that the concordance implies isotopy theorem [KS77, Essay I, Theorem 4.1]
means that the concordances in Theorems 5.16 and 5.18 can be upgraded to
isotopies of (smooth, PL) structures under the same hypotheses on dimensions,
that is $n\geq 6$ or $n=5$ and the structures already agree on $\partial M$.

##  6\. Tubular neighbourhoods

###  6.1. Submanifolds

Every smooth submanifold of a smooth manifold admits a normal vector bundle
and, by the smooth Tubular neighbourhood theorem, also admits a tubular
neighbourhood [Hir94, Sections 5 & 6][Wal16, Chapter 2.5]. However in the
topological category $n$-manifolds may not admit normal vector bundles, a
general problem we discuss further below and in Section 7 once we have
developed the necessary language. Curiously, in the special case of
$4$-manifolds these general problems do not exist, and familiar smooth results
hold true using an appropriate notion of normal vector bundles (Definition
6.15).

Before discussing tubular neighbourhoods and normal vector bundles in the
topological category, we give our convention for submanifolds. Recall that
$E_{n-1}\subset\mathbb{R}^{n}$ is the hyperplane $\\{x_{n}=0\\}$.

######  Definition 6.1.

Let $M$ be an $n$-dimensional manifold. We say a subset $X\subset M$ is a _$k$
-dimensional submanifold_ if given any $P\in X$ one of the holds:

  1. (1)

there exists a chart $\Phi\colon U\to V$ of type (i) for $M$ and $P$ such that

$\Phi(U\cap X)\,\,\subset\,\,\\{(0,\dots,0,x_{1},\dots,x_{k})\mid
x_{i}\in\mathbb{R}\\},$

  2. (2)

there exists a chart $\Phi\colon U\to V$ of type (ii) for $M$ and $P$ such
that $\Phi(P)$ lies in $E_{n-1}$ and

$\hskip 32.72049pt\Phi(U\cap
X)\,\,\subset\,\,\\{(0,\dots,0,x_{1},\dots,x_{k})\in\mathbb{R}^{n}\mid
x_{k}\geq 0\\},$

  3. (3)

or there exists a chart $\Phi\colon U\to V$ of type (i) for $M$ and $P$ such
that $\Phi(P)$ lies in $E_{n-1}$ and

$\hskip 32.72049pt\Phi(U\cap
X)\,\,\subset\,\,\\{(0,\dots,0,x_{1},\dots,x_{k})\in\mathbb{R}^{n}\mid
x_{k}\geq 0\\}.$

If for every $P\in X$ we can find charts as in (1) and (2), then we call $M$ a
_proper_ submanifold.

Figure 2.

######  Definition 6.2.

A map $f\colon X\to M$ from a $k$-manifold to an $m$-manifold $M$ is called a
(proper) _locally flat embedding_ if $f$ is a homeomorphism onto its image and
if the image is a (proper) submanifold of $M$.

Given any submanifold $X$ of $M$, the inclusion map $X\to M$ is a locally flat
embedding. Conversely, if $f\colon X\to M$ is a locally flat embedding, then
the image $f(X)$ is a submanifold.

######  Remark 6.3.

  1. (1)

Note that if $M$ is a $k$-manifold and $U$ is an open subset of
$\mathbb{R}^{k}$, then it follows from the invariance of domain that the image
any injective map $f\colon U\to M$ is an open subset of $M$. In particular
$f(U)$ is a submanifold of $M$. Put differently, $f$ is locally flat.

  2. (2)

In point set topology, one often defines a _topological embedding_ to be a map
$f\colon X\to Y$ of topological spaces that is a homeomorphism to its image.
The image of a topological embedding is not necessarily a submanifold and such
an image is sometimes called _wild_ due to the bizarre properties that such
objects can exhibit. For example, the famous Alexander horned sphere [Ale24]
is not a submanifold of $S^{3}$ under Definition 6.1, but it is the image of a
wild topological embedding $S^{2}\to S^{3}$.

  3. (3)

In the literature a compact subset $F$ of 4-manifold is often called a
_locally flat surface_ if $F$ is homeomorphic to a compact 2-dimensional
manifold with $\partial F=F\cap\partial M$ and if $F$ has the following
properties.

    1. (a)

Given any $P\in F\setminus\partial F$ there exists a topological embedding
$\varphi\colon D^{2}\times D^{2}\to M\setminus\partial M$ with
$\varphi(D^{2}\times D^{2})\cap F=\varphi(D^{2}\times\\{0\\})$ and with
$P\in\varphi(D^{2}\times\\{0\\})$.

    2. (b)

Given any $P\in\partial F$ there exists a topological embedding $\varphi\colon
D^{2}_{\geq 0}\times D^{2}\to M$ such that $\varphi(D^{2}_{\geq 0}\times
D^{2})\cap F=\varphi(D^{2}_{\geq 0}\times\\{0\\})$, $\varphi(D^{2}_{\geq
0}\times D^{2})\cap\partial M=\varphi(\partial_{y=0}D^{2}_{\geq 0}\times
D^{2})$, and with $P\in\varphi(D^{2}_{\geq 0}\times\\{0\\})$. Here, we used
the following abbreviations $D^{2}_{\geq 0}=\\{(x,y)\in D^{2}\,|\,y\geq 0\\}$
and $\partial_{y=0}D^{2}=\\{(x,0)\in D^{2}\\}$.

It follows easily from the definitions that $F\subset M$ is a locally flat
surface if and only if $F$ is proper submanifold of $M$.

The following is a common justification for requiring slice discs to be
locally flat in the theory of knot concordance.

######  Proposition 6.4.

Given a knot $K\subset S^{3}$ the corresponding cone

$\operatorname{Cone}(K):=\\{r\cdot Q\mid Q\in K\mbox{ and
}r\in[0,1]\\}\,\,\subset\,\,D^{4}.$

is locally flat if and only if $K$ is the unknot.

###### Proof.

Consider the specific unknot $U$ that is the equator of the equator
$U=S^{1}\subset S^{2}\subset S^{3}=\partial D^{4}$. Taking the cone radially
inwards to the origin of $D^{4}$ exhibits $\operatorname{cone}(U)$ as a
locally flatly (properly) embedded disc. Any other unknotted $K\subset S^{3}$
is related to $U$ by a homeomorphism of $S^{3}$. By the Alexander trick
5.4(1), this homeomorphism extends radially inwards to a homeomorphism of
$D^{4}$ fixing the origin. Thus the cone on any other unknot $K$ is locally
flatly embedded, as we obtain a chart as in Definition 6.1(1) at the origin of
$D^{4}$.

Conversely, suppose $K\subset S^{3}$ is a knot such that
$C:=\operatorname{Cone}(K)$ is locally flat. This implies that there is a
chart $\Phi\colon D^{4}\to D^{4}$ such that $\Phi(0)=P$ and such that
$\Phi(D^{2}\times\\{0\\})=\Phi(D^{4})\cap C$. We introduce the following
notation.

  1. (i)

Given $I\subset[0,1]$ we write $D_{I}:=\\{v\in D^{4}\mid\|v\|\in I\\}$.

  2. (ii)

Given $I\subset[0,1]$ we write $N_{I}:=\Phi(D_{I})$.

An elementary argument shows that there exist $s_{1}<t_{1}<s_{2}<t_{2}<s_{3}$
such that $D_{[0,s_{1}]}\subset N_{[0,t_{1}]}\subset D_{[0,s_{2}]}\subset
N_{[0,t_{2}]}\subset D_{[0,s_{3}]}$. We make the following observations:

  1. (1)

For any $I\subset[0,1]$ we have homeomorphisms $D_{I}\setminus
C\cong(S^{3}\setminus K)\times I$ and $N_{I}\setminus
C\xrightarrow{\Phi^{-1}}D_{I}\setminus U\cong(S^{3}\setminus U)\times I$.

  2. (2)

For any inclusion $I\subset J$ of intervals the inclusion induced maps
$D_{I}\to D_{J}$ and $N_{I}\to N_{J}$ are homotopy equivalences.

We consider the following commutative diagram where all maps are induced by
inclusions

---
$\textstyle{\pi_{1}(S^{3}\setminus K)\cong\pi_{1}(D_{\\{t_{2}\\}}\setminus
C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\pi_{1}(N_{[t_{1},t_{2}]}\setminus
C)\cong\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(D_{[t_{1},t_{3}]}\setminus
C).}$

Since the inclusion $D_{\\{t_{2}\\}}\setminus C\to D_{[t_{1},t_{3}]}\setminus
C$ is a homotopy equivalence we see that the left diagonal map is an
isomorphism. Thus we see that we have an automorphism of
$\pi_{1}(S^{3}\setminus K)$ that factors through $\mathbb{Z}$. Since the
abelianisation of $\pi_{1}(S^{3}\setminus K)$ is isomorphic to $\mathbb{Z}$ we
see that $\pi_{1}(S^{3}\setminus K)\cong\mathbb{Z}$. It follows from the Loop
Theorem that $K$ is in fact the unknot [Rol90, Theorem 4.B.1]. ∎

In some applications one needs the following refinement of the Collar
neighbourhood theorem 2.5.

######  Theorem 6.5 (Collar neighbourhood theorem for proper submanifolds).

Let $M$ be a manifold and let $X\subset M$ be a proper submanifold. There
exists a collar neighbourhood $\partial M\times[0,1]$ such that $(\partial
M\times[0,1])\cap X$ is a collar neighbourhood for $\partial X\subset X$.

