#### by Laurent Siebenmann

French to English translation by ~~Min Hoon Kim and Mark Powell~~

#### Introduction

At the end of the summer of 1981, in San Diego, M. Freedman proved that every
smooth homotopy 4-sphere __\( M^4 \)__ is homeomorphic to __\( S^4 \)__. Our main goal is
to give an exposition of his proof. In this paper, every manifold will be
metrisable and finite dimensional. We do not know yet whether such an __\( M^4 \)__
is always diffeomorphic to __\( S^4 \)__. On the other hand, Freedman proved that
every *topological* homotopy 4-sphere __\( M^4 \)__ (without any given smooth
structure) is actually homeomorphic to __\( S^4 \)__ (see below).

H. Poincaré
conjectured that every smooth, homotopy __\( n \)__-sphere __\( M^n \)__
is diffeomorphic to __\( S^n \)__. The first nontrivial case, dimension 3,
remains open (in 1982) despite the efforts of countless mathematicians. An
amusing detail is that the counterexample of
J. H. C. Whitehead
[e1]
to his own erroneous proof of this conjecture will
play a large role in this lecture (see Section 2).

J. Milnor
[e4]
discovered smooth manifolds __\( M^7 \)__ which are
homeomorphic to __\( S^7 \)__ but not diffeomorphic to __\( S^7 \)__ (such *exotic
spheres* exist in dimension __\( \geq 7 \)__
[e13]).
Therefore the above
Poincaré conjecture has to be revised for dimension __\( \geq 7 \)__.
S. Smale
[e9]
established his theory of handles to prove that every smooth
homotopy __\( n \)__-sphere is homeomorphic to __\( S^n \)__ for __\( n\geq 6 \)__. His technical
result, the __\( h \)__-cobordism theorem (see below) is more precise. Combining
this with surgery techniques of
Kervaire–Milnor
[e13]
establishes
the __\( n=5 \)__ and 6 cases of the above Poincaré conjecture.
M. Newman
adapted the *engulfing* method of
J. Stallings
to prove the purely
topological version, that is, every topological homotopy __\( n \)__-sphere is
homeomorphic to __\( S^n \)__ if __\( n\geq 5 \)__. (Smale’s surgery method has also
been adapted to the topological category
[e29].)
In summary,
the
Poincaré conjecture is essentially resolved in dimension __\( \geq 5 \)__,
is not resolved in dimension 3, and is partially resolved in dimension 4.

We sketch a proof of Freedman’s theorem which implies the topological
classification of smooth, simply connected
closed 4-manifolds and many
other results of
fundamental importance. Let __\( V \)__ and __\( V^{\prime} \)__ be two such
manifolds. Suppose that there is an isomorphism
__\[ \Theta : H_2(V)\to
H_2(V^{\prime}) \]__
which preserves the intersection forms. (Note that __\( V \)__ is a
homotopy 4-sphere if and only if __\( H_2(V)=0 \)__.)

__\( \Theta \)__is realised by a homeomorphism

__\( V\to V^{\prime} \)__.

*Proof*.
It is not difficult to realise __\( \Theta \)__ by a homotopy
equivalence __\( g : V\to V^{\prime} \)__
[e25].
Surgery theory
[e14],
[e21]
gives a compact 5-manifold __\( W \)__ with boundary
__\( \partial W=V\sqcup -V^{\prime} \)__ such that the inclusions __\( V\to W \)__ and __\( V^{\prime}\to W \)__
are homotopy equivalences and such that the restriction __\( r|_V : V\to V^{\prime} \)__
of the retraction __\( r : W\to V^{\prime} \)__ is homotopic to __\( g \)__. The compact triad
__\( (W;V,V^{\prime}) \)__ is called an __\( h \)__-*cobordism*. Smale’s theory of handles
tries to improve a Morse function
__\[ f : (W;V,V^{\prime})\to ([0,1];0,1) \]__
to obtain a situation where __\( f \)__ has no critical points, that is, __\( f \)__ is a
smooth submersion. Then __\( W \)__ is a fibre bundle over __\( [0,1] \)__ (a remark of
Ehresmann)
and hence __\( W \)__ is diffeomorphic to __\( V\times [0,1] \)__. We are going
to find a topological submersion __\( f \)__ which shows that __\( W \)__ is a topological
fibration on __\( I \)__ (see
[e23],
Section 6)
so that __\( W \)__ is homeomorphic to __\( V\times [0,1] \)__.
◻

In particular, we will prove the simply connected, topological 5-dimensional
__\( h \)__-cobordism theorem.

__\( h \)__-cobordism

__\( (W;V,V^{\prime}) \)__is topologically trivial. That is,

__\( W \)__is homeomorphic to

__\( V\times [0,1] \)__.

For __\( n\geq 6 \)__, instead of 5, Smale’s __\( h \)__-cobordism theorem gives the
stronger conclusion that __\( W \)__ is diffeomorphic to __\( V\times [0,1] \)__. In
dimension 5, his methods apply, but leaving to prove that __\( W \)__ is
diffeomorphic to __\( V\times [0,1] \)__. The following problem is not yet resolved.

__(Unresolved in February 1982.)__Let

__\[ S=S_1\sqcup\cdots \sqcup S_k \quad\textit{and}\quad S^{\prime}=S_1^{\prime}\sqcup\cdots \sqcup S_k^{\prime} \]__be two families of disjointly embedded 2-spheres in a simply connected 4-manifold

__\( M \)__

__(__in fact

__\( f^{-1}(\text{a point}) \)__

__)__in such a way that the homological intersection number

__\( S_i\cdot S_j^{\prime}=\pm \delta_{i,j} \)__. Can one reduce

__\( S\cap S^{\prime} \)__to

__\( k \)__points of intersection

__(__smooth and transverse

__)__by a smooth isotopy of

__\( S \)__

__\( M \)__?

Similarly, to obtain the fact that __\( W \)__ is homeomorphic to __\( V\times [0,1] \)__,
we claim (see
[e15]
and
[e29],
Essay III)
that it suffices to solve the following problem.

__(Resolved here.)__With the data of the smooth problem, reduce

__\( S\cap S^{\prime} \)__to

__\( k \)__points by a topological isotopy of

__\( S \)__in

__\( M \)__, that is given by an ambient isotopy

__\( h_t , \)__

__\( 1\leq t\leq 1 \)__, of

__\( \,\operatorname{Id}|_M \)__fixing a neighbourhood of

__\( k \)__-points of

__\( S\cap S^{\prime} \)__.

Whitney
introduced a natural method for solving these problems. In the
model __\( (\mathbb{R}^2;A,A^{\prime}) \)__,
(this is a straight line __\( A \)__ cutting a parabola __\( A^{\prime} \)__ in two points), we can
disengage __\( A \)__ from __\( A^{\prime} \)__ by a smooth isotopy *with compact support*
(that is, fixing a neighbourhood of __\( \infty \)__). One eliminates thus the two
intersection points.
We deduce that in the stabilised Whitney model,
__\[
(\mathbb{R}^4;A_+^{\vphantom{^{\prime}}},A_+^{\prime})=(\mathbb{R}^2\times
\mathbb{R}^2;A\times 0\times \mathbb{R}, A^{\prime}\times \mathbb{R}\times0),
\]__
there is an isotopy with compact support that makes the plane
__\( A_+^{\vphantom{^{\prime}}} \)__ disjoint from the plane __\( A_+^{\prime} \)__, deleting the two
transverse intersection points between __\( A_+^{\vphantom{^{\prime}}} \)__ and __\( A_+^{\prime} \)__.

We call a smooth (resp. topological) *Whitney process*, a smooth
embedding (resp. a topological embedding) of a disjoint union of copies of
the model __\( (\mathbb{R}^4;A_+^{\vphantom{^{\prime}}},A_+^{\prime}) \)__, whose image contains
__\( S\cap S^{\prime}\smallsetminus (k \text{ points}) \)__. Such a procedure would
clearly give the demanded isotopy to resolve the remaining smooth problem
(respectively, the remaining topological problem).

__(Casson–Freedman.)__In this context, after a preliminary smooth isotopy of

__\( S \)__

__\( \text{in } M \)__,

__(__adding intersection points with

__\( S^{\prime} \)__by finger moves, far from

__\( S\cap S^{\prime} \)__

__)__, the topological Whitney process becomes possible.

The first step of the proof (1973–1976) is due to
A. Casson.
Let
__\( B \)__ be a smooth, compact 2-disc in the boundary component
of __\( \mathbb{R}^2\smallsetminus A\cup A^{\prime} \)__. The product __\( B\times
\mathbb{R}^2 \)__ is an open, embedded 2-handle (as a closed submanifold)
in the Whitney model, and disjoint from __\( A_+^{\vphantom{^{\prime}}}\cup A_+^{\prime} \)__. In
__\( B\times \mathbb{R}^2 \)__, Casson constructed certain open sets __\( H=B\times
\mathbb{R}^2\smallsetminus \Omega \)__ with boundary __\( \partial H=\partial
B\times \mathbb{R}^2 \)__, that we call *open Casson handles*. (See
Section 2 for the precise definition). We are again unable (in
February 1982) to decide whether __\( H \)__ is diffeomorphic to __\( B\times \mathbb{R}^2 \)__ or
not. Replacing __\( B\times \mathbb{R}^2 \)__ by __\( H\subset B\times \mathbb{R}^2 \)__
in this Whitney model __\( (\mathbb{R}^4;A_+^{\vphantom{^{\prime}}},A_+^{\prime}) \)__ we have an
open set __\( (\mathbb{R}^4\smallsetminus \Omega;A_+^{\vphantom{^{\prime}}},A_+^{\prime}) \)__,
that we call the Whitney–Casson model. By a remarkable infinite process,
Casson proved the following.

__(Casson__

__[e38]__,

__compare__

__[e33]__.

__)__After a preliminary smooth isotopy of

__\( S \)__in

__\( M \)__, one can find in

__\( (M;S,S^{\prime}) \)__smoothly embedded, disjoint Whitney–Casson models so that the models contain all the points of

__\( S\cap S^{\prime} \)__except the

__\( k \)__intersection points.

The theorem of Casson and Freedman now follows from the theorem that we will discuss.

__(Freedman, 1981.)__Every open Casson handle is homeomorphic to

__\( B^2\times \mathbb{R}^2 \)__. Therefore, the Whitney model

__\( (\mathbb{R}^4;A_+^{\vphantom{^{\prime}}},A_+^{\prime}) \)__is homeomorphic to

__\( (\mathbb{R}^4\smallsetminus \Omega;A_+,A_+^{\prime}) \)__.

The noncompact version of Theorem B is also important.

__\( (W;V,V^{\prime}) \)__be a simply connected, proper smooth 5-dimensional

__\( h \)__-cobordism with a finite number of ends and a trivial

__\( \pi_1 \)__-system at each end. Then

__\( W \)__is homeomorphic to

__\( V\times [0,1] \)__.

The difficult proof proposed by Freedman (October 1981) initiates the
proof of the proper __\( s \text{-cobordism} \)__ theorem sketched in
[e18],
while avoiding performing two Whitney processes, in view of the loss of
differentiability occasioned by Theorem C.

This gives (compare
[1]
and
[e34])
the topological
classification of closed, simply connected topological 4-manifolds that
admit (do they all?) a smooth structure in the complement of a point. They
are classified by their intersection form on __\( H_2 \)__, together with the
Kirby–Siebenmann obstruction __\( x \)__
[e29];
every unimodular forms
over __\( \mathbb{Z} \)__ is realised, as well as every __\( x\in \mathbb{Z}_2 \)__, except
that for even forms, __\( x\in \sigma/8\in \mathbb{Z}_2 \)__. Every topological
4-manifold __\( V \)__ which is homotopy equivalent to __\( S^4 \)__ is in this class,
because __\( V\smallsetminus \{\text{point}\} \)__ is contractible and thus
__\( V\smallsetminus\{\text{point}\} \)__ can be immersed into __\( \mathbb{R}^4 \)__
(compare
[e29]).

It also follows (see
[1],
[e34])
that every smooth homology
3-sphere __\( V \)__ (that is, __\( H_*(V)\cong H_*(S^3) \)__) is the boundary of a
contractible topological 4-manifold __\( W \)__.

##### Report

Mike Freedman announced his proof of the topological Poincaré conjecture in August 1981 at the AMS conference at UCSB where D. Sullivan was giving a lecture series on Thurston’s hyperbolization theorem. His argument was very brilliant, but not yet completely watertight.

A large group of experts then formulated certain objections, which led to the statement of the approximation theorem (Theorem 5.1). However, Freedman already had in his head his trick of replication, and in a few days, his imposing formal proof was born.