###### Proof.

By the earlier Collar neighbourhood theorem 2.5 we can pick a collar
neighbourhood $\partial M\times[0,2]$ for $\partial M$ and we can also pick a
collar neighbourhood $\partial X\times[0,2]$ for $\partial X$. Given
$t\in[0,1]$ we consider the obvious homeomorphisms

$f_{t}\colon M=(M\setminus(\partial M\times[0,2)))\cup(\partial
M\times[0,2])\,\to\,(M\setminus(\partial M\times[0,2)))\cup(\partial
M\times[t,2])$

and

$g_{t}\colon X=(X\setminus(\partial X\times[0,2)))\cup(\partial
X\times[0,2])\,\to\,(X\setminus(\partial X\times[0,2)))\cup(\partial
X\times[t,2]).$

Next we consider the following proper locally flat isotopy:

$\begin{array}[]{rcl}h\colon X\times[0,1]&\to&M\\\
(x,t)&\mapsto&\left\\{\begin{array}[]{rl}(y,s)\in\partial M\times[0,t],&\mbox{
if }x=(y,s)\mbox{ with }y\in\partial X,s\in[0,t],\\\
f_{t}(g_{t}^{-1}(x)),&\mbox{ otherwise.}\end{array}\right.\end{array}$

Note that the collar neighborhood $\partial M\times[0,1]$ is of the desired
form for the proper submanifold $h_{1}(X)$. By the Isotopy Extension Theorem
2.10 we can extend $h$ to a isotopy $H$ of $M$. Thus $H_{1}^{-1}(\partial
M\times[0,1])$ is the desired collar neighborhood for $M$. ∎

Figure 3. Illustration of the proof of the Collar neighbourhood theorem 6.5.

###  6.2. Tubular neighbourhoods: existence and uniqueness

In the literature one can find many different definitions of tubular
neighbourhoods for smooth manifolds. We will give a definition for manifolds
that is modelled on the definition provided by Wall [Wal16] for smooth
manifolds. To do so we first need one extra definition.

######  Definition 6.6.

Let $M$ be an $n$-dimensional manifold. We say a subset $W\subset M$ is a _$k$
-dimensional submanifold with corners_ if given any $P\in W$ there exists a
chart of the type (1), (2) or (3) as in Definition 6.1 above, or if

  1. (4)

there exists a chart $\Phi\colon U\to V$ of type (ii) for $M$ such that

$\Phi(U\cap W)\,\,\subset\,\,\\{(0,\dots,0,x_{1},\dots,x_{k})\mid
x_{i}\in\mathbb{R}\mbox{ with }x_{k-1}\geq 0\mbox{ and }x_{k}\geq 0\\}$

and with $\Phi(P)\in\\{(0,\dots,0,x_{1},\dots,x_{k-2},0,0)\mid
x_{i}\in\mathbb{R}\\}$.

If $W$ is an $n$-dimensional submanifold with corners we write

$\partial_{0}W\,\,:=\,\,W\cap\overline{M\setminus
W},\quad\partial_{1}W\,\,:=\,\,W\cap\partial M,\quad\mbox{ and we note that
}\quad\operatorname{Int}W\,\,=\,\,W\setminus\partial_{0}W.$

Figure 4.

######  Definition 6.7.

Let $M$ be an $n$-manifold and let $X$ be a compact proper $k$-dimensional
submanifold. A _tubular neighbourhood_ for $X$ is a pair $(N,p\colon N\to X)$
with the following properties:

  1. (1)

$N$ is a codimension zero submanifold with corners.

  2. (2)

The map $p\colon N\to X$ is a linear $D^{n-k}$-bundle such that $p(x)=x$ for
all $x\in X$.

  3. (3)

We have $\partial_{1}N=p^{-1}(\partial X)$.

Here linear means that there exists an atlas of trivialisations such that the
transition maps take values in $O(n-k)$ instead of
$\operatorname{Homeo}(D^{n-k})$.

In the topological category, tubular neighbourhoods do not always exist.
Indeed it is shown in [Hir68, Theorem 4] that there exists a $4$-dimensional
submanifold of $S^{7}$ that does not admit a tubular neighbourhood.

Fortunately, for submanifolds of $4$-manifolds, tubular neighbourhoods exist
and they are unique in the appropriate sense.

######  Theorem 6.8 (Tubular neighbourhood theorem).

Every compact proper submanifold $X$ of a $4$-manifold $M$ admits a tubular
neighbourhood.

######  Theorem 6.9.

_(Uniqueness of tubular neighbourhoods)_ Let $M$ be a $4$-manifold and let $X$
be a compact proper $k$-dimensional submanifold. Furthermore let $p_{i}\colon
N_{i}\to X$, $i=1,2$ be two tubular neighbourhoods of $X$, with inclusion maps
$\iota_{i}\colon N_{i}\to M$. Then there exists an isomorphism $\Psi\colon
N_{1}\to N_{2}$ of linear disc bundles such that $\iota_{2}\circ\Psi\colon
N_{1}\to M$ and $\iota_{1}\colon N_{2}\to M$ are ambiently isotopic rel $X$.

The proofs of the above two theorems rely on the existence and uniqueness
results for normal vector bundles in [FQ90, Section 9], which we discuss
further in Section 6.3. Thus we postpone the proofs of the above two theorems
to Section 6.4.

Right now, let us first observe some nice consequences of the existence and
uniqueness of tubular neighbourhoods.

######  Remark 6.10.

Let $X$ be a compact proper submanifold of a $4$-manifold $M$. By Theorem 6.8
we can pick a tubular neighbourhood $p\colon N\to X$. We refer to
$E_{X}:=M\setminus\operatorname{Int}N$ as the _exterior of $X$_. By Theorem
6.9 the homeomorphism type of the exterior is well-defined.

######  Lemma 6.11.

Let $X$ be a compact proper submanifold of a $4$-manifold $M$. The exterior
$E_{X}$ of $X$ is a deformation retract of the complement $M\setminus X$.

###### Proof.

Let $p\colon N\to X$ be a tubular neighbourhood for $X$. Using the fact that
$p$ is a _linear_ bundle, introduce compatible radial coordinates in the
fibres and isotope radially outwards. This implies that $\partial_{0}N$ is a
deformation retract of $N\setminus X$. But this also implies that the exterior
$E_{X}=M\setminus N$ is a deformation retract of $M\setminus X$. ∎

######  Corollary 6.12.

Let $X$ be a submanifold of a compact $4$-manifold $M$. If $X$ is compact,
then $\pi_{1}(M\setminus X)$ and $H_{*}(M\setminus X)$ are finitely generated.

###### Proof.

It follows from Lemma 6.11 that $M\setminus X$ is homotopy equivalent to the
exterior of $X$ which in our case is a compact 4-manifold since we assume that
$M$ is compact. The corollary is now a consequence of Corollary 4.8. ∎

######  Proposition 6.13.

Let $X\subset M$ be a $2$-dimensional orientable submanifold of a compact
orientable $4$-manifold $M$, such that each connected component of $X$ has
nonempty boundary. Then the tubular neighbourhood of Theorem 6.8 is
homeomorphic to $X\times D^{2}$.

###### Proof.

A linear $D^{n}$-bundle is the unit disc bundle of a vector bundle. Connected
surfaces with boundary are homotopy equivalent to wedges of circles. Every
orientable vector space bundle over a wedge of circles is trivial, and so are
their unit disc bundles. ∎

###  6.3. Normal vector bundles

The reader will be familiar with the definition of a normal vector bundle when
working in the smooth category: if $X\subset M$ is a smooth submanifold of a
smooth manifold, then the normal vector bundle is the quotient vector bundle
$TM|_{X}/TX$. This definition uses the smooth structure to ensure the
existence of tangent vector bundles, and vector bundles are a strong enough
bundle technology to ensure the existence the perpendicular subspaces required
to form the quotient bundle. While some (weaker) canonical tangential
structures do exist in the topological category (see Section 7), the idea of
‘quotient bundle’ no longer makes sense for them.

In the topological category, following [FQ90, Section 9], we will use a
definition of normal vector bundle that is much closer to the geometry of
tubular neighbourhoods. We begin with a definition that is almost what we need
but suffers from a slight technical problem, which we then remedy.

######  Definition 6.14.

Let $M$ be a $n$-manifold and let $X$ be a proper $k$-dimensional submanifold.
An _internal linear bundle over $X$_ is a pair $(E,p\colon E\to X)$ with the
following properties:

  1. (1)

$E$ is a codimension zero submanifold of $M$.

  2. (2)

The map $p\colon E\to X$ is an $(n-k)$-dimensional vector bundle such that
$p(x)=x$ for all $x\in X$.

  3. (3)

We have $\partial E=p^{-1}(\partial X)$.

An internal linear bundle $(E,p\colon E\to X)$ is intended to recover, from
the smooth category, an _open_ tubular neighbourhood of $X$. As such, the
definition as stands suffers from the potential technical problem that the
closure of $E$ in $M$, which should be a closed tubular neighbourhood, may be
an immersion; see Figure 5. As in [FQ90, p. 137], we use the following
additional idea to rule out this problem.