In the meantime, R. D. Edwards had found a mistake in the shrinking arguments (see Section 4) and, being an expert in this method, had repaired the mistake even before pointing it out. (I think that he introduced in particular the relative shrinking arguments.) At the end of October 1981, Freedman explained the details of his proof, with charm and patience, at a special conference at University of Texas at Austin (the school of R. L. Moore) before an audience of specialists, including, in the place of honour, Casson and RH Bing, creators of the two theories essential in the proof.

This paper relates the proof given in Texas, with improvements
in detail added in behind the scenes. Already in 1981,
R. Ancel
[2]
had clarified and improved the complexities
in bookkeeping of the approximation theorem (Theorem 5.1). In particular, he was
able to reduce a hypothesis of Freedman demanding that the preimages of
the singular point constitute a null decomposition, showing that __\( S(f) \)__
countable or of *dimension* 0
[e26]
suffices.
J. Walsh
contributed certain simplifications to the shrinking arguments (end of
Section 4).
W. Eaton
suggested to me the 4-balls that help to understand
relative shrinking (Lemma 4.9 and Proposition 4.11). I proposed a global
coordinate system of a Casson handle. (It was initially necessary to embed
the frontier of a handle in there.)

My exposition (January 1982) does not seem to have changed essentially
from my memories of Texas. Only my construction of corrective 2-discs (the
__\( D(\alpha) \)__ of Section 3.9) deviates, probably for reasons of taste. I am
indebted to
A. Marin
for his brotherly and insightful comments.

#### 1. Terminology

__\( d \)__. Maps are all continuous. The support of a map

__\( f : X\to X \)__is the closure of

__\( \{x\in X\mid f(x)\neq x\} \)__. The support of a homotopy, or an isotopy

__\( f_t : X\to X \)__

__\( (0\leq t\leq 1) \)__is the closure of

__\[ \{x\in X\mid f_t(x)\neq x\text{ for some }t\in [0,1]\}. \]__For a subset

__\( A \)__, define the closure

__\( \bar{A} \)__, the interior

__\( \mathring{A} \)__and the frontier

__\( \delta A \)__, always with respect to the understood ambient space (the largest involved). If

__\( A \)__is a manifold, it is often necessary to distinguish

__\( \mathring{A} \)__from its formal interior

__\( \operatorname{Int} A \)__and

__\( \delta A \)__from the formal boundary

__\( \partial A \)__.

A decomposition __\( \mathcal{D} \)__ of a space __\( X \)__ will be a collection of compact
disjoint subsets in __\( X \)__ that is
USC (upper semi continuous); the quotient
space __\( X/\mathcal{D} \)__ is obtained by identifying each element of __\( \mathcal{D} \)__
to a point (see
[e27]
for a metric). The quotient map
__\( X\to
X/\mathcal{D} \)__ is closed, which is exactly equivalent to the USC property.

The set of connected components of a space __\( X \)__ is denoted by __\( \pi_0(A) \)__. If
__\( A \)__ is compact, __\( \pi_0(A) \)__ is at the same time a decomposition of __\( A \)__ for
which the quotient __\( A/\pi_0(A) \)__ is a compact set of dimension 0 (totally
discontinuous), that is identified with __\( \pi_0(A) \)__ as a set. If __\( A\subset X \)__,
__\( \pi_0(A) \)__ gives a decomposition of __\( X \)__ whose quotient space is denoted by
__\( X/\pi_0(A) \)__. The *endpoint compactification* will appear in Section 2.

The manifolds and submanifolds mentioned will be (unless otherwise
indicated) smooth. For manifolds, we adopt the usual convention
([e29], Essay I);
in particular, __\( \mathbb{R}^n \)__ is the Euclidean space
with the metric __\( d(x,y)=|x-y| \)__; __\( B^n=\{x\in \mathbb{R}^n\mid |x|\leq 1\} \)__;
__\( I=[0,1] \)__. A *multidisc* is a disjoint union of finitely many
discs
(each are diffeomorphic to __\( B^2 \)__). Similarly, for multihandle,
etc. The symbols __\( \cong \)__, __\( \approx \)__ and __\( \simeq \)__ indicate a diffeomorphism,
a homeomorphism and a homotopy equivalence, respectively.

#### 2. Casson tower and Freedman’s mitosis

__\( B^2 \)__and

__\( D^2 \)__of the standard smooth 2-disc

__\[ \{(x,y)\in \mathbb{R}^2\mid x^2+y^2\leq 1\} .\]__The standard 2-handle is

__\( (B^2\times D^2,\partial B^2\times D^2) \)__; its

*attaching region*

__\( \partial_- \)__is

__\( \partial B^2\times D^2 \)__; its

*skin*

__\( \partial_+ \)__is

__\( B^2\times \partial D^2 \)__, its

*core*is

__\( B^2\times 0 \)__. A 2-

*handle*is a pair

__\[ (H^4,\partial_-H) \ \text{ diffeomorphic to }\ (B^2\times D^2,\partial B^2\times D^2) .\]__An open 2-handle is a manifold diffeomorphic to

__\( B^2\times \mathring{D^2} \)__. For a 2-handle (possibly open), the

*attaching region*, the

*skin*and the

*core*are defined by a diffeomorphism with the standard 2-handle (perhaps the open one). In this paper, we can allow ourselves to omit the prefix “2-”; handles of index

__\( \neq 2 \)__appear rarely. Also, we write

__\( \mathring{D^2} \)__where we ought strictly to write

__\( \operatorname{Int}D^2 \)__.

A *defect* __\( X \)__ in a handle __\( (H^4, \partial_-H) \)__ is a compact submanifold
__\( X \)__ of __\( H^4\smallsetminus \partial_-H \)__ such that:

__\( (X, X\cap \partial_+H) \)__is a handle where__\( \partial_+H \)__is the skin of the handle__\( (H,\partial_-H) \)__;__\( (\partial_+H,X\cap \partial_+H) \)__is (degree__\( \pm 1 \)__) diffeomorphic to the Whitehead double__\[ (B^2\times S^1,i(B^2\times S^1)) \]__illustrated in Figure 1;- in the 4-ball
__\( H^4 \)__(with rounded corners), the core__\( A^2 \)__of the handle__\( (X, X\cap \partial_+H) \)__is an unknotted disc, that is,__\( (H,A) \)__is diffeomorphic to__\( (B^4, B^2) \)__.

A *multidefect* __\( X \)__ in a handle __\( (H^4, \partial_-H) \)__ is a finite sum
and union of defects
such that for an identification __\( (H^4,\partial_-H) \)__
with
__\[ (B^2\times D^2, \partial B^2\times D^2) ,\]__
project to __\( B^2 \)__ the same
number of disjoint discs in __\( \operatorname{Int} B^2 \)__.
A multi-defect __\( X \)__ in a handle __\( (H^4,\partial H) \)__ is a finite,
disjoint union __\( \bigsqcup_i X(i) = X \)__ of
__\( \geq 1 \)__ defects __\( X(i) \)__, that, for a suitable identification
__\[ (H^4,\partial H) \cong (B^2,\partial B^2)
\times D^2 ,\]__
are sent, under the projection __\( B^2 \times D^2 \to B^2 \)__,
to a disjoint union of discs in
__\( B^2 \)__. A *multihandle*
__\( (H^4,\partial_-H^4) \)__ is a disjoint, finite sum of handles. A multiple
defect __\( X\subset H^4 \)__ in a multiple handle is a compact subset that gives
rise, by intersection, to a multidefect in each handle. With this data,
we have the following.

__\( (H^4\smallsetminus \mathring{X};\partial_-H,\delta X) \)__determines

__\( H^4 \)__and

__\( X \)__in the following sense. If

__\( X^{\prime} \)__is a multidefect in a handle

__\( (H^{\prime},\partial_-H^{\prime}) \)__and

__\[ \theta : (H\smallsetminus \mathring{X};\partial_-H,\delta X)\to (H^{\prime}\smallsetminus \mathring{X^{\prime}};\partial_-H^{\prime},\delta X^{\prime}) \]__is a diffeomorphism, there exists a diffeomorphism

__\( \Theta : H\to H^{\prime} \)__extending

__\( \theta \)__.

*Sketch of proof* (*see*
[e38]).
If we attach a
multihandle __\( (X^{\prime},\partial_-X^{\prime}) \)__ to __\( H\smallsetminus \mathring{X} \)__ along
the frontier __\( \delta X \)__, in such a way that there exists no extension
of __\( \theta \)__ to a diffeomorphism
__\[ \Theta : H\to (H\smallsetminus
\mathring{X})\cup X^{\prime}=H^{\prime} ,\]__
we claim that __\( (\partial H^{\prime},\partial_-H) \)__ is
diffeomorphic to __\( (S^3,\text{solid torus}) \)__ where the solid torus is tied
in a nontrivial knot — in fact, a connected sum of __\( k \)__ nontrivial twist
knots, __\( 1\leq k\leq |\pi_0(X)| \)__.
◻

A *residual defect* __\( \Omega \)__ in a handle __\( (H^4,\partial_-H^4) \)__ is the
intersection of a sequence
__\[X_1\supset \mathring{X_1}\supset X_2\supset \mathring{X_2}\supset
X_3\supset \cdots\]__
of compact submanifolds of __\( H^4\smallsetminus \partial_-H^4 \)__ such that,
for all __\( k \)__, __\( (X_k,\delta X_k) \)__ is a multihandle in which __\( X_{k+1} \)__ is
a multidefect. The sequence __\( X_1\supset X_2\supset \cdots \)__ is called a
Russian doll of defects.

A *Casson handle* is a pair
__\[ (H_\infty^4,\partial_-H_\infty^4) \]__
such that there exists a handle __\( (H,\partial_-H) \)__ with a residual defect
__\( \Omega\subset H \)__ and an open smooth embedding
__\[ i_\infty : H_\infty
\to H \]__
with image __\( H\smallsetminus \Omega \)__, which induces a diffeomorphism
__\[ i_\infty| : \partial_-H_\infty\to \partial_-H .\]__
In other words,
__\( (H_\infty, \partial_-H_\infty) \)__ is diffeomorphic to __\( (H\smallsetminus
\Omega,\partial_-H) \)__.

The data of __\( (H,\partial_-H) \)__, the Russian doll of defects __\( X_i \)__
and __\( i_\infty : H_\infty \to H \)__, constitute what we will call a
*presentation of a Casson handle* __\( (H_\infty,\partial_-H_\infty) \)__. We
will also denote
__\[ H_k=i_\infty^{-1}(H\smallsetminus \mathring{X_k})
\quad\text{and}\quad
\partial_- H_k=\partial_-H_\infty .\]__
Then, __\( H_\infty=\bigcup_k
H_k \)__. The manifold __\( H_k \)__ is called a tower of height __\( k \)__, its stages
are
__\[ E_j=i_\infty^{-1}(X_{j-1}\smallsetminus X_j) \]__
for __\( j\leq k \)__. The
restriction of __\( i_\infty \)__ to __\( H_k \)__ will be denoted __\( i_k : H_k\to H \)__.

The skin of __\( (H_\infty, \partial_-H_\infty) \)__ is
__\[ \partial_+H_\infty=i_\infty^{-1}(\partial_+H) ;\]__
moreover, by taking
intersection with __\( \partial_+H_\infty \)__, we define the skin __\( \partial_+H_k \)__
of __\( H_k \)__ and __\( \partial_+E_k \)__ ~~of ~~ Similarly __\( E_k \)__.__\( \partial_+X_k=X_k\cap
\partial_+H . \)__

A Casson handle __\( (H_\infty,\partial_-H_\infty) \)__ is never compact; we
will often encounter the endpoint compactification __\( \widehat{H}_\infty \)__
of __\( H_\infty \)__. Recall that the endpoint compactification __\( \widehat{M} \)__
of a connected, locally connected and locally compact space __\( M \)__ is the
Freudenthal compactification that adds to __\( M \)__ the compact 0-dimensional space
__\( \operatorname{Ends}(M) \)__ which is the (projective) limit of an inverse system
__\[
\{\pi_0(M\smallsetminus K)\mid K\subset M \text{ such that } K \text{ is compact}\}.
\]__

By __\( i_\infty \)__, __\( \widehat{H}_\infty \)__ is identified with the quotient of __\( H^4 \)__
obtained by crushing each connected component of __\( \Omega \)__ to a point. (To
verify this, note that __\( \pi_0(\Omega) \)__ with the compact topology is the
(projective) limit of an inverse system
__\( \{\pi_0(U)\mid U \)__ is an
open subset of __\( H \)__ containing __\( \Omega\} \)__.)

We remark that __\( \widehat{H}_\infty \)__ is the Alexandroff compactification by a
point, exactly when __\( \Omega\subset H \)__ is connected, or if each successive
multiple defect __\( X_i \)__ is a single defect. The reader who feels discombobulated
by all the complexities to come may be interested in restricting themselves
at first to this case, which already contains all the geometric ideas.