######  Definition 6.15.

Let $M$ be an $n$-manifold, let $X$ be a proper $k$-dimensional submanifold,
and let $(E,p\colon E\to X)$ be an internal linear bundle over $X$. Suppose
that given any $(n-k)$-dimensional vector bundle $(F,q\colon F\to X)$, any
radial homeomorphism from an open convex disc bundle of $F$ to $E$ can be
extended to a homeomorphism from the whole of $F$ to a neighbourhood of $E$.
Then we say $(E,p\colon E\to X)$ is _extendable_.

In Figure 5 we illustrate an example of a non-extendable internal linear
bundle.

Figure 5.

Now we can define the notion of a normal vector bundle.

######  Definition 6.16 ([FQ90, p. 137]).

Let $M$ be a $n$-manifold and let $X$ be a proper $k$-dimensional submanifold.
A _normal vector bundle_ for $X$ is an internal linear bundle that is
extendable.

######  Theorem 6.17 (Existence of normal vector bundles).

Every proper submanifold of a compact 4-manifold admits a normal vector
bundle.

######  Remark 6.18.

Generally, the existence of normal vector bundles is peculiar to when the
submanifolds has low dimension or low codimension. We refer the reader to
[FQ90, Section 9.4] for a discussion of the other known situations where these
objects always exist. Here is a summary of the known cases. A submanifold of
dimension at most 3 in a closed manifold of dimension at least 5 has a normal
bundle, and codimension one submanifolds have normal bundles [Bro62]. That
every codimension two submanifold of a manifold of dimension not equal to four
have normal bundles was shown in [KS75], and this was extended to include
dimension four in [FQ90, Section 9.3]. It is striking that, while among smooth
manifolds dimension 4 exhibits worse than usual behaviour, in the topological
category the existence of normal vector bundles seems to show it is among the
better behaved of the dimensions.

For the proof of Theorem 6.17 we will essentially appeal to [FQ90, Theorem
9.3A] and [FQ90, Theorem 9.3D]; the former deals with existence while the
latter deals with uniqueness. We reproduce this theorem here for the benefit
of the reader.

######  Theorem 6.19.

Let $N$ be a submanifold of a $4$-manifold $M$, with a closed subset
$K\subseteq N\setminus\partial N$ and a normal bundle over some neighbourhood
of $K$ in $N$. Then there is a normal bundle over $N$ that agrees with the
given one over the neighbourhood of $K$. Moreover this extension is unique up
to ambient isotopy relative to some neighbourhood of $K$.

###### Proof.

Let $X$ be a proper submanifold of a compact 4-manifold $M$. The case that $X$
has no boundary is dealt with in [FQ90, Theorem 9.3A]. The case that $X$ has
nonempty boundary follows also from [FQ90, Theorem 9.3A] if we apply more
care. We sketch the argument.

First, in dimension three the topological and the smooth category are the
same. Thus we can view the submanifold $\partial X\subset\partial M$ as a
smooth submanifold. Hence it has a smooth normal vector bundle e.g. [Kos93,
Chapter III.2] or [Lan02b, Section IV.5].

Next use the Collar neighbourhood theorem 6.5 to obtain a collar $\partial
M\times[0,1]\subset M$ that restricts to a collar $\partial X\times[0,1]$ for
the boundary of $X$. Extend the smooth tubular neighbourhood of $\partial
X\subset\partial M$ into the collar by taking a product with $[0,1]$.

Finally, consider the 4-manifold without boundary
$M^{\prime}:=M\setminus(\partial M\times[0,\frac{1}{2}])$. What remains of $X$
is a submanifold $N:=X\setminus(\partial X\times[0,\frac{1}{2}])$. The
submanifold $N$ already has a preferred normal vector bundle on the closed
subset $K:=\partial X\times(1/2,1]$. Now apply [FQ90, Theorem 9.3A] to the
triple $(M^{\prime},N,K)$ to obtain a normal vector bundle $E\to N$ agreeing
with the given one on $K$. The normal vector bundles over $N$ and $\partial
X\times[0,1]$ agree on the overlap $K$. Thus they define a normal vector
bundle on all of $X$. ∎

Figure 6. Illustration of the proof of Theorem 6.17.

Next we turn to the uniqueness of normal vector bundles.

######  Theorem 6.20 (Uniqueness of normal vector bundles).

Let $M$ be a compact 4-manifold and let $X$ be a proper submanifold of $M$.
Suppose we are given two normal vector bundles $p_{i}\colon E_{i}\to X$,
$i=1,2$ for $X$. For $i=1,2$ let $\iota_{i}\colon E_{i}\to M$ be the inclusion
map. Then there exists a bundle isomorphism $f\colon
E_{1}\xrightarrow{\cong}E_{2}$ such that $\iota_{2}\circ f$ and $\iota_{1}$
are ambiently isotopic rel. $X$.

###### Proof.

If $X$ has no boundary, then the theorem is an immediate consequence of [FQ90,
Theorem 9.3D]. Now suppose that $X$ has nonempty boundary.

First we claim that any normal vector bundle of $X$ is obtained by the
construction outlined in the proof of Theorem 6.17. To see this, let $p\colon
E\to X$ be a normal vector bundle. Pick a collar neighbourhood $\partial
X\times[0,2]\subset X$. Since $p$ is extendable, we can view $p$ as the
interior of a disc bundle $q\colon F\to X$ in $M$. Write $C:=q^{-1}(\partial
X)\subseteq\partial M$. The disc bundle $q\colon q^{-1}(\partial
X\times[0,2])\to\partial X\times[0,2]$ defines a collar neighbourhood
$C\times[0,2]$ for the compact submanifold $C$ of $\partial M$. By the collar
neighbourhood theorem 2.5 we can extend the collar neighbourhood
$C\times[0,1]$ of $C$ to a collar neighbourhood $\partial M\times[0,1]$. With
this choice of collar neighbourhood of $\partial M$, the construction in the
proof of Theorem 6.17, with further appropriate choices, gives rise to the
normal vector bundle $p\colon E\to X$. This completes the proof of the claim.

After this long preamble it suffices to prove the theorem for any two normal
vector bundles obtained as in the proof of Theorem 6.17. Uniqueness follows by
arguing that each step in the proof of existence of normal vector bundles was
essentially unique. The proofs of uniqueness in the three steps make use of
the following ingredients.

First, apply the uniqueness statement for normal vector bundles of
submanifolds of smooth manifolds to $\partial X\subseteq\partial M$ e.g.
[Kos93, Chapter III.2] or [Lan02b, Section IV.5].

Next use the uniqueness of collar neighbourhoods as formulated in Theorem 2.7,
applied to the two collar neighbourhoods of $\partial M$ subordinate to the
given normal bundles of $X$.

Finally apply the full relative version of [FQ90, Theorem 9.3D] to extend the
normal vector bundle uniquely over the rest of $X$. ∎

###  6.4. Tubular neighbourhoods: proofs

Now we will use the results from the previous section to prove the existence
and uniqueness of tubular neighbourhoods. First we show how one can obtain
tubular neighbourhoods from normal vector bundles.

######  Definition 6.21.

Let $p\colon E\to X$ be a vector bundle. Given $x\in X$, write
$E_{x}:=p^{-1}(x)$. A _positive definite form_ $g=\\{g_{x}\\}_{x\in X}$
consists of a positive definite form $g_{x}$ for every $E_{x}$ such that
$g_{x}$ changes continuously with $x$.

The following lemma follows from standard techniques, so we leave it to the
reader to fill in the details.

######  Lemma 6.22.

Let $X$ be a compact manifold and let $p\colon E\to X$ be an $n$-dimensional
vector bundle. Then the space of positive definite forms on $E$ is nonempty
and convex. Furthermore, let $g=\\{g_{x}\\}_{x\in X}$ be a positive definite
form on $E$ and consider the map

$p\colon E(g)\,:=\,\bigcup\limits_{x\in X}{}\\{v\in E_{x}\mid g_{x}(v,v)\leq
1\\}\,\,\to\,\,X.$

This map has the following properties:

  1. (1)

The map $p\colon E(g)\to X$ is a linear $D^{n}$-bundle.

  2. (2)

Given two different positive definite forms $g$ and $h$ on $E$ there exists an
isotopy of the vector bundle $E$ sending $E(g)$ to $E(h)$, and restricting to
the identity on the 0-section and outside some compact subset of $E$.

Let $X$ be a compact manifold and let $p\colon E\to X$ be a vector bundle.
Given any positive definite form $g$ we refer to

$p\colon E^{\prime}\,\,:=\,\,\bigcup\limits_{x\in X}{}\\{v\in E_{x}\mid
g_{x}(v,v)\leq 1\\}\,\,\to\,\,X$

as a _corresponding disc bundle_. It follows from Lemma 6.22 that for most
purposes the precise choice of $g$ is irrelevant.

We can now prove the existence of tubular neighbourhoods.

###### Proof of the tubular neighbourhood theorem 6.8.

Let $X$ be a compact proper submanifold of a $4$-manifold $M$. By Theorem 6.17
there exists a normal vector bundle $p\colon N\to X$ for $X$. A choice of
corresponding disc bundle is easily seen to be a tubular neighbourhood. ∎

The uniqueness proof for tubular neighbourhoods also requires us to associate
a normal vector bundle to a tubular neighbourhood.

######  Lemma 6.23.

Let $M$ be a compact $4$-manifold and let $X$ be a compact proper
$k$-dimensional submanifold. Let $p\colon N\to X$ be a tubular neighbourhood
for $X$. There exists a normal vector bundle $q\colon E\to X$ and a positive
definite form $g$ such that $N=E(g)$ and $p\colon N\to X$ equals $q\colon
E(g)\to X$.

###### Proof.

Let $p\colon N\to X$ be a tubular neighbourhood for $X$. Recall that we have
$\operatorname{Int}N=N\setminus\partial_{0}N$. Consider
$W:=M\setminus\operatorname{Int}N$. This is a compact $4$-manifold. Pick a
collar neighbourhood $\partial W\times[0,1]$ and set
$E:=N\cup\partial_{0}N\times[0,\frac{1}{2})$. We have an obvious projection
map $q\colon E\to X$ turning $q$ into a bundle map where the fibre is given by
the open $(4-k)$-ball of radius $\frac{3}{2}$. We leave it to the reader to
turn $q\colon N\to X$ into an internal linear bundle, to show that it is in
fact extendable (at this point one has to use that in the definition of $E$ we
only used “half” of the collar neighbourhood $\partial_{0}N\times[0,1]$), and
to equip $N$ with a positive definite form $g$ such that $N=E(g)$. ∎

We conclude the section with the proof of the uniqueness theorem for tubular
neighbourhoods.

###### Proof of Theorem 6.9.

Let $M$ be a $4$-manifold and let $X$ be a compact proper $k$-dimensional
submanifold. Furthermore let $p_{i}\colon N_{i}\to X$, $i=1,2$ be two tubular
neighbourhoods of $X$. For $i=1,2$, let $q_{i}\colon E_{i}\to X$ be two
corresponding normal vector bundles and let $g_{i}$ be the positive definite
forms provided by Lemma 6.23. It follows from Theorem 6.20 that there exists a
bundle isomorphism $f\colon E_{1}\xrightarrow{\cong}E_{2}$ such that
$\iota_{2}\circ f$ and $\iota_{1}$ are ambiently isotopic rel $X$. It follows
from the definitions that $N_{2}$ is equivalent to the disc bundle defined by
$f^{*}g_{2}$ on $E_{1}$. But Lemma 6.22 (3) implies that $f^{*}g_{2}$ and
$g_{1}$ define equivalent tubular neighbourhoods. ∎

##  7\. Background on bundle structures

In this section we recall the bundle technologies we will need to use in later
sections. First we recall the three standard types of fibre bundle with fibre
$\mathbb{R}^{n}$: $\operatorname{O}$, $\operatorname{PL}$ and
$\operatorname{TOP}$.

######  Definition 7.1.

Let $\operatorname{TOP}(n)$ be the subgroup of homeomorphisms of
$\mathbb{R}^{n}$ that fix the origin, topologised using the compact open
topology. A principal $\operatorname{TOP}(n)$-bundle has an associated fibre
bundle with fibre $\mathbb{R}^{n}$ and a preferred 0-section. Call such a
bundle a _topological $\mathbb{R}^{n}$-bundle_. Let $\operatorname{TOP}$ be
the colimit
$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\scriptscriptstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits\operatorname{TOP}(n)$
under the inclusions
$-\times\operatorname{Id}_{\mathbb{R}}\colon\operatorname{TOP}(n)\to\operatorname{TOP}(n+1)$.
Write $\operatorname{BTOP}(n)$ and $\operatorname{BTOP}$ for the corresponding
classifying spaces.

Let $\operatorname{O}(n)$ be the orthogonal homeomorphisms of $\mathbb{R}^{n}$
that fix the origin. Similarly to the case of homeomorphisms, define
$\operatorname{BO}(n)$, $\operatorname{O}$, and $\operatorname{BO}$.

The definition of the analogous spaces for $\operatorname{PL}$ is a little
more involved, using semi-simplicial groups. For a gentle introduction to
simplicial sets, see [Fri12]. The canonical reference for classifying spaces
constructed using simplicial groups is [May75].

######  Definition 7.2.

A homeomorphism $f\colon K\to L$ between two simplicial complexes $K$ and $L$
is a _$\operatorname{PL}$ -homeomorphism_ if there are subdivisions
$K^{\prime}$ of $K$ and $L^{\prime}$ of $L$ such that $f\colon K^{\prime}\to
L^{\prime}$ is a simplicial map.

For background on piecewise linear topology see [RS72]. The next definition
comes from [RS68]. Let $\Delta^{k}$ be the standard $k$-simplex.

######  Definition 7.3.

Let $\operatorname{PL}(n)_{\bullet}$ be the semi-simplicial group defined as
follows.

  1. (i)

The group $\operatorname{PL}(n)_{k}$ assigned to the $k$-simplex is the group
of $\operatorname{PL}$ homeomorphisms
$f\colon\mathbb{R}^{n}\times\Delta^{k}\to\mathbb{R}^{n}\times\Delta^{k}$ over
$\Delta^{k}$, such that $f|_{\mathbb{R}^{n}\times\\{t\\}}$ fixes the origin in
$\mathbb{R}^{n}$ for every $t\in\Delta^{k}$. That is, with
$p\colon\mathbb{R}^{n}\times\Delta^{k}\to\Delta^{k}$ the projection, the
diagram

$\textstyle{\mathbb{R}^{n}\times\Delta^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{f}$$\textstyle{\mathbb{R}^{n}\times\Delta^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\Delta^{k}}$

commutes.

  2. (ii)

The $i$th face map is given by restricting to the $i$th face of $\Delta^{k}$.

Now define $\operatorname{BPL}(n)$ by first using the level-wise bar
construction to obtain a semi-simplicial space, and then geometrically
realising to obtain a space $\operatorname{BPL}(n)$. Define
$\operatorname{PL}$ and $\operatorname{BPL}$ as colimits analogously to
Definition 7.1.

######  Remark 7.4.

It is interesting that we do not define $\operatorname{PL}(n)$ using the
subspace topology from $\operatorname{TOP}(n)$. Note that we also do not
define $\operatorname{Diff}(n)=\operatorname{Diff}(\mathbb{R}^{n})$ in this
way. But for defining $\operatorname{Diff}(n)$ as a topological group, and
using this topology to define $\operatorname{BDiff}(n)$, we have the bespoke
Whitney topology. In the absence of an analogous topology for
$\operatorname{PL}(n)$, we use the simplicial strategy. In fact this
simplicial method could be used to define all three of
$\operatorname{TOP}(n)$, $\operatorname{PL}(n)$ and $\operatorname{Diff}(n)$,
giving a uniform treatment. But only in the $\operatorname{PL}$ case do we
really know of no other method that works.

All smooth manifolds have tangent vector bundles and all smooth submanifolds
have normal vector bundles. This is one reason that vector bundles,
corresponding to the structure group $\operatorname{O}(n)$, are the de facto
bundle technology in the smooth category. A general difficulty we will face
when talking about manifold transversality in Section 10 is that we will need
to use some well-defined notion of normal structure for a submanifold and,
outside of the smooth category, submanifolds do not necessarily admit normal
vector bundles. However, various weaker bundle technologies have been
developed, which replace this crucial concept in the topological category.

The rest of this section is devoted to a discussion of _microbundles_ [Mil64].
The existence and uniqueness of microbundles leads to the existence and
uniqueness of tangent and (stable) normal $\mathbb{R}^{n}$-bundles for
$\operatorname{TOP}(n)$, as discussed below. Source material on microbundles
is not hard to find in the literature, but has been included here for the
convenience of the reader, in order for this survey to be more self-contained.

The interaction between the weaker fibre automorphism groups
$\operatorname{PL}(n)$ and $\operatorname{TOP}(n)$ for tangent and (stable)
normal $\mathbb{R}^{n}$-bundles, and the topological/piecewise linear/smooth
structures on the manifold itself are the topic of _smoothing theory_ , to
which we turn in Section 8.

######  Definition 7.5.

An $n$-dimensional _microbundle_ $\xi$ consists of a base space $B$ and a
total space $E$ sitting in a diagram

$B\xrightarrow{i}E\xrightarrow{r}B,$

such that $r\circ i=\operatorname{Id}_{B}$, and that is _locally trivial_ in
the following sense: for every point $b\in B$, there exists an open
neighbourhood $U$, an open neighbourhood $V$ of $i(b)$ and a homeomorphism
$\phi_{b}\colon V\to U\times\mathbb{R}^{n}$ such that

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{\phi_{b}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\times\\{0\\}}$$\textstyle{U}$$\textstyle{U\times\mathbb{R}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{1}}$

commutes.

Note that we only require a neighbourhood of $i(b)$ to be trivial, and not all
of the fibre $r^{-1}(b)$. In fact, we only care about neighbourhoods
$i(B)\subset E$, and declare two microbundles $B\to E\to B$ and $B\to
E^{\prime}\to B$ to be _equivalent_ , if $i(B)$ and $i^{\prime}(B)$ have
homeomorphic neighbourhoods such that the homeomorphism commutes with both the
inclusion map and the restriction of the retraction map.