__\( \widehat{H}_\infty \)__ has all the local homological properties of a manifold;
it is what we call a *homology manifold*. But its formal boundary,
the closure of __\( \partial H_\infty \)__, is not a topological manifold
near its ends. For example, if __\( \Omega \)__ is connected, by definition,
__\( \partial H_\infty \)__ (which is homeomorphic to __\( \partial H\smallsetminus
\partial_+\Omega \)__) is one of the contractible 3-manifolds of
J. H. C. Whitehead
[e1],
[e2],
with a nontrivial
__\( \pi_1 \)__-system at infinity.
__\[ \partial_+\Omega\subset \partial H\cong S^3 \]__
is a *Whitehead compactum*. In the general case, __\( \partial_+\Omega \)__ is
called a *ramified Whitehead compactum*. Thus, __\( (\widehat{H\smallsetminus
\Omega},\partial_-H) \)__ has no chance of being a topological handle. On the
other hand,
__\[ H\smallsetminus (\partial_+H\cup \Omega) \]__
is homeomorphic to
__\( B^2\times \mathbb{R}^2 \)__; this will be the central result of this paper.

__(Freedman, 1981.)__Every open Casson handle

__\( M \)__is homeomorphic to

__\( B^2\times \mathbb{R}^2 \)__.

The proof of Theorem 2.2 starts with a result of 1979, when Freedman was
able to construct a smooth 4-manifold __\( M \)__ without boundary which is not
homeomorphic to __\( S^3\times \mathbb{R} \)__ that is however the image of a proper
map of degree __\( \pm 1 \)__, __\( S^3\times \mathbb{R}\to M \)__ (see
[1]
and
[e34]).

A Casson tower of height __\( k \)__, or more briefly __\( C_k \)__, is a pair diffeomorphic
to __\( (H\smallsetminus \mathring{X_k},\partial_-H) \)__ where __\( X_1\supset X_2\supset
\cdots \)__ is a Russian doll of defects in a handle __\( (H,\partial_-H) \)__.

__(Mitosis, a finite version.)__Let

__\( (H_6,\partial_-H_6) \)__be a Casson tower

__\( C_6 \)__of height 6. There is a Casson tower

__\( C_{12} \)__of height 12, or

__\( (H_{12}^{\prime},\partial_-H_{12}^{\prime}) \)__, such that

__\( \partial_-H_{12}^{\prime}=\partial_-H_6 \)__.__\( H_{12}^{\prime}\smallsetminus \partial_-H_6\subset \operatorname{Int}H_6 \)__.__\( H_{12}^{\prime}\smallsetminus H_6^{\prime} \)__is contained in a disjoint union of balls in__\( \operatorname{Int}H_6 \)__, one ball for each connected component.

Condition (3) is related to the fact that, for each Casson tower
__\[ (H_k,\partial_-H_k) ,\]__
the manifold __\( H_k \)__ can be expressed as a regular
neighbourhood of a 1-complex, compare
[e38].
Figure 5 shows a schematic
diagram of Freedman which summarises Theorem 2.3.

In Section 3, Figure 6 will represent a __\( C_6 \)__, and Figure
7 will represent a __\( C_{12} \)__, etc. From the point of view of
the representation of corners on the boundary, it might be better to use
Figure 8.

The method of Freedman [1] (compare [e34]) allows one to give a proof of Theorem 2.3. However, it is slightly more detailed than the analogues in [1], [e34]. We will not cover this point in this paper (see [e37] for an excellent write up of the mitosis theorem (finite version, Theorem 2.3).

__\( (k,2k) \)__,

__\( k > 6 \)__, in place of

__\( (6,12) \)__gives a statement that one can deduce without too much pain and sorrow that we could use in place of Theorem 2.3 in what follows.

Since we are going to use Theorem 2.3 often, it is convenient to make the following:

__\( H_k \)__and

__\( X_k \)__in place of

__\( H_{6k+6} \)__and

__\( X_{6k+6} \)__,

__\( k=0,1,2,\dots\, \)__.

__(__Also the meaning of

__\( E_k=H_k\smallsetminus H_{k-1} \)__,

__\( i_k \)__, etc. is changed.

__)__

__(Mitosis, an infinite version.)__Let

__\( (H_\infty,\partial_-H_\infty) \)__be a Casson handle presented as above, and let

__\( k\geq 0 \)__be an integer. There exists another Casson handle

__\( (H_\infty^{\prime},\partial_-H_\infty)\subset (H_\infty, \partial_-H_\infty) \)__satisfying the conditions

__:__

__\( H_{k-1}^{\prime}=H_{k-1} \)__if__\( k\geq 1 \)__.__\( \overline{H^{\prime}_\infty}\smallsetminus H_{k-1}^{\prime}\subset (\operatorname{Int} H_k)\smallsetminus H_{k-1} \)__.- The closure
__\( \overline{H_\infty^{\prime}} \)__of__\( H_\infty^{\prime} \)__in__\( H_\infty \)__is the endpoint compactification of__\( H_\infty^{\prime} \)__.

This infinite version, Theorem 2.5, follows from the finite version, Theorem 2.3, by an infinite repetition. One sufficiently shrinks balls given by Theorem 2.3 to ensure the condition (3) of Theorem 2.5.

#### 3. Architecture of topological coordinates

__\( B^2\times \mathbb{R}^2 \)__, to two theorems on approximation by homeomorphisms. For Casson handles, we will use the terminology of Section 2, under the modified form in Change of Notation 2.4 (by a reindexing).

The open Casson handle __\( M \)__ will be identified with __\( N\smallsetminus
\partial_+N \)__ where __\( (N,\partial_-N) \)__ is a Casson handle (not open). Let
__\( \widehat{N} \)__ be the endpoint compactification __\( \text{of } N \)__. Subtracting from __\( N \)__ the
(topological) interior of a collar neighbourhood of __\( \partial_+N \)__ in __\( N \)__,
very pinched towards the ends __\( \text{of } N , \)__ we obtain a Casson handle
__\[ (H_\infty,
\partial_-H_\infty)\subset (M,\partial M)\subset (N,\partial_-N) \]__
whose
closure in __\( \widehat{N} \)__ is the endpoint compactification __\( \widehat{H}_\infty \)__
of __\( H_\infty \)__. We fix a presentation of __\( (H_\infty, \partial_-H_\infty) \)__.

We will construct a ramified system of Casson handles in __\( (N,\partial_-N) \)__,
that, in some way, explores its interior.

##### 3.1. Construction

__\( (a_1,\ldots,a_k) \)__in

__\( \{0,1\} \)__(finite dyadic sequence), we can define a presented Casson handle

__\[ (H_\infty(a_1,\ldots,a_k),\partial_-H_\infty) \]__contained in

__\( (H_\infty,\partial_-H_\infty) \)__, whose presentation consists of an embedding

__\[ i_\infty(a_1,\ldots,a_k) : H_\infty(a_1,\ldots,a_k)\to B^2\times D^2, \]__and a Russian doll of defects

__\( X_i(a_1,\ldots,a_k) \)__, in the standard handle

__\( B^2\times D^2 \)__such that (for (1)–(5), see the right figure of Figure 10):

__\( H_\infty=H_\infty(\emptyset) \)____(__case__\( k=0 \)____)__as a presented Casson handle.__\( H_\infty(a_1,\ldots,a_k,1)=H_\infty(a_1,\ldots,a_k) \)__.__\( H_k(a_1,\ldots,a_k,0)=H_k(a_1,\ldots,a_k) \)____(__recall that__\( H_k \)__are sets of 6-stages__)__.- The closure
__\[ \overline{H}_\infty(a_1,\ldots,a_k,0) \quad\text{in}\quad \widehat{H}_\infty \]__is an endpoint compactification of__\( H_\infty(a_1,\ldots,a_k,0) \)__. __\( \overline{H}_\infty(a_1,\ldots,a_k,0)\smallsetminus H_k(a_1,\ldots,a_k)\subset \mathring{H}_{k+1}(a_1,\ldots,a_k)\smallsetminus H_k(a_1,\ldots,a_k) \)__.__\( i_k(a_1,\ldots,a_k,0)=i_k(a_1,\ldots,a_k) \)__, so__\( X_k(a_1,\ldots,a_k,0)=X_k(a_1,\ldots,a_k) \)__.- The intersection of
__\( X_{k+1}(a_1,\ldots,a_k,0) \)__and__\( X_{k+1}(a_1,\ldots,a_k) \)__is empty, and their union is a multiple defect in__\( X_k(a_1,\ldots,a_k) \)__. - (Without Change of Notation 2.4)
We also require a coherence condition on the total Russian doll assumed by
(7), that is to say
__\( \{X_k\} \)__, where__\( X_k=\bigcup X_k(a_1,\ldots,a_k) \)__. To formulate it, we momentarily suspend the reindexing convention (Change of Notation 2.4) and write__\( T_k=\partial_+X_k \)__. The condition is that there exists an interval__\( J\subset \partial D^2 \)__such that, for all__\( t\in J \)__, the meridional disc__\( B_t=B^2\times t \)__of the solid torus__\( B^2\times \partial D \)__meets the multiple solid tori__\( T_k \)__ideally, in the sense that each connected component of__\( B_t\cap T_k \)__is a meridional disc of__\( T_k \)__, that meets__\( T_{k+1} \)__in an ideal fashion illustrated in the left figure of Figure 10.

*Execution of Construction 3.1* (*by induction on \( k \)*).
We start with

__\( H_\infty(\emptyset)=H_\infty \)__. Having defined a presented handle for every sequence of length

__\( \leq k \)__, we define them for every sequence

__\( (a_1,\ldots,a_k,1) \)__by (2). Next, we define

__\( H_\infty(a_1,\ldots,a_k,0) \)__by the mitosis theorem (infinite version, Theorem 2.5). This assures that conditions (3), (4) and (5) are met. It remains to define the presentation of the Casson handle

__\( (H_\infty(a_1,\ldots,a_n,0),\partial_-H_\infty) \)__in such a fashion that the two last conditions (6) and (7) are satisfied. To define

__\( i_\infty(a_1,\ldots,a_k,0) \)__, it is convenient to graft, onto

__\( i_k(a_1,\ldots,a_k) \)__, a presentation the near part of the Casson handle

__\[ (H_\infty(a_1,\ldots,a_k,0), \, \partial_-H_\infty) ,\]__to know the Casson multihandle

__\[ \bigl(H_\infty(a_1,\ldots,a_k,0)\smallsetminus\mathring{H}_k(a_1,\ldots,a_k,0),\delta H_k(a_1,\ldots,a_k,0)\bigr), \]__where exceptionally

__\( \ \mathring{}\ \)__and

__\( \delta \)__denote the interior and the frontier in

__\( H_\infty(a_1,\ldots,a_k,0) \)__rather than in

__\( \widehat{N} \)__. The grafting is done with the help of Lemma 2.1. The last condition (7) is assured afterwards by an isotopy in

__\( \mathring{X}_k(a_1,\ldots,a_k) \)__. Having (1) to (7), the reader will know how to arrange that (8) is also satisfied. ◻

__\( (a_1,a_2,\dots) \)__is an infinite sequence in

__\( \{0,1\} \)__, the union

__\[ H_\infty(a_1,a_2,\dots)=\bigcup_k H_\infty (a_1,a_2,\ldots,a_k) \]__gives a Casson handle with an obvious presentation. Moreover, the closure

__\[ \overline{H}_\infty(a_1,a_2,\dots) \]__is the endpoint compactification

__(__exercise

__)__. Thus, we have a vast collection of Casson handles in

__\( N \)__, conveniently nested.

Of the system of handles
__\[ (H_\infty(a_1,\ldots,a_k), \partial_-H_\infty) ,\]__
we especially use their skins __\( \partial_+H_\infty(a_1,\ldots,a_k) \)__. The union
__\[ P^3=\bigcup\partial_+H_\infty(a_1,\ldots,a_k) \]__
of the skins is what one calls
a *branched manifold* in __\( N^4 \)__, since near every point __\( P^3\smallsetminus
\partial_-H_\infty \)__, the pair __\( (N^4,P^3) \)__ is __\( C^1 \)__-isomorphic (same as
__\( C^\infty \)__-isomorphic, after some work that we leave to the reader) to the
product of __\( \mathbb{R}^2 \)__ with the model of branching __\( (\mathbb{R}^2,Y^1) \)__
where __\( Y^1 \)__ is the union of two smooth curves (isomorphic to __\( \mathbb{R}^1 \)__),
properly embedded in __\( \mathbb{R}^2 \)__ and which have in common exactly
one closed half-line. One observes without difficulty that the closure
__\( \overline{P} \)__ of __\( P \)__ in __\( \widehat{N} \)__ is the endpoint compactification of __\( P \)__.