######  Definition 7.6.

Let $r\colon E\to B$ be a microbundle $\xi$ and let $f\colon A\to B$ be a map.
The _pullback_ of $\xi$ under $f$ is the microbundle $f^{*}\xi$ with total
space

$f^{*}E=\big{\\{}(a,e)\in A\times E\mid f(a)=r(e)\big{\\}},$

retraction $(f^{*}r)(a,e)=a$, and section
$(f^{*}s)(a)=\big{(}a,s(f(a))\big{)}$. In the case that $f$ is an inclusion,
also consider the microbundle $\xi|_{A}$, which has total space
$r^{-1}(A)\subset E$, and retraction $r_{A}\colon r^{-1}(A)\to A$ and section
$s_{A}\colon A\to r^{-1}(A)$ are both the restrictions of $r$, $s$. In this
case, the map of total spaces $(a,e)\mapsto e$ gives a preferred isomorphism
$f^{*}\xi$ to $\xi|_{A}$.

A topological $\mathbb{R}^{n}$-bundle clearly has an underlying microbundle.
Kister and Mazur proved independently the surprising result that _every_
microbundle is equivalent to such an underlying microbundle [Kis64, Theorem
2].

######  Theorem 7.7 (Kister-Mazur).

Let $B$ be a manifold and $B\xrightarrow{i}E\xrightarrow{r}B$ be an
$n$-dimensional microbundle $\xi$. Then there exists an open set $F\subset E$
containing $i(B)$ such that $r\colon F\to B$ is the projection map of a
topological $\mathbb{R}^{n}$-bundle, whose $0$-section is $i$ and whose
underlying microbundle is $\xi$. Moreover, if $F_{1}$ and $F_{2}$ are any two
topological $\mathbb{R}^{n}$-bundles over $B$ such that the underlying
microbundles are isomorphic, then $F_{1}$ and $F_{2}$ are isomorphic as
topological $\mathbb{R}^{n}$-bundles.

Every manifold admits a tangent microbundle.

######  Definition 7.8 (Tangent microbundle).

The _tangent microbundle_ of an $n$-dimensional manifold $M$ is the
microbundle $M\xrightarrow{\Delta}M\times
M\xrightarrow{\operatorname{pr}_{1}}M$ where $\Delta$ is the diagonal map. The
Kister-Mazur theorem implies this corresponds to a unique _topological tangent
bundle_ $\tau_{M}\colon M\to\operatorname{BTOP}(n)$, with corresponding
_stable topological tangent bundle_ $\tau_{M}\colon M\to\operatorname{BTOP}$.

More subtle is the concept of a normal microbundle.

######  Definition 7.9 (Normal microbundle).

A _normal microbundle_ of a submanifold $S$ of a manifold $M$ is a microbundle
$S\to E\to S$ such that $E$ is a neighbourhood of $S$ in $M$.

It is immediate from the definition of normal microbundle that the local
flatness in the definition of a submanifold $S$ is a necessary condition for
the existence of a normal microbundle. Wild submanifolds that are not
submanifolds do not admit normal microbundles. Indeed, it is generally far
from straightforward to prove the existence of normal microbundles at all.
Here is an existence and uniqueness result due to Stern [Ste75, Theorem‘4.5].
See also [Hir66],[Hir68, p. 65], and [KS77, Essay IV, Appendix A].

######  Theorem 7.10.

Let $M^{n+q}$ be a manifold, and let $N^{n}\subset M^{n+q}$ be a proper
submanifold of codimension $q$. Suppose that $n\leq q+1+j$ and $q\geq 5+j$ for
some $j=0,1,2$. Then $N$ admits a normal microbundle restricting to a normal
microbundle of $\partial N\subset\partial M$.

If in addition $n\leq q+j$, then this normal microbundle is unique up to
isotopy.

######  Remark 7.11 (Unique up to isotopy).

For a submanifold $N\subset M$ we say a normal microbundle $\nu(N)$ is unique
up to isotopy if whenever there is another normal microbundle
$\nu^{\prime}(N)$, there exists a microbundle equivalence $f$ between $\nu(N)$
and $\nu^{\prime}(N)$ such that $\pi^{\prime}\circ f$ is isotopic to $\pi$
relative to $N$.

We exploit these theorems to define a stable normal structure on any manifold,
that will play an important role in Section 8. Consider that any closed
$n$-manifold $M$ can be embedded as a submanifold $M\subset\mathbb{R}^{m}$ for
large $m$ [Hat02, Corollary A.9]. For large enough $m$, any two such
embeddings are isotopic. For large enough $m$, Theorem 7.10 implies there is a
normal microbundle $\xi$. After possibly increasing $m$ further, the last
sentence of Theorem 7.10 implies this normal microbundle $\xi$ is unique. By
the Kister-Mazur Theorem this defines a unique topological
$\mathbb{R}^{m-n}$-bundle. We remove the dependence on $m$ by passing to the
stable bundle $\operatorname{TOP}(m-n)\subset\operatorname{TOP}$. Thus the
process described gives a well-defined classifying map $\nu_{M}\colon
M\to\operatorname{BTOP}$. Summarising, we have the following.

######  Definition 7.12.

Given any closed $n$-manifold, the topological $\mathbb{R}^{\infty}$-bundle
$\nu_{M}\colon M\to\operatorname{BTOP}$, described above, is called the
_stable topological normal bundle_. It is well-defined and unique.

The next example shows that outside the hypotheses of Theorem 7.10, we should
expect that normal microbundles can be very badly behaved.

######  Example 7.13.

Normal microbundles do not necessarily exist. Rourke and Sanderson [RS67,
Example 2] construct $S^{19}$ as a submanifold of a certain $28$-dimensional
$\operatorname{PL}$ manifold $M$ in such a way that it does not admit a
topological normal microbundle. The embedding is even piecewise linear.

Even when topological normal microbundles exist, they are not always unique:
Rourke and Sanderson consider the smooth standard embedding $S^{18}\subset
S^{27}$ [RS68, Theorem 3.12] and construct a certain normal microbundle $\xi$
of $S^{18}\subset S^{27}$. The construction of $\xi$ is such that if $\xi$
were concordant to the trivial normal microbundle, this concordance would
induce a normal microbundle structure back on the embedding $S^{19}\subset
M^{28}$ of the previous example. As this is not possible, $\xi$ is non-
trivial. Note that the smooth normal bundle $\nu S^{18}$ of the standard
embedding is trivial, so $S^{18}\subset S^{27}$ admits at least two different
normal microbundles.

The following theorem ensures the issues of the previous example are not seen
in dimension 4.

######  Theorem 7.14.

Let $X$ be a codimension $n$ proper submanifold of a $4$-manifold $M$. Then
$X$ admits a normal microbundle. Moreover, if $\xi$ is a normal microbundle of
$X$, it is the underlying microbundle to a normal vector bundle.

###### Proof.

The existence of normal microbundles in ambient dimension 4 is an immediate
consequence of the existence of normal vector bundles (Theorem 6.17).

Given a normal microbundle $\xi$, we apply the Kister-Mazur theorem 7.7 to
obtain an embedded $\mathbb{R}^{n}$-bundle with underlying microbundle $\xi$.
For $n\leq 2$, the homotopy fibre $\operatorname{TOP}(n)/\operatorname{O}(n)$
for the forgetful map $\operatorname{BO}(n)\to\operatorname{BTOP}(n)$ is
contractible, and for $k\leq 4$ we have
$\pi_{k}(\operatorname{TOP}(3)/\operatorname{O}(3))=0$ [KS77, Essay V,
Theorems 5.8 and 5.9]. Using these facts, and checking the obstructions in
each of the cases $n=0,1,2,3,4$, we see in each case the embedded topological
$\mathbb{R}^{n}$-bundle can be upgraded to an embedded vector bundle. Choose
such a vector bundle refinement. By restricting to an open disc bundle and
rescaling we can ensure this internal linear bundle is extendable and thus a
normal vector bundle in the sense of Definition 6.15. ∎

We will make use of our discussion of normal microbundles in Section 10 on
topological transversality.

##  8\. Smoothing 4-manifolds

We present three theorems which associate a smooth manifold to a given
4-manifold. Often these theorems can be used to reduce proofs about
4-manifolds to the case of smooth 4-manifolds, where the standard tools of
differential topology are available.

###  8.1. Smoothing non-compact 4-manifolds

The first of our smoothing theorems [Qui82, Corollary 2.2.3], [FQ90, p. 116]
says that noncompact connected 4-manifolds admit a smooth structure.

######  Theorem 8.1.

Every connected, noncompact 4-manifold is smoothable. Thus every $4$-manifold
$M$ has a smooth structure in the complement of any closed set that has at
least one point in each compact component of $M$.

There are some related statements in the literature on smoothing 4-manifolds
in the complement of a point, that appeared prior to Freedman’s work [Fre82]
and prior to [Qui82]. We discuss them briefly here. For the case of
$\operatorname{PL}$ structures on noncompact 4-manifolds, given a lift of the
(unstable) tangent microbundle classifying map $M\to\operatorname{BTOP}(4)$ to
the $\operatorname{PL}$ category $M\to\operatorname{BPL}(4)$ (see Section 7),
the proof can be found in [Las70, p. 54] and [KS77, Essay V, Addendum 1.4.1].
This result was stated for smooth bundle structures and smooth structures on
manifolds in [Las71]. Alternatively, [HM74], [FQ90, Theorem 8.3B] apply to
improve a $\operatorname{PL}$ structure to a smooth structure, unique up to
isotopy, for any manifold of dimension at most six. Again, in [Las71] Lashof
assumes a lift of the (unstable) tangent microbundle classifying map
$M\to\operatorname{BTOP}(4)$ to a map $M\to\operatorname{BO}(4)$. For
noncompact connected 4-manifolds, such a lift always exists, as was later
shown by Quinn [Qui82, Qui84], [FQ90, p. 116] using the full disc embedding
theorem [Fre82], and giving rise to Theorem 8.1.