The branched manifold __\( P \)__ splits along the singular points into compact
manifolds:
__\begin{align*}
P_k(a_1,\ldots,a_k) &=\partial_+E_k(a_1,\ldots,a_k)\\
&=E_k(a_1,\ldots,a_k)\cap
\partial_+H_\infty(a_1,\ldots,a_k).
\end{align*}__
Thus, __\( P_k(a_1,\ldots,a_k) \)__ is the skin of the __\( k \)__-th stage of
__\( (H_\infty(a_1,\ldots,a_k),\partial_-H_\infty) \)__.

##### 3.2. Construction of the design __\( G^4 \)__ __(__see Figure 11__)__

For __\( P^3 \)__, we construct a neighbourhood __\( G^4 \)__ in __\( N^4 \)__ called the
*design*, which has a decomposition __\( \mathcal{I} \)__ of __\( G^4 \)__ into disjoint
intervals, satisfying the following.

- For every interval
__\( I_\alpha \)__of__\( \mathcal{I} \)__, the intersection__\[ I_\alpha\cap \partial_-N \]__is__\( I_\alpha \)__or the empty set. A neighbourhood of__\( I_\alpha \)__in__\( (G^4,P^3;\mathcal{I}) \)__is isomorphic to the product of__\( \mathbb{R}^2 \)__with an open 2-dimensional model__\( (G^2,P^1;\mathcal{I}^{\prime}) \)__as in Figure 12. - The closure
__\( \overline{G} \)__of__\( G \)__in__\( \widehat{N} \)__is its endpoint compactification, and hence coincides with__\( G\cup \overline{P} \)__.

It follows by combining, quite naively, two bicollars of genuine submanifolds
of __\( P^3 \)__. On the other hand, we clearly are permitted to suppose that __\( G^4 \)__
contains the collar __\( N\smallsetminus \mathring{H}_\infty \)__ of __\( \partial_+N \)__.

The design __\( (G^4,\mathcal{I}) \)__ decomposed
into intervals splits in a canonical fashion
(along the 3-manifold formed by the exceptional intervals of __\( \mathcal{I} \)__
having interior points on __\( \partial G^4 \)__) into genuine
trivial __\( I \)__-bundles
__\[ I(a_1,\ldots,a_k)\times P_k(a_1,\ldots,a_k) ,\]__
where __\( I(a_1,\ldots,a_k) \)__ is a 1-simplex and
__\[
(\text{its centre})\times P_k(a_1,\ldots,a_k)\subset G^4
\]__
is nearly the natural inclusion __\( P_k(a_1,\ldots,a_k)\subset G^4 \)__. More
precisely, the two embeddings are isotopic in __\( G^4 \)__ by an isotopy which moves
only a collar of the boundary of __\( P_k(a_1,\ldots,a_k) \)__. It is convenient
to give a normal orientation to __\( P^3 \)__ in __\( N^4 \)__ (towards the exterior),
to deduce from it the orientation of the 1-simplices __\( I(a_1,\ldots,a_k) \)__.

##### 3.3. Construction of __\( g : G^4\to B^2\times D^2 \)__

This __\( g \)__ will be a smooth embedding which will reveal the structure of
__\( G^4 \)__. We choose, by recurrence, linear embeddings __\( I(a_1,\ldots,a_k)\subset
(0,1] \)__ conserving the orientation. To start, __\( I(\emptyset)\subset (0,1] \)__
ends at 1. Suppose now these embeddings have been defined for all
sequences of length __\( \leq k \)__. Then, we embed __\( I(a_1,\ldots,a_k,0) \)__ and
__\( I(a_1,\ldots,a_k,1) \)__ respectively on the initial third and the final third
of the interval __\( I(a_1,\ldots,a_k)\subset (0,1] \)__.

The central third of __\( I(a_1,\ldots,a_k) \)__ is a closed interval that we
may call __\( J(a_1,\ldots,a_k) \)__. The complement in __\( I(\emptyset) \)__ of all the
open intervals __\( \mathring{J}(a_1,\ldots,a_k) \)__ is then a compact Cantor set
in __\( (0,1] \)__.

On the other hand, we claim that the
embeddings
__\[
i_k(a_1,\ldots,a_k)| :
\partial_+H_k(a_1,\ldots,a_k)\to B^2\times \partial D^2
\]__
define together
a smooth map __\( i : P\to B^2\times \partial D^2 \)__.
Let
__\[
\varphi :
(0,1]\times B^2\times \partial D^2\to B^2\times D^2
\]__
be the embedding
__\( (t,x,y)\mapsto (x,ty) \)__. We will have the tendency to identify domain and
codomain by __\( \varphi \)__.

We define __\( g : G^4\to B^2\times D^2 \)__ on
__\[ I(a_1,\ldots,a_k)\times P_k(a_1,\ldots,a_k) \]__
by the rule that __\( (t,x)\mapsto \varphi(t,i(x)) \)__.
For that definition to make sense, we have to first adjust, by isotopy,
the trivialisation given by the __\( I \)__-fibres
__\[ I(a_1,\ldots,a_k)\times
P_k(a_1,\ldots,a_k) \]__
in __\( (G^4,\mathcal{I}) \)__, a routine task that is left
to the reader.

##### 3.4. Construction of __\( g_0 : G_0^4\to B^2\times D^2 \)__

__\( G_0^4 \)__be the union of

__\( G^4 \)__and a small collar neighbourhood

__\( C^4 \)__of

__\( \partial_-N \)__in

__\( N \)__that respects

__\( \delta G^4 \)__(see Figure 11 for Section 3.2). Let us extend

__\( g \)__to an embedding

__\[ g_0 : G_0^4\to B^2\times D^2 .\]__By uniqueness of collars, we can arrange

__\( g \)__and

__\( g_0 \)__so that

__\( g_0 \)__sends

__\( C^4\smallsetminus \mathring{G}^4 \)__to

__\[ (B^2\smallsetminus \lambda B^2)\times \mu D^2 ,\]__where

__\( \lambda\in (0,1] \)__is near to 1 and

__\( \mu \)__to the initial point of

__\( I(\emptyset) \)__. This completes the construction of

__\( g_0 : G_0^4\to B^2\times D^2 \)__. Looking near

__\( g_0 \)__and its image, we will claim that we have completely described the closure

__\( \overline{G_0^4} \)__of

__\( G_0^4 \)__in

__\( \widehat{N}^4 \)__.

##### 3.5. The image __\( g_0(G_0^4)\subset B^2\times D^2 \)__

__\( T(a_1,\ldots,a_k)\equiv T_k(a_1,\ldots,a_k)=\partial_+X(a_1,\ldots,a_k) \)__, a multisolid torus__\( \subset B^2\times \partial D^2 \)__.__\( T_*(a_1,\ldots,a_k)=\varphi(J(a_1,\ldots,a_k)\times T(a_1,\ldots,a_k))\subset B^2\times \mathring{D}^2 \)__, a radially thickened copy of__\( T(a_1,\ldots,a_k) \)__, called a*hole*.__\( B_*=\lambda B^2\times \mu D^2 \)__(see definition of__\( g_0 \)__), called the*central hole*.__\( F_k=\bigcup\{\varphi(I(a_1,\ldots,a_{k-1})\times T(a_1,\ldots,a_k))\mid k \text{ fixed}\} \)__; the frontiers__\( \delta F_k \)__,__\( k\geq 2 \)__, are indicated in dashed lines in the right-hand figure below.__\( (B^2\times D^2)_0=(B^2\times D^2\smallsetminus \mathring{B}_*)\smallsetminus \bigcup\{\mathring{T}_*(a_1,\ldots,a_k)\} \)__, called the*holed standard handle*.__\( W_0=\bigcap_k F_k \)__, a compactum in__\( (B^2\times D^2)_0 \)__.

With this notation, we
claim that the image
__\( g_0(G_0^4) \)__ is __\( (B^2\times D^2)_0\smallsetminus W_0 \)__.

##### 3.6. The main diagram

__\( \mathcal{W}_0 \)__,

__\( g_1 \)__,

__\( \mathcal{D}^{\prime} \)__,

__\( g_2 \)__,

__\( \mathcal{D} \)__,

__\( g_3 \)__,

__\( \mathcal{D}_+ \)__,

__\( f \)__. The proof that

__\[ (B^2\times \mathring{D}^2)/\mathcal{D}_+ \]__is homeomorphic to

__\( B^2\times \mathring{D}^2 \)__(by the methods of Bing) will appear in Section 4. The proof that

__\( f \)__is approximable by homeomorphisms is postponed to Section 5.

##### 3.7. Construction of __\( \mathcal{W}_0 \)__ and __\( g_1 \)__

__\( \mathcal{W}_0 \)__is the decomposition of the compact set

__\( (B^2\times D^2)_0 \)__, where nondegenerated elements are the connected components

__\( W \)__of the compact set

__\( W_0\subset (B^2\times D^2)_0 \)__. Each

__\( W\in \mathcal{W}_0 \)__is a Whitehead compactum in a single level

__\( \varphi(t\times B^2\times \partial D^2) \)__. We check naively that the inclusion

__\[ (B^2\times D^2)_0\smallsetminus W_0\to (B^2\times D^2)_0/\mathcal{W}_0 \]__induces a homeomorphism

__\[ ((B^2\times D^2)_0\smallsetminus W_0)^\wedge\to (B^2\times D^2)_0/\mathcal{W}_0. \]__We already know that

__\( \widehat{G}_0 \)__is identified with

__\( \overline{G}_0\subset \widehat{N} \)__. We define the homeomorphism

__\( g_1 \)__as a composition of homeomorphisms:

__\[ g_1 : \overline{G}_0\to \widehat{G}_0\xrightarrow{\hat{g}_0} ((B^2\times D^2)_0\smallsetminus W_0)^\wedge\to (B^2\times D^2)_0/\mathcal{W}_0. \]__

##### 3.8. Construction of __\( \mathcal{D}^{\prime} \)__ and __\( g_2 \)__

__\( \mathcal{D}^{\prime} \)__be the decomposition of

__\( B^2\times D^2 \)__given by the

__\( B_* \)__,

__\( T_*(\alpha) \)__(

__\( \alpha \)__can be any finite dyadic sequence), and the elements of

__\( \mathcal{W} \)__which are disjoint from them. To define

__\[ g_2 : \widehat{N}\to (B^2\times D^2)/\mathcal{D}^{\prime} ,\]__we must extend

__\[ q_1g_1 : \overline{G}_0\to (B^2\times D^2)/\mathcal{D}^{\prime} \]__to each connected component

__\( Y \)__of

__\( \widehat{N}\smallsetminus \overline{G}_0 \)__. Its frontier

__\( \delta Y \)__is identified by

__\( g_1 \)__to the quotient in

__\[ (B^2\times D^2)_0/\mathcal{W}_0 ,\]__either of

__\( \partial B_* \)__, or of a boundary of a connected component of a hole

__\( T_*(a_1,\ldots,a_k) \)__. By definition,

__\( g_2(Y) \)__is the image in

__\( (B^2\times D^2)/\mathcal{D} \)__of this boundary. It is easy to check the continuity of

__\( g_2 \)__.

Next, __\( g_3 \)__ and __\( \mathcal{D} \)__ in the main diagram are defined by
restriction. The design __\( G^4 \)__ has
led us inexorably to define
__\[ g_3 :
M^4\to B^2\times \mathring{D}^2/\mathcal{D} ,\]__
which compares the open Casson
handle __\( M^4 \)__ with a very explicit quotient of the open handle __\( B^2\times
\mathring{D}^2 \)__.

The decomposition __\( \mathcal{D} \)__ which specifies this quotient has noncellular
elements, that is, the holes __\( T_*(a_1,\ldots,a_k) \)__, each of which has
the homotopy type of a circle. Therefore the quotient map
__\[ B^2\times
\mathring{D}^2\to B^2\times \mathring{D}^2/\mathcal{D} \]__
is certainly not
approximable by homeomorphisms. One can also check that the
Čech cohomology
__\( \check{H}^2 \)__ of the quotient is of infinite type.

The construction of __\( \mathcal{D}_+ \)__ below repairs this terrible defect;
it will be constructed by hand; __\( \mathcal{D}_+ \)__ will be less fine than
__\( \mathcal{D} \)__, which will enable us to define __\( f=q_3\circ g_3 \)__ without effort.