Due to the seminal nature of Freedman’s Field’s medal winning paper [Fre82],
it is well worth clarifying the details of some citations therein. In the
proof of Corollary 1.2, in the proof of Theorem 1.5 on page 369, in the proof
of Theorem 1.6, and at the start of Section 10, Freedman uses that smoothing
theory is available for noncompact 4-manifolds. In particular, smoothing for
noncompact contractible 4-manifolds plays a vital rôle in Freedman’s proof of
the topological 4-dimensional Poincaré conjecture [Fre82, Theorem 1.6].
Freedman cites [KS77] for this fact, however [KS77, Essay V, Remarks 1.6 (A)]
specifically excludes smooth structures (but for a stronger result).
Nevertheless, as mentioned above, Lashof [Las71, p. 156] proved the smooth
version of [KS77, Essay V, Addendum 1.4.1], or one can use $\operatorname{PL}$
smoothing theory [HM74], [FQ90, Theorem 8.3B] to improve a PL structure from
[KS77, Essay V, Addendum 1.4.1] to a smooth structure, essentially uniquely.

Freedman only applies smoothing theory in cases, such as for contractible $M$,
that he can ensure the existence of a lift of $\tau_{M}\colon
M\to\operatorname{BTOP}(4)$ to $\operatorname{BO}(4)$. Later, Quinn [Qui82,
Corollary 2.2.3] showed that such a lift always exists for connected
noncompact 4-manifolds. In fact, he showed that the map
$\operatorname{TOP}(4)/\operatorname{O}(4)\to\operatorname{TOP}/\operatorname{O}\simeq
K(\mathbb{Z}/2,3)$ is 5-connected [FQ90, Theorem 8.7A], where only 3-connected
is needed for Theorem 8.1. In other words, it was shown prior to Freedman’s
work that homotopy 4-spheres admit a smooth structure in the complement of a
point, so the results that Freedman required were indeed known. However,
smoothing in the complement of a point was not known for general connected
4-manifolds until after the work of Quinn in 1982. Further discussion can also
be found in Quinn [Qui84] and Lashof-Taylor [LT84].

Below we will give applications of Theorem 8.1, see e.g. the proof of Theorem
10.9.

###  8.2. The Kirby-Siebenmann invariant and stable smoothing of 4-manifolds

The formulation of the other two statements on smoothing 4-manifolds make use
of the Kirby-Siebenmann invariant. The Kirby-Siebenmann invariant
$\operatorname{ks}(M)\in\mathbb{Z}/2$ of a compact 4-manifold is defined in
[FQ90, Section 10.2B], or alternatively by [KS77, p. 318] or [Rud16,
Definition 3.4.2], and we describe the construction now.

The homotopy fibre $\operatorname{TOP}/\operatorname{PL}$ of the forgetful map
$\operatorname{BPL}\to\operatorname{BTOP}$ has the homotopy type of a
$K(\mathbb{Z}/2,3)$ and has the structure of a loop space, permitting the
definition of the delooping
$\operatorname{B}(\operatorname{TOP}/\operatorname{PL})$ [BV68, Theorem C],
[BV73] which is an Eilenberg-Maclane space of type $K(\mathbb{Z}/2,4)$. A
connected topological 4-manifold has a unique smooth structure on its
boundary. Using the homotopy fibre sequence

$\operatorname{TOP}/\operatorname{PL}\to\operatorname{BPL}\to\operatorname{BTOP}\to\operatorname{B}(\operatorname{TOP}/\operatorname{PL}),$

the unique obstruction to a lift of the classifying map $\tau_{M}\colon
M\to\operatorname{BTOP}$ of the stable tangent microbundle to
$\operatorname{BPL}$ is therefore a homotopy class in

$[(M,\partial
M),(\underset{=K(\mathbb{Z}/2,4)}{\underbrace{\operatorname{B}(\operatorname{TOP}/\operatorname{PL})}},*)]\cong
H^{4}(M,\partial M;\mathbb{Z}/2)=\mathbb{Z}/2.$

Here, we used again that 4-manifolds have the homotopy type of a CW-complex.
We refer to the corresponding element of $\mathbb{Z}/2$ as the Kirby-
Siebenmann $\operatorname{ks}(M)$ invariant of the compact, connected manifold
$M$. For disconnected compact 4-manifolds, $M=\bigsqcup_{i=1}^{n}M_{i}$,
define

$\operatorname{ks}(M)=\sum_{i=1}^{n}\operatorname{ks}(M_{i})\in\mathbb{Z}/2.$

In the following theorem we summarise some key properties of the Kirby-
Siebenmann invariant.

######  Theorem 8.2.

Let $M$ and $N$ be compact 4-manifolds.

  1. (1)

If $M\times\mathbb{R}$ admits a smooth structure $($e.g. if $M$ admits a
smooth structure$)$, then $\operatorname{ks}(M)=0$.

  2. (2)

The Kirby-Siebenmann invariant gives rise to a surjective homomorphism
$\Omega_{4}^{\operatorname{TOP}}\to\mathbb{Z}/2$. In particular for $M$ a
closed 4-manifold that bounds a compact 5-manifold, $\operatorname{ks}(M)=0$.

  3. (3)

The Kirby-Siebenmann invariant is additive under the connected sum operation.

  4. (4)

The forgetful map
$\Omega^{\operatorname{Spin}}_{4}\to\Omega_{4}^{\operatorname{TOPSpin}}$ fits
into a short exact sequence

$0\to\Omega^{\operatorname{Spin}}_{4}\to\Omega_{4}^{\operatorname{TOPSpin}}\xrightarrow{\operatorname{ks}=\sigma/8}\mathbb{Z}/2\to
0$

with the last map given by the signature divided by 8, modulo 2, which equals
the Kirby-Siebenmann invariant
$\Omega_{4}^{\operatorname{TOPSpin}}\to\Omega_{4}^{\operatorname{STOP}}\xrightarrow{\operatorname{ks}}\mathbb{Z}/2$.
This sequence does not split, so
$\Omega_{4}^{\operatorname{TOPSpin}}\cong\mathbb{Z}$.

  5. (5)

If $S\subset\partial M$ and $T\subset\partial N$ are compact codimension zero
submanifolds with $S\cong T$, then

$\operatorname{ks}(M\cup_{S\cong
T}N)=\operatorname{ks}(M)+\operatorname{ks}(N).$

  6. (6)

If there exists a compact 5-manifold with $\partial W=M\cup_{\partial
M=\partial N}N$, then $\operatorname{ks}(M)=\operatorname{ks}(N)$.

We could not find explicit proofs of these facts in the literature, so we give
some details.

###### Proof of Theorem 8.2.

Let us prove (1). The tangent bundle of $M\times\mathbb{R}$ is isomorphic to
$\tau_{M}\oplus\varepsilon$, where $\tau_{M}$ is the tangent microbundle of
$M$ and $\varepsilon$ denotes a rank one trivial vector bundle over $M$. If
$M\times\mathbb{R}$ admits a smooth structure, then there is a lift
$\tau_{M\times\mathbb{R}}^{\operatorname{Diff}}\colon
M\to\operatorname{BO}(5)$, the smooth tangent bundle to $M\times\mathbb{R}$.
Let $p\colon\operatorname{BO}(5)\to\operatorname{BTOP}(5)$ be the canonical
map. Then
$\tau_{M}\oplus\varepsilon=p\circ\tau_{M\times\mathbb{R}}^{\operatorname{Diff}}$.
Passing to the stable classifying spaces, we obtain a lift
$M\to\operatorname{BO}$ whose composition with the canonical map
$\operatorname{BO}\to\operatorname{BTOP}$ agrees with
$\tau_{M}\oplus\varepsilon^{\infty}$, the stable tangent microbundle of $M$.
Since the map $\operatorname{BO}\to\operatorname{BTOP}$ factors through
$\operatorname{BPL}\to\operatorname{BTOP}$, we have a stable lift of
$\tau_{M}$ and so $\operatorname{ks}(M)=0$. This completes the proof of (1).

Now to prove (2), suppose that a closed 4-manifold
$M=\bigsqcup_{i=1}^{k}M_{i}$ bounds a compact 5-manifold $W^{\prime}$. Perform
0 and 1-surgeries on $W^{\prime}$ to obtain a path connected, simply
connected, compact 5-manifold with $\partial W=M$. We prove that
$\operatorname{ks}(M):=\sum_{i=1}^{k}\operatorname{ks}(M_{i})=0$. Consider the
diagram

$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ks}(M_{i})}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ks}(M)}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{W}}$$\scriptstyle{\operatorname{ks}(W)}$$\textstyle{\operatorname{BTOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{B}(\operatorname{TOP}/\operatorname{PL})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{K(\mathbb{Z}/2,4).}$

The restriction $M_{i}\to W\to\operatorname{BTOP}$ equals the stable tangent
microbundle of $M$, since $M$ has a collar $M\times[0,1]\subset W$ by Theorem
2.5. Therefore the diagram commutes. It follows that the top left horizontal
map in the next diagram sends $\operatorname{ks}(W)$ to
$(\operatorname{ks}(M_{1}),\dots,\operatorname{ks}(M_{k}))$, so the map
$H^{4}(W;\mathbb{Z}/2)\to\mathbb{Z}/2$ sends $\operatorname{ks}(W)$ to
$\sum_{i=1}^{k}\operatorname{ks}(M_{i})=\operatorname{ks}(M)$.