##### 3.9. Construction of __\( \mathcal{D}_+ \)__

__\[ W=W_0\cap (B^2\times \mathring{D}^2)=W_0\smallsetminus (B^2\times \partial D^2). \]__Its connected components define a decomposition

__\( \mathcal{W} \)__of

__\( B^2\times \mathring{D}^2 \)__. We have known since the 1950s how to show that

__\( B^2\times \mathring{D}^2/\mathcal{W} \)__is homeomorphic to

__\( B^2\times \mathring{D}^2 \)__, see Section 4.

For the requirements of the next paragraph, the
quotient __\( (B^2\times
\mathring{D}^2)/\mathcal{D}_+ \)__ must be a quotient of __\( B^2\times
\mathring{D}^2/\mathcal{W} \)__ by a decomposition whose elements are the
connected components of
__\[
\bigcup \{q(T_*(\alpha))\cup E(\alpha)\mid \alpha\text{ a finite dyadic
sequence}\}.
\]__
Here __\( \{E(\alpha)\} \)__ is a collection of disjoint, topologically *flat*
multi-2-discs such that for each finite dyadic sequence __\( \alpha \)__, the
intersection
__\[ E(\alpha)\cap \biggl(\bigcup_{\alpha^{\prime}}q(T_*(\alpha^{\prime}))\!\biggr) \]__
is

- the boundary
__\( \partial E(\alpha) \)__; and - a multilongitude of
__\( \partial T_*(\alpha) \)__far from__\( W \)__(each connected component of__\[ q(T_*(\alpha))\cup E(\alpha) \]__is then contractible).

Moreover, we want that the diameter of the connected components of
__\( E(a_1,\ldots,a_k) \)__ tends towards 0 (on each compact set) as __\( k\to
\infty \)__. Section 4 does not demand any more than this and visibly,
__\( \{E(\alpha)\} \)__ specifies __\( \mathcal{D}_+ \)__.

The specification of __\( \{E(\alpha)\} \)__ is unfortunately tedious. __\( E(\alpha) \)__
will be the faithful image __\( q(D(\alpha)) \)__ of a multidisc in __\( B^2\times
\mathring{D}^2 \)__. For fundamental group reasons, the multidisc __\( D(\alpha) \)__
is obliged to meet __\( W \)__, but, to assure flatness of __\( q(D(\alpha)) \)__ (proved
in Section 4), it must be a well behaved meeting, permitted by (7) and
(8) of Construction 3.1.

We have __\( T_k=\bigcup_\alpha T_k(\alpha) \)__; conditions (6) and (7) of
Construction 3.1 assure that __\( T_k \)__ is a multisolid torus of which certain
connected components constitute __\( T_k(\alpha) \)__. We have
__\[ \bigcap_k T_k=p(W) ,\]__
which is a ramified Whitehead compactum in __\( B^2\times \partial D^2 \)__.

To start, we specify (simultaneously and independently) in __\( B^2\times
\partial D^2 \)__, (topologically) immersed, locally flat discs __\( D^{\prime}(\alpha) \)__
which will be the projection __\( p(D(\alpha))=D^{\prime}(\alpha) \)__. We assume easily
the two properties (a) and (b), where (b) uses (8) of Construction 3.1.

(a) __\( D^{\prime}(a_1,\ldots,a_k) \)__ is a disjoint union of immersed discs in __\( T_{k-1} \)__,
with as their only singularities, an arc of double points for each,
above
__\[ T_k(a_1,\ldots,a_k) .\]__
The boundary
__\[ \partial D^{\prime}(a_1,\ldots,a_k) \]__
is formed from one longitude of each connected component of __\( \partial
T_k(a_1,\ldots,a_k) \)__. The double points of __\( D^{\prime}(a_1,\ldots,a_k) \)__ are outside
__\( \mathring{T}_k(a_1,\ldots,a_k) \)__.

(b) For each __\( l\geq k \)__, the intersection
__\[ \mathring{D}^{\prime}(a_1,\ldots,a_k)\,\cap\,T_l \]__
is a multidisc (embedded in __\( T_k(a_1,\ldots,a_k) \)__) of which each
connected component __\( D_0 \)__ is a meridional disc of __\( T_l \)__ that meets the solid
tori of the next generator (__\( T_{l+1/6} \)__ with our revised indexing of Change of
Notation 2.4) ideally (see the left-hand figure of Figure 10).

By resolving the double points of __\( D^{\prime}(\alpha) \)__, which we have to embed
in
__\[
(0,1)\times B^2\times \partial D^2\subset B^2\times D^2,
\]__
specifying the
first coordinate by a convenient function __\( \rho(\alpha) : D(\alpha)\to
(0,1) \)__.

We will embed a single __\( D(\alpha) \)__ at a time (following some chosen
order). We embed first __\( D(a_1,\ldots,a_k) \)__ closer and closer (by a secondary
induction).
Some notation:
__\begin{equation*}
\eqalign{
T_*^+(a_1^{\prime},\ldots,a_l^{\prime})&=J(a_1^{\prime},\ldots,a_l^{\prime})\times T(a_1^{\prime},\ldots,a_{l-1}^{\prime}),\cr
F_l^*=p^{-1}(p(F_l))&=(0,1)\times T_l,\cr
W^+=p^{-1}(p(W))&=(0,1)\times \biggl(\bigcap_k T_k\biggr).
}
\end{equation*}__

One can easily check that, for
__\[ D(a_1,\ldots,a_k) ,\]__
the properties (c) and
(d) for __\( l > k \)__, of which (d) for __\( l \)__ is only provisional.

(c) __\( D(a_1,\ldots,a_k) \)__ is embedded, is contained in __\[ I(a_1,\ldots,a_{k-1})\times
T(a_1,\ldots,a_{k-1}), \]__ and is disjoint from __\( B_* \)__ and from
__\[ \bigcup\{T_*^+(\alpha^{\prime})\mid \alpha^{\prime}\neq (a_1,\ldots,a_k)\} .\]__ The boundary
__\( \partial D (a_1,\ldots,a_k) \)__ is in a single level __\( t\times B^2\times
\partial D^2 \)__, where __\( t\in \mathring{J}(a_1,\ldots,a_k) \)__.

(d) Each connected component of the multidisc
__\[ F_l^+\cap D(a_1,\ldots,a_k) \]__
is
in a single level __\( t\times B^2\times \partial D^2 \)__; this level is disjoint
from each box __\( T_*(\alpha^{\prime}) \)__, and does not contain any other connected
component of __\( F_l^+\cap D(a_1,\ldots,a_k) \)__.

For __\( l=k \)__ and __\( k+1 \)__, here are the illustrations of the graph of __\( \rho \)__
in a simple case.

We observed that in pushing
__\[ D(a_1,\ldots,a_k) \]__
vertically, as small as we
want, and only on
__\[ \mathring{F}_l^+\cap D(a_1,\ldots,a_k) ,\]__
we can pass from
(d) for __\( l \)__ to (d) for __\( l+1 \)__, without losing (c). Therefore, without losing
(c), we can pass to the next property.

(e) For each integer __\( l > k \)__, the connected
components of the multidisc
__\[ F_l^+\cap
D(a_1,\ldots,a_k) \]__
project onto as many disjoint intervals of
radius
in __\( (0,1) \)__.

This condition assures that, for all __\( W\in \mathcal{W} \)__, the
intersection
__\[ W\cap D(a_1,\ldots,a_k) \]__
is an intersection of discs
(and so cellular). Therefore __\( q(D(a_1,\ldots,a_k)) \)__ is certainly a disc
(compare Theorem 4.4).
In Section 4, we will prove by hand that it is a
*flat disc*. If, before __\( D(a_1,\ldots,a_k) \)__, we have already defined
(for the main induction) a finite collection of discs
__\[ D(\alpha_1),\ldots,
D(\alpha_n) ,\]__
we follow the same construction as above, always staying in a
neighbourhood of __\( T_*^+(a_1,\ldots,a_k) \)__ (guaranteed by (c)), disjoint from
__\( D(\alpha_1)\cup \cdots\cup D(\alpha_n) \)__ and for all elements of __\( \mathcal{W} \)__
that touch __\( D(\alpha_1)\cup\cdots\cup D(\alpha_n) \)__.

Thus the family __\( \{D(\alpha)\} \)__ of *disjoint* 2-discs is defined by a
double induction and satisfies the properties (a), (b), (c) and (e) with
__\[ p(D(\alpha))=D^{\prime}(\alpha) .\]__
Next __\( \{D(\alpha)\} \)__ defines __\( \mathcal{D}_+ \)__
as already indicated. One easily checks all the properties wanted for
__\[ q(D(\alpha))=E(\alpha)
\quad\text{in}\quad
(B^2\times \mathring{D}^2)/\mathcal{W} ,\]__
except local flatness of __\( E(\alpha) \)__ which is postponed to Section 4.

##### 3.10. End of the proof that __\( M \)__ is homeomorphic to __\( B^2\times\mathring{D}^2 \)__ __(__modulo Sections 4 and 5__)__

__\( (B^2\times \mathring{D}^2)/\mathcal{D}_+ \)__is homeomorphic to

__\( B^2\times \mathring{D}^2 \)__, we show modulo Section 5 the approximability by homeomorphisms of

__\[ f : M^4\to (B^2\times \mathring{D}^2)/\mathcal{D}_+ \]__in the following fashion. We form the commutative diagram

__\( \operatorname{Int} M\subset S^4 \)__exists since

__\( M \)__embeds in

__\( B^2\times D^2 \)__(the experts also know that

__\( \operatorname{Int} M \)__is diffeomorphic to

__\( \mathbb{R}^4 \)__[e38]), and where

__\[ f_*(S^4\smallsetminus\operatorname{Int} M^4)=\infty .\]__Therefore,

__\[ S(f_*)=\{y\in S^4\mid f^{-1}_*(y)\neq \text{ a point}\} \]__is visibly a contractible set.

Also __\( S(f_*) \)__ is nowhere dense.
[Here
is a proof. The restriction __\( f_*| \)__
is the same as
__\[
q_3\circ q_1\circ g_1| : M\cap \overline{G}_0^4\to (B^2\times
\mathring{D}^2)/\mathcal{D}_+,
\]__
which is already surjective and __\( f_*^{-1}(S(f_*)) \)__ is contained in the
nowhere dense set of __\( M\cap \overline{G}_0^4 \)__ given
by
__\( (\partial G_0)\cup( \)__ends of __\( G_0^4)\cup g_1^{-1}\bigl(\bigcup_\alpha E(\alpha)\bigr) \)__.]

Therefore, according to Theorem 5.1, the map __\( f_* \)__ is approximable by
homeomorphisms. Next, by Proposition 4.2 (localisation principle), the
restriction
__\[ \operatorname{Int} M^4\to S^4\smallsetminus \{\infty\} \]__ is also
approximable by homeomorphism. Finally, by Proposition 4.3 (globalisation
principle), the map
__\[f : M\to (B^2\times \mathring{D}^2)/\mathcal{D}_+\]__
is approximable
by homeomorphisms. Thus Theorem 2.2 is proved modulo Sections 4 and 5.

__\( \overline{S(f_*)}\subset S^4 \)__is in fact a compactum of dimension

__\( \leq 1 \)__, because it is the union of a contractible set

__\( S(f_*) \)__with a set of dimension 0, that is, the ends of

__\( G_0^4 \)__which are not in the frontier of a connected component

__\( Y \)__of

__\( M^4\smallsetminus G_0^4 \)__. For reasons of cohomology,

__\( \dim \overline{S(f_*)}\geq 1 \)__. Therefore it is a compactum of dimension exactly 1.

#### 4. Bing shrinking

__\( B^2\times \mathring{D}^2/\mathcal{D}_+ \)__defined in Section 3 is homeomorphic to

__\( B^2\times \mathring{D}^2 \)__. The necessary techniques come from a series of articles of RH Bing from the 1950s (see especially [e3], [e5], [e6]), which made his reputation as a great virtuoso of geometric topology.

We consider a proper surjective map __\( f : X\to Y \)__ between metrisable,
locally compact spaces __\( X \)__, __\( Y \)__. Let
__\[ \mathcal{D}=\{f^{-1}(y)\mid y\in
Y\} \]__
be the decomposition associated with __\( f \)__. When is __\( f \)__ (strongly)
*approximable by homeomorphisms*, in the sense that for all open
coverings __\( \mathcal{V} \)__ of __\( Y \)__,
the __\( \mathcal{V} \)__-neighbourhood
__\[
N(f,V)=\{g : X\to Y\mid \text{for all }x\in X,\text{ there exists
}V\in\mathcal{V}\text{ such that }f(x), g(x)\in V\}
\]__
contains a homeomorphism?

Since __\( f \)__ induces a homeomorphism __\( \varphi : X/\mathcal{D}\to Y \)__, we
see easily that __\( f \)__ is approximable by homeomorphisms if and only if one can
find maps __\( g : X\to X \)__ such that
__\[ \mathcal{D}=\{g^{-1}(x)\mid x\in X\} \]__
and that __\( f\circ g \)__ approximates __\( f \)__ (in effect, __\( \varphi \)__ translates __\( g \)__
into a homeomorphism __\( g^{\prime} : Y\to X \)__). This observation makes the following
theorem plausible.

__(Bing shrinking criterion.)__

__\( f \)__is approximable by homeomorphism if and only if, for every covering

__\( \mathcal{U} \)__of

__\( X \)__and

__\( \mathcal{V} \)__of

__\( Y \)__, there exists a homeomorphism

__\( h : X\to X \)__such that

__\( h(\mathcal{D}) < \mathcal{U} \)__, and for all compact

__\( D\in \mathcal{D} \)__,

__\( D \)__and

__\( h(D) \)__are

__\( f^{-1}(\mathcal{V}) \)__-near in the sense that there exists an

__\( f^{-1}(V)\in f^{-1}(\mathcal{V}) \)__that contains

__\( D\cup h(D) \)__.

We then say that __\( \mathcal{D} \)__ is *shrinkable*. We can show a proof
by hand
[e28],
or by Baire category
[e35],
[e11]
(the idea is to find a homeomorphism __\( h : X\to Y \)__ that converges towards
__\( g \)__ that determines __\( \mathcal{D} \)__). The proof also gives:

__\( h \)__respects

__(__or fixes

__)__a closed set

__\( A\subset X \)__, then

__\( f \)__is approximable by homeomorphisms that send

__\( A \)__on

__\( f(A) \)__

__(__or which coincide on

__\( A \)__with

__\( f \)__

__)__, and reciprocally.

__(Localisation principle.)__If

__\( f : X\to Y \)__is approximable by homeomorphisms and

__\( Y \)__is a manifold (or

__\( Y \)__satisfies the principle of deformability by homeomorphisms coming from [e20],

__\( \mathcal{D}_1 \)__of [e23]), then, for each open set

__\( V \)__of

__\( Y \)__, the restriction

__\[ f_V : f^{-1}(V)\to V \]__of

__\( f \)__is approximable by homeomorphisms.

*Proof* (*indication*).
To approximate __\( f_V \)__, we combine
(by the principle __\( \mathcal{D}_1 \)__) a series of approximations of __\( f \)__;
compare
([e23], Section 3.5).
I believe that this lemma is not in
the literature because, for dimension __\( \neq 4 \)__, we have stronger results
[e22],
[e32].
However, upon reflection, the
complicated argument of
[e22]
works. In each case that interests
us, the reader will be able to find an ad hoc proof that is easier.
◻

__\( X \)__and

__\( Y \)__are

__\[ \operatorname{Cantor}\times [0,1]=2^{\mathbb{N}}\times [0,1] ,\]__and

__\( f=g\times \operatorname{Id}_{[0,1]} \)__, where

__\[ g(1,a_2,a_3,\dots)=(a_2,a_3,\dots), \quad g(0,a_2,a_3,\dots)=(0,0,0,\dots). \]__

__(Globalisation principle.)__Let

__\( f : X\to Y \)__be a proper map such that, for an open set

__\( V\subset Y \)__, the restriction

__\[ f_V : f^{-1}(V)\to V \]__is approximable by homeomorphisms. Then,

__\( f \)__is approximable by proper maps

__\( g \)__such that

__\( g^{-1}(V)=f^{-1}(V) \)__,__\( g_V : g^{-1}(V)\to V \)__is a homeomorphism, and__\( g= f \)__on__\( X\smallsetminus f^{-1}(V) \)__.

This principle is easy to establish, because if __\( \mathcal{V} \)__ is the covering
of __\( V \)__ by open balls centred on __\( y\in V \)__ and of
radius
__\[
\inf\{ d(y,z)\mid
z\in Y\smallsetminus V\},
\]__
then every map
__\[ \gamma : f^{-1}(V)\to V \]__
that
is in __\( N(f_V,\mathcal{V}) \)__, extends by __\( f \)__ to a map __\( g : X\to Y \)__. In
the very special case that __\( \mathcal{D} \)__ is __\( \pi_0(K) \)__ for a compact set
__\( K\subset X \)__, the Bing shrinking criterion simplifies as follows. (Then,
__\( \mathcal{D} \)__ consists of connected components of __\( K \)__ and the image of __\( K \)__
in __\( X/\mathcal{D} \)__ is 0-dimensional and is identified with __\( \pi_0(K) \)__.)

__(Criterion.)__Under these conditions,

__\( \mathcal{D} \)__is shrinkable if for all

__\( \epsilon > 0 \)__and for all open

__\( \mathcal{D} \)__-saturated

__\( U \)__of

__\( X \)__such that

__\( U\cap K \)__is compact, there is a homeomorphism

__\( h : X\to X \)__with support in

__\( U \)__

__(__respectively

__\( A\subset X) \)__such that

__\( h(K\cap U) \)__lies in a finite disjoint union of compact sets, each of diameter

__\( < \epsilon \)__.

This condition, modulo localisation principle (Proposition 4.2), is clearly necessary.

For all __\( \epsilon > 0 \)__, one can
consider
__\[
\mathcal{D}_\epsilon=\{D\in
\mathcal{D}\mid \operatorname{diam}D\geq \epsilon\}.
\]__
We say that __\( \bigcup_{D\in
\mathcal{D}_\epsilon} D \)__ is a closed subset of __\( X \)__. Here is a remarkable
but disturbing example where __\( \mathcal{D} \)__ is null, __\( \mathcal{D}_\epsilon \)__
is shrinkable for any __\( \epsilon > 0 \)__, but __\( \mathcal{D} \)__ is not shrinkable. The
elements of __\( \mathcal{D} \)__ are the connected components of a compact set
__\( X=\bigcap_n F_n \)__ where __\( F_0 \)__ and __\( F_1 \)__ are as illustrated. This image is
suitably replicated in each solid torus; __\( F_n \)__ is then __\( 2^n \)__ solid tori. Each
__\( D\in \mathcal{D} \)__ is clearly cellular, hence __\( \mathcal{D}_\epsilon \)__ is
shrinkable by Lemma 5.2. But, with the help of cyclic covers, one can check
that __\( \mathcal{D} \)__ is not shrinkable (see
[e12],
[e1]).

There are thankfully properties of individual elements, a little stronger than
cellularity, which discards this sort of example. For a compact __\( A\subset
X \)__, we consider the property __\( \mathcal{R}(X,A) \)__: for each __\( \epsilon > 0 \)__,
for every null decomposition __\( \mathcal{D} \)__ of __\( X \)__ containing __\( A \)__, and
for all neighbourhoods __\( U \)__ of __\( A \)__, there is a map __\( f : X\to X \)__ with
support in __\( U \)__ that shrinks at least __\( A \)__, (that is, __\( f(A) \)__ is a point and
__\( f|_U : U\to U \)__ is approximable by homeomorphisms), such that, for
all
__\( D\in \mathcal{D} \)__,
__\[
\operatorname{diam} f(D)\leq \max (\operatorname{diam}
D,\epsilon).
\]__
If __\( \mathcal{D} \)__ is fixed in advance, we call the (weaker)
property __\( \mathcal{R}(X,A;\mathcal{D}) \)__.

__\( U \)__of

__\( A \)__, we have

__\( \mathcal{R}(X,A) \)__is equivalent to

__\( \mathcal{R}(U,A) \)__. Moreover,

__\( \mathcal{R}(X,A) \)__is independent of the metric.

__\( \mathcal{D} \)__is null, and

__\( \mathcal{R}(X,D;\mathcal{D}) \)__is satisfied for all

__\( D\in \mathcal{D} \)__, then

__\( \mathcal{D} \)__is shrinkable.

*Proof*.
The proof is an edifying exercise.
◻

__\( \mathcal{R}(X,A) \)__is satisfied if

__\( A \)__is a topological flat disc of any codimension in the interior of the manifold.

*Proof of Proposition 4.6*.
This is
__\( \mathcal{R}(\mathbb{R}^n,B^k) \)__ for __\( k\leq n \)__. The proof of
__\( \mathcal{R}(\mathbb{R}^2,B^1) \)__ which is indicated by Figure 16.

In (a), every element of __\( \mathcal{D} \)__ that meets the big rectangle has
already diameter __\( < \epsilon/4 \)__; if __\( D\in \mathcal{D} \)__ meets a gap
between successive rectangles, it is disjoint from the rectangle after. We
set __\( f(B^1)=0 \)__, and __\( f=\operatorname{Id} \)__ outside the biggest rectangle
(which is in __\( U \)__); __\( f \)__ is linear on each vertical interval in a rectangle
of (b) and also linear on each 1-cell of the rectangular cellulation in
(b) of (big rectangle__\( \smallsetminus B^1 \)__). Moreover, __\( p\circ f=p \)__ where
__\( p \)__ is the projection to the __\( y \)__-axis (the __\( \mathbb{R}^{n-k} \)__ normal to
__\( B^k \)__). Finally, the size of the image of each of the *vertical*
rectangle is __\( < \epsilon/4 \)__.
◻

We consider the Whitehead
pair
__\[
(B^2\times S^1,j(B^2\times S^1))=(T,T^{\prime}),
\]__
and the *thickened* pair
__\[
(\mathbb{R}\times T,[0,1]\times T^{\prime}) .
\]__

__\( \epsilon > 0 \)__, there exists an isotopy

__\( h_t \)__

__\( (t\in [0,1]) \)__of

__\( \operatorname{Id}|_{\mathbb{R}\times T} \)__with compact support in

__\[ (-\epsilon,1+\epsilon)\times \operatorname{Int} T \]__such that we have

__\[ \operatorname{diam}(h_1(t\times T^{\prime})) < \epsilon \quad\text{and}\quad h_1(t\times T^{\prime})\subset [t-\epsilon,t+\epsilon]\times T \]__for all

__\( t\in [0,1] \)__.

*Idea of proof*.
It is suggested by Figure 18.
◻

By this lemma, one can shrink many decompositions related to Whitehead
compacta. For example, let __\( \mathcal{W}\subset \mathbb{R}^3 \)__ be a Whitehead
compactum and let
__\[\mathcal{D}=\{t\in W\mid t\in [0,1], W\in \mathcal{W}\}\]__
be the decomposition __\( I\times \mathcal{W} \)__ of __\( \mathbb{R}\times
\mathbb{R}^3=\mathbb{R}^4 \)__. Then __\( \mathcal{D} \)__ is shrinkable by Lemma
4.7 applied to the solid tori __\( T \)__, __\( T^{\prime} \)__, __\( T^{\prime\prime} \)__, …whose intersection is
__\( \mathcal{W} \)__. Therefore __\( \mathbb{R}^4/\mathcal{D} \)__ is homeomorphic to
__\( \mathbb{R}^4 \)__. Moreover, by Proposition 4.2 (localisation principle),
we have that
__\[ (0,1)\times \mathbb{R}^3/\mathcal{W} \]__
is homeomorphic to
__\( (0,1)\times \mathbb{R}^3 \)__. Hence we have the following celebrated fact.

__(Celebrated fact [e16].)__

__\( \mathbb{R}\times (\mathbb{R}^3/\mathcal{W})=\mathbb{R}^4 \)__.

This is a result of
Andrews
and
Rubin
[e16]
in 1965, proved after
analogous results, but more difficult, of
Bing
[e5]
in 1959,
which is a curious anachronism. There is a good explanation!
A. Shapiro,
at the time when he succeeded in turning __\( S^2 \)__ inside out in __\( S^3 \)__ by a regular
homotopy, compare
[e31],
had also established Theorem 4.8. In
any case, Bing tells me that
D. Montgomery
had communicated to him this
claim without being able himself to justify it except by giving an easier
argument (see Lemma 4.9) showing that
__\[ \mathbb{R}\times (S^3\smallsetminus
\mathcal{W}) \]__
is homeomorphic to __\( \mathbb{R}^4 \)__, compare
[e6].
Did
the proof of Shapiro from the 50s disappear without a trace?

To establish the flatness of the discs __\( \{E(\alpha)\} \)__ constructed in
Section 3.9, we will also need a lemma that is easier than Lemma 4.7,
treating again the Whitehead pair __\( (T,T^{\prime}) \)__. Let __\( D \)__ be a meridional disc
of __\( T \)__ that cuts __\( T^{\prime} \)__ transversally
in two discs.

__\( \mathbb{R}\times T \)__a topological 4-ball

__\( B \)__, such that

__\[ \operatorname{Int} B\supset [0,1]\times T^{\prime} \]__and

__\( B\cap (\mathbb{R}\times D) \)__is an equatorial 3-ball of the form

__\[ (\text{interval})\times D_0\subset \mathbb{R}\times D .\]__

*Proof of Lemma 4.9*.
This has nothing to do with the proof of
Lemma 4.7! We find __\( B \)__ easily from a 2-disc immersed in __\( T \)__ like in Figure
20 (compare Section 3.9).
◻

To establish that
__\[ (B^2\times \mathring{D}^2)/\mathcal{D}_+ \]__
is homeomorphic
to __\( B^2\times \mathring{D}^2 \)__, we will now use the construction of Section 3.