$\textstyle{H^{4}(W;\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H^{4}(M;\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{\bigoplus_{i=1}^{k}H^{4}(M_{i};\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{\bigoplus_{i=1}^{k}\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,\dots,1)}$$\textstyle{\mathbb{Z}/2}$$\textstyle{H_{1}(W,M;\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}(M;\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\bigoplus_{i=1}^{k}H_{0}(M_{i};\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$

The left square of this diagram commutes by Poincaré-Lefschetz duality. The
middle square and the triangle commute trivially. But since $W$ is connected
and simply connected, every element of $H_{1}(W,M;\mathbb{Z}/2)$ can be
represented by a (possibly empty) union of arcs with boundary on $M$. Thus the
image of $\operatorname{ks}(W)$ in
$\bigoplus_{i=1}^{k}H_{0}(M_{i};\mathbb{Z}/2)$ is nonzero in evenly many
summands, and therefore its image in $\mathbb{Z}/2$ on the far right is zero.
By commutativity of the diagram it follows that $\operatorname{ks}(M)=0$, as
desired.

Now (2) follows. First note that the addition on
$\Omega_{4}^{\operatorname{TOP}}$ is by disjoint union, so $\operatorname{ks}$
is additive by definition. We have just shown that the map
$\operatorname{ks}\colon\Omega_{4}^{\operatorname{TOP}}\to\mathbb{Z}/2$ is
well-defined, since for $M$ a closed 4-manifold that bounds a compact
5-manifold, $\operatorname{ks}(M)=0$. Therefore
$\operatorname{ks}\colon\Omega_{4}^{\operatorname{TOP}}\to\mathbb{Z}/2$ is a
homomorphism as desired.

The $E_{8}$ manifold, whose existence was established by Freedman [Fre82,
Theorem 1.7] as a key step in the proof of the Classification Theorem 11.2,
has $\operatorname{ks}(E_{8})=1$. To see this note that $E_{8}$ cannot be
smoothed, even after adding copies of $S^{2}\times S^{2}$, by Rochlin’s
theorem (Theorem 3.4) that every closed spin smooth 4-manifold has signature
divisible by 16. Whereas if $\operatorname{ks}(E_{8})=0$, then $E_{8}$ would
be stably smoothable by Theorem 8.6. Therefore
$\operatorname{ks}\colon\Omega_{4}^{\operatorname{TOP}}\to\mathbb{Z}/2$ is
surjective.

Now we can prove (3) easily. Observe that a disjoint union $M\sqcup N$ is
cobordant to $M\\#N$ via the cobordism

$(M\times I\sqcup N\times I)\cup_{S^{0}\times D^{4}}(D^{1}\times D^{4}),$

with $\\{-1\\}\times D^{4}$ embedded in the interior of $M\times\\{1\\}$, and
$\\{1\\}\times D^{4}$ embedded in the interior of $N\times\\{1\\}$. Then we
have just shown that the Kirby-Siebenmann invariant vanishes on $M\\#N\sqcup
M\sqcup N$ and therefore
$\operatorname{ks}(M\\#N)=\operatorname{ks}(M)+\operatorname{ks}(N)\in\mathbb{Z}/2$.

To prove (4), we consider the following diagram. The maps between bordism
groups are structure forgetting maps, so the diagram commutes.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{4}^{\operatorname{Spin}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
16}$$\textstyle{\Omega_{4}^{\operatorname{TOPSpin}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{4}^{\operatorname{SO}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{4}^{\operatorname{STOP}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Recall that $\Omega_{4}^{\operatorname{SO}}\cong\mathbb{Z}$ given by the
signature and generated by $\mathbb{CP}^{2}$. The signature provides a
splitting homomorphism, so
$\Omega_{4}^{\operatorname{STOP}}\cong\mathbb{Z}\oplus\mathbb{Z}/2$. Also
$\Omega_{4}^{\operatorname{Spin}}\cong\mathbb{Z}$ given by the signature
divided by 16 and generated by the $K3$ surface.

Both sequences are exact: a smooth manifold has vanishing $\operatorname{ks}$
invariant, and vanishing $\operatorname{ks}(M)$ implies smoothable after
adding copies of $S^{2}\times S^{2}$ by Theorem 8.6 below. Since
$M\\#S^{2}\times S^{2}$ is (spin) bordant to $M$, the sequences are exact at
their middle terms. Finally, a topological null bordism of a compact smooth
4-manifold can be smoothed by high dimensional smoothing theory, so the left
hand maps are injective.

We claim that the sequence in the upper row does not split. Consider the $K3$
surface generating $\Omega_{4}^{\operatorname{Spin}}\cong\mathbb{Z}$. By the
down-then-left route, $[K3]$ maps to
$(16,0)\in\mathbb{Z}\oplus\mathbb{Z}/2\cong\Omega_{4}^{\operatorname{STOP}}$.
On the other hand the $E_{8}$-manifold represents a class in
$\Omega_{4}^{\operatorname{TOPSpin}}$ and maps to
$(8,1)\in\mathbb{Z}\oplus\mathbb{Z}/2\cong\Omega_{4}^{\operatorname{STOP}}$.

It follows that $2\cdot[E_{8}]$ maps to $0\in\mathbb{Z}/2$ so has trivial
$\operatorname{ks}$ invariant, and therefore by exactness of the top row lies
in the image of $\Omega_{4}^{\operatorname{Spin}}$. Let $N$ be a closed spin
smooth 4-manifold $\operatorname{TOPSpin}$-bordant to $E_{8}\\#E_{8}$. Since
$\sigma(E_{8}\\#E_{8})=16=\sigma(K3)$, we have
$[N]=[K3]\in\Omega_{4}^{\operatorname{Spin}}$. It follows that $K3$, the
generator of $\Omega_{4}^{\operatorname{Spin}}\cong\mathbb{Z}$, maps to
$2\cdot[E_{8}]\in\Omega_{4}^{\operatorname{TOPSpin}}$. Thus we have a diagram
with exact rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
2}$$\scriptstyle{=}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\mapsto[E_{8}]}$$\textstyle{\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\mapsto
2\cdot[E_{8}]}$$\textstyle{\Omega_{4}^{\operatorname{TOPSpin}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since $\operatorname{ks}(E_{8})=1$, the diagram commutes. Then by the five
lemma, $\Omega_{4}^{\operatorname{TOPSpin}}\cong\mathbb{Z}$, generated by
$E_{8}$, and the sequence does not split, as claimed. The diagram

$\textstyle{\Omega_{4}^{\operatorname{TOPSpin}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\sigma/8}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\mapsto(8,1)}$$\textstyle{\Omega_{4}^{\operatorname{STOP}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{Z}\oplus\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ks}}$$\textstyle{\mathbb{Z}/2,}$

which commutes by computing on the generator $E_{8}$ of
$\Omega_{4}^{\operatorname{TOPSpin}}\cong\mathbb{Z}$, shows that
$\operatorname{ks}(M)=\sigma(M)/8\in\mathbb{Z}/2$ for $\operatorname{TOPSpin}$
manifolds $M$. This completes the proof of (4).

To prove (5), it was suggested by Jim Davis to consider the exact sequence

$\Omega_{4}^{\operatorname{O}}\to\Omega_{4}^{\operatorname{\operatorname{TOP}}}\to\Omega_{4}^{\\{{\operatorname{O}}\to\operatorname{\operatorname{TOP}}\\}}\to\Omega_{3}^{\operatorname{O}}=0.$

Here
$\Omega_{4}^{\\{{\operatorname{O}}\to\operatorname{\operatorname{TOP}}\\}}$ is
represented by compact topological 4-manifolds with smooth boundary, up to
5-dimensional cobordism relative to a smooth cobordism on the boundary. That
is, 4-manifolds with boundary $(M,\partial M)$ and $(N,\partial N)$ are
equivalent if there is a compact 5-manifold with boundary

$\partial W=M\cup_{\partial M}\partial_{\operatorname{vert}}W\cup_{\partial
N}N,$

for some smooth 4-dimensional cobordism $\partial_{\operatorname{vert}}W$ with
boundary $\partial M\sqcup\partial N$.

By the exact sequence,
$\Omega_{4}^{\\{{\operatorname{O}}\to\operatorname{TOP}\\}}$ is isomorphic to
the cokernel of
$\Omega_{4}^{\operatorname{O}}\to\Omega_{4}^{\operatorname{\operatorname{TOP}}}$.
We claim that this cokernel is isomorphic to $\mathbb{Z}/2$ via the Kirby-
Siebenmann invariant. To see this, by (2) there is a surjective homomorphism
$\operatorname{ks}\colon\Omega_{4}^{\operatorname{TOP}}\to\mathbb{Z}/2$. If
$\operatorname{ks}(M)=0$ then $M$ is stably smoothable by Theorem 8.6, so $M$
is bordant to a smooth manifold and therefore lies in the image of
$\Omega^{\operatorname{O}}_{4}$. If $M$ is smooth, then $\operatorname{ks}(M)$
is zero, so the sequence
$\Omega_{4}^{\operatorname{O}}\to\Omega_{4}^{\operatorname{\operatorname{TOP}}}\xrightarrow{\operatorname{ks}}\mathbb{Z}/2\to
0$ is exact, and we may identify this sequence with the given sequence.