__\( \mathcal{W} \)__of

__\( B^2\times \mathring{D}^2 \)__is shrinkable.

*Proof of Proposition 4.10*.
We apply Theorem 4.4, Lemma 4.7 (or
Lemma 4.9, without exploiting the last condition of Lemma 4.9). For this,
it is convenient to remark first that for all open __\( \mathcal{W} \)__-saturated
__\( U \)__ in __\( B^2\times D^2 \)__, __\( W\cap U \)__ is contained in an open subset of __\( U \)__
that is a disjoint union of open sets of the form
__\[ \mathring{I}^{\prime}\times
\mathring{T}(a_1,\ldots,a_k) ,\]__
where __\( I^{\prime} \)__ is an interval.
◻

Our next goal is the flatness of the
discs
__\[
E(\alpha)=q(D(\alpha))\subset
(B^2\times \mathring{D}^2)/\mathcal{W}.
\]__
Let
__\[\mathcal{W}(\alpha)=\{w\in \mathcal{W}\mid w\cap D(\alpha)\neq
\emptyset\},\]__
and let __\( W(\alpha)=\bigcup \mathcal{W}(\alpha) \)__.

__\( \mathcal{W}(\alpha) \)__is shrinkable respecting

__\( D(\alpha) \)__. Therefore, the quotient

__\( q_\alpha(D(\alpha)) \)__of

__\( D(\alpha) \)__is flat in

__\[ (B^2\times \mathring{D}^2)/\mathcal{W}(\alpha) .\]__

*Proof of Proposition 4.11*.
We apply Lemma 4.9 and the relative
criteria (Theorem 4.4). For every open __\( \mathcal{W}_\alpha \)__-saturated __\( U \)__
of __\( B^2\times \mathring{D}^2 \)__, the intersection __\( W_\alpha\cap U \)__ is trivially
contained in an open set which, for some integer __\( l \)__, is a disjoint union of
open sets of the form
__\[ \mathring{I}^{\prime}\times \mathring{T}^{\prime}\subset U ,\]__
where
__\( T^{\prime} \)__ is a connected component of multiple solid tori __\( T_l(b_1,\ldots,b_l) \)__
and __\( i^{\prime} \)__ is an interval.

Condition (d) of Section 3.9 allows us to choose these sets so that in addition, for each:

__\( D(\alpha)\cap (I^{\prime}\times T^{\prime}) \)__is a single 2-disc, which is projected onto a meridional disc__\( D \)__of__\( T^{\prime} \)__which is also a connected component of__\( D^{\prime}(\alpha)\cap T^{\prime} \)__; see Section 3.9.

By condition (b) of Section 3.9 the meridional disc __\( D \)__ ideally chopped off
__\( T_{l+1/6}\cap T^{\prime} \)__, so Lemma 4.9 gives us disjoint 4-balls __\( B_1,\ldots,B_s \)__
in __\( \mathring{I}^{\prime}\times \mathring{T}^{\prime} \)__, such that

- each intersection
__\( B_i\cap D(\alpha) \)__is a diametral 2-disc and not knotted in__\( B_i \)__, and __\( \mathring{B}_1\cup\cdots\cup \mathring{B}_s \)__contains the compact set__\[ W^+\cap (\mathring{I}^{\prime}\times \mathring{T}^{\prime})\supset W_\alpha\cap (\mathring{I}^{\prime}\times \mathring{T^{\prime}}). \]__

For all compact __\( K \)__ in __\( \mathring{B}_i \)__ and all __\( \epsilon > 0 \)__, we can
easily find a homeomorphism __\( h : B_i\to B_i \)__ with compact support which
respects __\( \mathring{B}_i\cap D(\alpha) \)__ and such that __\( \operatorname{diam}
h(K) < \epsilon \)__. The criteria of Theorem 4.4 (respecting __\( D(\alpha) \)__) is
therefore satisfied.
◻

__\( q(D(\alpha))=E(\alpha) \)__is flat in

__\( (B^2\times \mathring{D}^2)/\mathcal{W} \)__.

*Proof of Proposition 4.12*.
The open
set
__\[
U_\alpha=(B^2\times \mathring{D}^2)\smallsetminus
(W_\alpha\cup D(\alpha))
\]__
is clearly homeomorphic
to
__\[
(B^2\times
\mathring{D}^2)/\mathcal{W}_\alpha-q_\alpha(D(\alpha))
\]__
by
__\( q_\alpha \)__. Therefore, by Propositions 4.2 and 4.3, the quotient
map
__\[
q_\alpha^{\prime} : (B^2\times \mathring{D}^2)/\mathcal{W}_\alpha\to
(B^2\times \mathring{D}^2)/\mathcal{W}
\]__
is approximable by homeomorphisms
fixing __\( q_\alpha^{\prime} \)__ on the flat disc __\( q_\alpha(D(\alpha)) \)__.
Therefore,
__\[
q(D(\alpha))=q^{\prime}(\alpha)\circ q(\alpha)(D(\alpha))
\]__
is flat.
◻

We now propose to finish by showing that the quotient
maps
__\begin{align*}
B^2\times \mathring{D}^2 &\xrightarrow{\approx}(B^2\times
\mathring{D}^2)/\mathcal{W}\\
&\xrightarrow{p_1}((B^2\times
\mathring{D}^2)/\mathcal{W})/\{E(\alpha)\}\\
&\xrightarrow{p_2} (B^2\times
\mathring{D}^2)/\mathcal{D}_+
\end{align*}__
are approximable by homeomorphisms.

__\( p_1 \)__is approximable by homeomorphisms.

*Proof of Proposition 4.13*.
This follows from Propositions 4.12,
4.6 and 4.5.
◻

To approximate __\( p_2 \)__ by homeomorphisms, we need a little
preparation. According to Propositions 4.13 and 4.10, there is a shrinking
map
__\[ r : B^2\times\mathring{D}^2\to B^2\times \mathring{D}^2 \]__
inducing the same decomposition as the quotient map
__\[ ((B^2\times
\mathring{D}^2)/\mathcal{W})/\{E(\alpha)\} ;\]__
we can identify the domain of
__\( p_2 \)__ with __\( B^2\times \mathring{D}^2 \)__ by __\( r \)__.

The decomposition __\( \mathcal{P} \)__
constituted of the preimages
__\[ p_2^{-1}(y)=\{\text{a point}\} \]__
is the countable collection of natural
quotients
of connected components of holes
__\( T_*(\alpha) \)__ and __\( B_* \)__, which now
identify __\( r(T_*(\alpha)) \)__ and
__\[ r(B_*)\subset B^2\times
\mathring{D}^2 .\]__
We observe
that __\( \mathcal{P} \)__ is null. The quotient
map
__\[
\lambda B^2\times \mu D^2=B_*\to rB_*
\]__
shrinks the Whitehead
compactum
__\[
\mathcal{W}(\partial
B_*)=\{w\in \mathcal{W}\mid w\subset B_*\},
\]__
and these compact sets lie in
__\[ \lambda B^2\times \mu \partial D^2\subset \partial B_* .\]__

__\( r(\partial B_*) \)__has a bicollar neighbourhood

__\( V \)__in

__\( B^2\times \mathring{D}^2 \)__, that is,

__\( (V,r(\partial B_*)) \)__is homeomorphic to

__\[ (\mathbb{R}\times r(\partial B_*) , 0\times r(\partial B_*)) .\]__

This will result in the following proposition.

__\( \partial B_* \)__in

__\[ (B^2\times \mathring{D}^2)/\mathcal{W}(\partial B_*) \]__admits a bicollar neighbourhood.

*Proof of Proposition 4.15*.
This is equivalent to the existence
of a bicollar neighbourhood in
__\[ (\mathbb{R}\times \partial B_*)/(0\times
\mathcal{W}(\partial B_*)) .\]__
However, by (slightly generalised) Theorem
4.8 and Propositions 4.2 and 4.3, the quotient map of the latter space on
__\[ (\mathbb{R}\times \partial B_*)/(\mathbb{R}\times \mathcal{W}(\partial
B_*)) \]__
is approximable by homeomorphisms, fixing the quotient of __\( 0\times
\partial B_* \)__.
◻

*Proof of Proposition 4.14.*
The map __\( r \)__ factorises into __\( r^{\prime\prime}\circ
r^{\prime} \)__ where __\( r^{\prime} \)__ factors through __\( \mathcal{W}(\partial B_*) \)__. However,
Proposition 4.15 ensures a bicollar neighbourhood of __\( r^{\prime}(\partial B_*) \)__ in
__\[ (B^2\times \mathring{D}^2)/\mathcal{W}(\partial B_*) .\]__
Propositions 4.2 and
4.3 ensure that __\( r^{\prime\prime} \)__ is approximable by homeomorphisms fixing __\( r^{\prime}(\partial
B_*) \)__. Therefore, the
pair
__\[
((B^2\times \mathring{D}^2)/\mathcal{W}(\partial B_*), r^{\prime}(\partial B_*))
\]__
(with the bicollar) is homeomorphic to __\( (B^2\times
\mathring{D}^2,r(\partial B_*)) \)__.
◻

__\( \mathcal{R}(B^2\times \mathring{D}^2, r(B_*);\mathcal{P}) \)__is satisfied.

*Proof of Proposition 4.16*.
Given an open neighbourhood
__\( U \)__ of __\( r(B_*) \)__, there exists, by Proposition 4.14, a homeomorphism
__\[ h : B^2\times \mathring{D}^2\to B^2\times \mathring{D}^2 \]__
with
compact support in a bicollar __\( V \)__ of __\( r(\partial B_*) \)__ in __\( U \)__, such
that
__\[ h(r(B_*))\subset r(\mathring{B}_*) .\]__
Since __\( r(\mathring{B}_*) \)__
is homeomorphic to __\( \mathbb{R}^4 \)__, there exists a map __\( g \)__ with support
in __\( r(\mathring{B}_*) \)__ and approximable by homeomorphisms such that
__\( g\circ h\circ r(B_*) \)__ is a point in __\( r(\mathring{B}_*) \)__.
Let
__\[
f=g\circ
h : B^2\times \mathring{D}^2\to B^2\times \mathring{D}^2.
\]__
By uniform
continuity on the compact support __\( F\subset r(B_*)\cup V \)__ of __\( f \)__, we know
that, for a given __\( \epsilon > 0 \)__, there exists __\( \delta > 0 \)__ such that for all
sets
__\[ E\subset B^2\times \mathring{D}^2 \]__
of diameter less than __\( \delta \)__,
the diameter of __\( f(E) \)__ is less than __\( \epsilon \)__. By Lemma 4.17, there exists
a stretch homeomorphism __\( \theta : B^2\times \mathring{D}^2 \)__ fixing
__\( r(B_*) \)__ and with support in __\( V \)__ such that, for all __\( P\in \mathcal{P} \)__
distinct from __\( B_* \)__ such that
__\[ \theta(P)\cap F\neq \emptyset ,\]__
we have
__\( \operatorname{diam} \theta(P) < \delta \)__. Then
__\[ f=f_0\circ \theta \]__
satisfies
__\( \mathcal{R}(B^2\times \mathring{D}^2,B_*;\mathcal{P}) \)__.
◻

__(Stretch lemma.)__Let

__\( l \)__be a null decomposition

__\( X\times [0,\infty) \)__where

__\( X \)__is compact and all elements of

__\( l \)__is disjoint from

__\( X\times 0 \)__. For all

__\( \epsilon > 0 \)__, there exists a homeomorphism with compact support

__\[ \varphi : [0,\infty)\to [0,\infty) \]__such that

__\( \Phi=\varphi\times\operatorname{Id}_X \)__satisfies that, for all

__\( E\in l \)__such that

__\[ \Phi(E)\cap (X\times [0,1])\neq \emptyset, \]__we have

__\( \operatorname{diam}(\Phi(E)) < \epsilon \)__.

*Proof of Lemma 4.17* (*Indications*.)
Figure 21
completes the proof.
□

All elements of __\( \mathcal{P} \)__ distinct from __\( r(B_*) \)__ are of the form
__\( r(T_*^{\prime}(\alpha)) \)__ where __\( T_*^{\prime}(\alpha) \)__ is a connected component of a torus
__\( T_*(\alpha) \)__. Following the method of the proof of Proposition 4.16,
we establish similarly the following proposition.