To prove (5), we therefore need that the disjoint union $M\sqcup N$ is bordant
to $M\cup_{S=T}N$, where $S\subset\partial M$ and $T\subset\partial N$ are
compact codimension zero submanifolds with $S\cong T$. Here is a construction
of such a bordism. For $I=[0,1]$, take

$(M\times I)\,\sqcup\,(S\times I\times[1/2,1])\,\sqcup\,(N\times I),$

identify

$S\times\\{0\\}\times[1/2,1]\,\,\sim\,\,S\times[1/2,1]\,\,\subseteq\,\,(M\times[1/2,1]),$

and, using the identification $S\cong T$, identify

$S\times\\{1\\}\times[1/2,1]\,\sim\,T\times[1/2,1]\subseteq N\times[1/2,1].$

Let $W$ be the result of this gluing and some rounding of corners. The
boundary of $W$ is

$(M\sqcup N)\cup_{\partial M\,\sqcup\,\partial
N}\partial_{\operatorname{vert}}W\cup_{\partial(M\cup_{S=T}N)}M\cup_{S=T}N,$

where

$\displaystyle\partial_{\operatorname{vert}}W\,\,=\,\,$
$\displaystyle(\partial M\times[0,1/2])\cup(\overline{\partial M\setminus
S}\times[1/2,1])$ $\displaystyle\cup$ $\displaystyle(S\times
I\times\\{1/2\\})\cup(\partial S\times I\times[1/2,1])$
$\displaystyle(\partial N\times[0,1/2])\cup(\overline{\partial N\setminus
T}\times[1/2,1]).$

This shows that $M\sqcup N$ and $M\cup_{S=T}N$ are equal in
$\Omega_{4}^{\\{\operatorname{O}\to\operatorname{TOP}\\}}$, and therefore have
the same Kirby-Siebenmann invariants. Since $\operatorname{ks}(M\sqcup
N)=\operatorname{ks}(M)+\operatorname{ks}(N)$, this completes the proof of
(5).

Finally we prove (6). If $M\cup_{\partial M=\partial N}N$ bounds a compact
5-manifold, then by (2) we have that $\operatorname{ks}(M\cup_{\partial}N)=0$.
By (5),
$\operatorname{ks}(M)+\operatorname{ks}(N)=\operatorname{ks}(M\cup_{\partial}N)\in\mathbb{Z}/2$.
Therefore $\operatorname{ks}(M)=\operatorname{ks}(N)$ as required. This proves
(6) and therefore completes the proof of Theorem 8.2. ∎

The following theorem says that the converse to Theorem 8.2 (1) holds.

######  Theorem 8.3.

If $M$ is a compact, connected 4-manifold with vanishing Kirby-Siebenmann
invariant, then $M\times\mathbb{R}$ admits a smooth structure.

###### Proof.

The vanishing of the Kirby-Siebenmann invariant implies that there is a lift
of $\tau_{M}\colon M\to\operatorname{BTOP}$ to a map $M\to\operatorname{BPL}$.
Since $\operatorname{PL}/{\operatorname{O}}$ is 6-connected [FQ90, Theorem
8.3B], [HM74, Proof of 4.13], there is in fact a lift
$\widetilde{\tau}_{M}\colon M\to\operatorname{BO}$. This corresponds to a lift
$\widetilde{\tau}_{M}\oplus\varepsilon^{n}\colon M\to\operatorname{BO}(4+n)$,
for some $n$. This in turn corresponds to a lift

$\widetilde{\tau}_{M\times\mathbb{R}^{n}}\colon
M\times\mathbb{R}^{n}\to{\operatorname{BO}}(4+n)$

of the tangent microbundle $\tau_{M\times\mathbb{R}^{n}}\colon
M\times\mathbb{R}^{n}\to\operatorname{BTOP}$. By [KS77, Essay V, Theorem 1.4],
there exists a corresponding smooth structure on $M\times\mathbb{R}^{n}$. Then
apply the Product Structure Theorem 5.16 [KS77, Essay I, Theorem 5.1], to
deduce the existence of a smooth structure on $M\times\mathbb{R}$, using that
the dimension of $M\times\mathbb{R}$ is at least five. ∎

######  Example 8.4.

Here is an application of Theorem 8.3. By the classification of simply
connected, closed 4-manifolds [FQ90, Section 10.1] (see also our Theorem
11.2), there is a simply connected, closed spin 4-manifold $N$ with
intersection form $E_{8}\oplus E_{8}$. Since this form is not diagonalisable
over $\mathbb{Z}$, by Donaldson’s theorem [Don83] (Theorem 3.7) this
4-manifold does not admit a smooth structure. However the Kirby-Siebenmann
invariant of $N$ vanishes, since for a closed 4-manifold $M$ with even
intersection form, the Kirby-Siebenmann invariant $\operatorname{ks}(M)$
coincides with $\sigma(M)/8\mod{2}$, and $E_{8}\oplus E_{8}$ is rank 16 and
positive definite, with signature 16. Therefore $N\times\mathbb{R}$ admits a
smooth structure by Theorem 8.3, even though $N$ does not.

######  Construction 8.5.

Here is a construction of the Chern manifold $*\mathbb{C}\textup{P}^{2}$. This
manifold was first constructed in [Fre82, p. 370]. Attach a 2-handle
$D^{2}\times D^{2}$ to $D^{4}$ by identifying $S^{1}\times D^{2}$ with a
$+1$-framed trefoil in $\partial D^{4}=S^{3}$. The boundary of the resulting
manifold is an integral homology sphere. Freedman proved that every integral
homology sphere bounds a contractible 4-manifold [Fre82, Theorem 1.4′], [FQ90,
Corollary 9.3C]. Cap off $D^{4}\cup D^{2}\times D^{2}$ with this contractible
4-manifold, to obtain the closed 4-manifold $*\mathbb{C}\textup{P}^{2}$. By
the Rochlin invariant, every compact, smooth, spin 4-manifold with boundary
$+1$-surgery on the trefoil has $\sigma/8$ odd. Therefore the contractible
4-manifold (with signature zero) has Kirby-Siebenman invariant $1$. So by
Theorem 8.2 (5), we have $\operatorname{ks}(*\mathbb{C}\textup{P}^{2})=1$. The
Chern manifold $*\mathbb{C}\textup{P}^{2}$ is homotopy equivalent to
$\mathbb{C}\textup{P}^{2}$ but is not homeomorphic. For further discussion of
the star construction, see [FQ90, Section 10.4] and [Tei97].

The following theorem says in particular that given any compact $4$-manifold
$M$ there exists a closed orientable simply-connected $4$-manifold $N$ such
$M\\#N$ is smoothable.

######  Theorem 8.6.

Let $M$ be compact $4$-manifold. There exists a closed, orientable, simply
connected $4$-manifold $N$ such $M\\#N$ admits a smooth structure. If moreover
the Kirby-Siebenmann invariant of $M$ is zero, then there exists a
$k\in\mathbb{N}_{0}$ such that $M\\#_{k}S^{2}\times S^{2}$ admits a smooth
structure.

###### Proof.

Let $M$ be compact $4$-manifold. Perform the connected sum with an appropriate
number of copies of $*\mathbb{C}\textup{P}^{2}$, the closed oriented simply-
connected $4$-manifold with nontrivial Kirby-Siebenmann invariant, homotopy
equivalent to $\mathbb{C}\textup{P}^{2}$, constructed on [FQ90, p. 167], in
order to obtain a manifold with every connected component having zero Kirby-
Siebenmann invariant. It follows from the discussion on [FQ90, p. 164] and the
Sum-stable smoothing theorem [FQ90, p. 125], that performing the connected sum
with enough copies of $S^{2}\times S^{2}$ produces a manifold that admits a
smooth structure. ∎

######  Remark 8.7.

Given a lift of the classifying map of the (unstable) tangent microbundle of
$M$ to $\operatorname{BO}(4)$, Lashof-Shaneson [LS71] showed that there exists
a $k\in\mathbb{N}_{0}$ such that $M\\#_{k}S^{2}\times S^{2}$ admits a smooth
structure. The result quoted in the previous proof extended this to a lift of
the corresponding _stable_ maps.

##  9\. Tubing of surfaces

As an example of the use of the technology we have discussed thus far, we show
that one can tube together two locally flat embedded surfaces in a 4-manifold,
to obtain an embedding of the connected sum. This operation is standard in the
smooth category, but as ever in the topological category one should take some
care.

The following situation is by no means the most general such result possible.
We wish to illustrate two things. First, that operations on surfaces that can
be performed in the smooth category can usually also be performed in general
4-manifolds with locally flat surfaces (although performing these operations
in a parametrised way seems to be beyond current knowledge). Second, we want
to show the level of detail required to demonstrate that such operations work.

######  Proposition 9.1 (Tubing).

Let $S$ and $T$ be 2-dimensional proper submanifolds of a connected 4-manifold
$M$, that is $S$ and $T$ are locally flat embedded surfaces. Pick a point
$P\in S\setminus\partial S$ and $Q\in T\setminus\partial T$. Let $[\gamma]\in
H_{1}(M,\\{P,Q\\};\mathbb{Z})$ be a relative homology class. There is a
locally flat embedded arc $C$ joining $P$ and $Q$, satisfying the following.

  1. (i)