__\( \mathcal{R}(B^2\times \mathring{D}^2,r(T_*^{\prime}(\alpha));\mathcal{P}) \)__is satisfied.

*Proof of Proposition 4.18* (*indications*).
The quotient of
__\[ T_*^{\prime}(\alpha)=J(\alpha)\times T^{\prime}(\alpha) ,\]__
by the longitude __\( l(\alpha) \)__ that
is in __\( D(\alpha) \)__, is a cone whose centre is the quotient of __\( l(\alpha) \)__,
and the base is a solid torus.
__\[ \delta r(T_*^{\prime}(\alpha))-r(l(\alpha)) \]__
has a bicollar neighbourhood in __\( B^2\times \mathring{D}^2 \)__, compare
Proposition 4.13. The accumulation points of elements __\( P\neq \text{a
point} \)__ of __\( \mathcal{P} \)__ are the *centre* __\( r(l(\alpha)) \)__ and a compact
set
__\[ r(W\cap \partial T_*^{\prime}(\alpha)) \]__
far from __\( r(l(\alpha)) \)__.
◻

__\( p_2 \)__is approximable by homeomorphisms and hence

__\[ B^2\times \mathring{D}^2/\mathcal{D}_+\approx B^2\times \mathring{D}^2. \]__

*Proof*.
Apply Propositions 4.18, 4.16 and 4.5.
◻

#### 5. Freedman’s approximation theorem

__(Freedman’s approximation theorem.)__Suppose that

__\( X \)__and

__\( Y \)__are homeomorphic to the

__\( n \)__-sphere. Let

__\( f : X\to Y \)__be a surjective, continuous map such that the singular set

__\[S(f)=\{y\in Y\mid f^{-1}(y)\neq \text{a point}\}\]__is nowhere dense and at most countable. Then,

__\( f \)__can be approximated by homeomorphisms.

__\( \neq 4 \)__, there exist much stronger approximation theorems

__[e19]__,

__[e18]__,

__[e32]__. Therefore, in dimension 4, the problem of generalising Theorem 5.1 remains open.

In the case that __\( S(f) \)__ is finite, this theorem is
well known since it
constitutes the essential part of the celebrated Schönflies theorem
which was established around 1960 by
B. Mazur,
M. Brown
and
M. Morse.

Recall that a compact set __\( A \)__ in a topological __\( n \)__-manifold __\( M \)__ (without
boundary) is *cellular* if each neighbourhood of __\( A \)__ contains a
neighbourhood which is homeomorphic to __\( B^n \)__.

__\( A \)__be a compact, cellular set in the interior

__\( \operatorname{Int} M \)__of a manifold

__\( M \)__. Then, the quotient map

__\( q : M\to M/A \)__can be approximated by homeomorphisms which are supported in an arbitrarily given neighbourhood of

__\( A \)__.

Compare the Bing shrinking criterion, Theorem 4.1
[e8].
A direct
proof shrinks __\( A \)__ gradually to a point.

*Proof of Theorem 5.1 if \( S(f) \) is a point*.
Let

__\( y_0=S(f) \)__and

__\( A=f^{-1}(y_0) \)__, we have that

__\( X\smallsetminus A \)__is homeomorphic to

__\( \mathbb{R}^n \)__. Since

__\( X \)__is homeomorphic to

__\( S^n \)__, it follows that

__\( A \)__is cellular in

__\( X \)__(exercise). Then, we obtain approximations by applying Lemma 5.2. ◻

In the setting of Freedman’s ideas, the case where __\( S(f) \)__ is __\( n \)__ points,
__\( n\geq 2 \)__, is already as difficult as Theorem 5.1. However one can consult
[e8],
[e10]
for an easy proof. We recall the Schönflies
theorem.

__(Schönflies theorem.)__Let

__\( \Sigma^{n-1} \)__be a topologically embedded

__\( (n-1) \)__-sphere in

__\( S^n \)__such that there is a bicollar neighbourhood

__\( N \)__of

__\( \Sigma \)__in

__\( S^n \)__, that is,

__\( (N,\Sigma) \)__is homeomorphic to

__\[ (\Sigma\times [-1,1],\Sigma\times 0) .\]__Then the closure of each of the two components of

__\( S^{n}-\Sigma \)__is homeomorphic to the

__\( n \)__-ball

__\( B^n \)__.

*Proof of Theorem 5.3* (*starting from Theorem 5.1 for* __\( S(f) \)__
*consisting of two points*).
Let __\( X_1 \)__ and __\( X_2 \)__ be two connected components
of __\( S^n\smallsetminus \mathring{N} \)__ and __\( W_1 \)__ and __\( W_2 \)__ be the closures of
connected components of __\( S^n\smallsetminus \Sigma^{n-1} \)__ containing __\( X_1 \)__
and __\( X_2 \)__, respectively. It is necessary to show that __\( W_1 \)__ and __\( W_2 \)__
are homeomorphic to __\( B^n \)__.

Shrinking __\( X_1 \)__ and __\( X_2 \)__, we obtain a quotient map
__\[f : S^n\to S^n/\{X_1,X_2\}\approx (\Sigma\times [-1,1])/\{\Sigma\times
0,\Sigma\times 1\}\approx S^n\]__
that is approximable by homeomorphisms from Theorem 5.1 (the case of __\( S(f) \)__
is two points). So __\( X_1 \)__ and __\( X_2 \)__ are cellular in __\( S^n \)__. Apply Lemma 5.2
to __\( X_i\subset \mathring{W}_i \)__, we deduce that
__\[W_i\to W_i/X_i\approx \Sigma\times [0,1]/\{\Sigma\times 1\} \approx B^n\]__
is approximable by homeomorphisms.
◻

__\( \Sigma \)__bounds an

__\( n \)__-ball in

__\( S^n \)__, already arises from the case of Theorem 5.1 where

__\( S(f)=\{1\text{ point}\} \)__proved above. Freedman uses this case.

To prove Theorem 5.1, Freedman introduced a nice trick of iterated
replication of the approximation map, which vaguely reminds me of the
arguments of
Mazur
[e7].
This trick leads us to leave the
category of continuous maps and to instead work in the less familiar realm
of closed relations. It was during the seventies that closed relations
imposed themselves for the first time on geometric topology; they surfaced
implicitly in a very original article by
M. A. Stanko
[e24]
and have become essential since: I believe that it would be a herculean
task to prove, without closed relations, the subsequent theorem of
Ancel
and
Cannon
[e30]
that any topological embedding __\( S^{n-1}\to S^n \)__,
__\( n\geq 5 \)__, can be approximated by locally flat embeddings.

*closed relation*

__\( R : X\to Y \)__between metrisable spaces

__\( X \)__and

__\( Y \)__is a closed subset

__\( R \)__of

__\( X\times Y \)__. If

__\( S : Y\to Z \)__is a closed relation, the composition

__\( S\circ R : X\to Z \)__is

__\[S\circ R=\{(x,z)\in X\times Z \mid \text{there is }y\in Y\text{ such that } (x,y)\in R\text{ and }(y,z)\in S\},\]__which is also closed if

__\( Y \)__is compact. Therefore the collection of closed relations between compact spaces is a category.

A continuous map __\( f : X\to Y \)__ gives a closed relation
__\[ \{(x,f(x))\mid
x\in X\} \]__
(the graph of __\( f \)__) which we still call __\( f \)__. Reciprocally, provided
that __\( Y \)__ is compact, a closed relation __\( R : X\to Y \)__ is the graph of
a continuous function (which is uniquely determined) if __\( R\cap x\times Y \)__
is a point for all __\( x\in X \)__.

__\( [0,1)\to \mathbb{R}/\mathbb{Z} \)__is continuous and bijective; the inverse is discontinuous, but the graphs of both are closed.

By extending usual notions for continuous functions, for __\( A\subset X \)__
and __\( B\subset Y \)__, we have

- the
*image*__\( R(A)=\{y\in Y\mid \text{there exists }x\in A\text{ such that }(x,y)\in R\} \)__, - the
*restriction*__\( R|_{A} : A\to Y \)__is the closed subset__\( R\cap A\times Y \)__in__\( A\times Y \)__, - the
*inverse*__\( R^{-1} : Y\to X \)__such that__\( \{(y,x)\in Y\times X\mid (x,y)\in R\} \)__.

__\( R^{-1} \)__is the inverse of

__\( R \)__in the categorical sense if and only if

__\( R \)__is the graph of a bijection function

__(__if and only if the categorical inverse exists

__)__.

To exploit an analogy between a function __\( X\to Y \)__ and a relation __\( R :
X\to Y \)__, we will at any time assimilate __\( R \)__ to the function that associates
for each point __\( x\in X \)__ to a subset __\( R(x)\subset Y \)__.

*Proof of Theorem 5.1*.
Any submanifold of codimension 0 that
is introduced will be assumed to be topological and locally flat. Let __\( N \)__
be a neighbourhood of __\( f \)__ in __\( X\times Y \)__. The theorem asserts that there
exists a homeomorphism __\( H : X\to Y \)__ such that __\( H\subset N \)__.

By removing a small __\( n \)__-ball __\( D\subset Y\smallsetminus\overline{S(f)} \)__
from __\( Y \)__ and removing its preimage __\( f^{-1}(D) \)__ from __\( X \)__, we see that it
is permissible to adopt the following.

__(Change of data.)__Suppose

__\( X \)__and

__\( Y \)__are homeomorphic to

__\( B^n \)__rather than

__\( S^n \)__. Let

__\( f : X\to Y \)__be a surjective, continuous map such that the singular set

__\[S(f)=\{y\in Y\mid f^{-1}(y)\neq \text{a point}\}\]__is nowhere dense and at most countable and

__\( S(f)\subset \operatorname{Int} Y \)__. Then

__\( f \)__can be approximated by homeomorphisms.

It is easy to see that Theorem 5.4 implies Theorem 5.1 using the special
case of Theorem 5.3 (Schönflies theorem) where __\( \Sigma^{n-1} \)__ bounds
a ball (see observation after Theorem 5.3).

The first step of an inductive construction of __\( H \)__ is to apply the following
proposition to the triangle shown to the right.
Moreover, the neighbourhood __\( N \)__ of Proposition 5.5 becomes __\( N \)__ the above;
and __\( L \)__ becomes __\( Y \)__.

Suppose that __\( X \)__ and __\( Y \)__ are homeomorphic to __\( B^n \)__. A relation __\( R :
X\to Y \)__ is called *good* if it is closed, and satisfying the following
conditions:

__\( R\subset X\times Y \)__projects onto__\( X \)__and onto__\( Y \)__.__\( R(x) \)__is not a singleton set for at most countably many points in__\( X \)__and these exceptional points constitute a nowhere dense set contained in__\( \operatorname{Int} X \)__. The same holds for__\( R^{-1} \)__.

It is said that a good relation __\( R^{\prime} : X\to Y \)__ is *finer than* __\( R \)__
if
__\[ R^{\prime}\subset R\subset X\times Y .\]__

__(__which is possibly commutative

__)__shown to the right, where

__\( X \)__,

__\( Y \)__and

__\( Z \)__are homeomorphic to

__\( B^n \)__, and

__\( f \)__,

__\( g \)__are in addition continuous functions; a neighbourhood

__\( N \)__of

__\( R \)__in

__\( X\times Y \)__; and

__\( L\subset Z \)__an open subset

__(__called the gap

__)__. We impose the following conditions:

__\( R\subset (f^{-1}(\overline{L})\times g^{-1}(\overline{L}))\cup (f^{-1}(Z\smallsetminus L)\times g^{-1}(Z\smallsetminus L)) \)__; it is inevitable if the triangle switches.__\( R=g^{-1}\circ f \)__on__\( f^{-1}(\overline{L}) \)__.__\( R \)__is given by the intersection graph of a homeomorphism__\[ f^{-1}(Z\smallsetminus L)\to g^{-1}(Z\smallsetminus L) .\]__- The singular sets
__\( S(f) \)__and__\( S(g) \)__are separated on__\( L \)__, that is, there are two open disjoint sets__\( U \)__and__\( V \)__which contain__\( S(f)\cap L \)__and__\( S(g)\cap L \)__, respectively.

Then, for all __\( \epsilon > 0 \)__, we can modify the three data __\( g \)__, __\( R \)__, __\( L \)__
to __\( g_* \)__, __\( R_* \)__, __\( L_* \)__ so that in addition to the same conditions above
__(__with __\( g_* \)__, __\( R_* \)__, __\( L_* \)__ instead of __\( g \)__, __\( R \)__, __\( L \)____)__, we have __\( R_*=R \)__
on __\( f^{-1}(Z\smallsetminus L) \)__, __\( L_*\subset L \)__, and for all __\( y\in Y \)__,
__\( \operatorname{diam} R_*^{-1}(y) < \epsilon \)__.

__\( N_*\subset N \)__of

__\( R_* \)__in

__\( X\times Y \)__such that

__\[ \operatorname{diam}(N_*^{-1}(y)) < \epsilon \]__for all

__\( y\in Y \)__.

*Proof of Addendum*.
If the conclusion is false, then
there are two sequences of points of __\( X\times Y \)__, say __\( (x_k,y_k) \)__,
__\( (x_k,y_k^{\prime}) \)__, __\( k=1,2,3,\dots \)__, which converge in compact __\( R_* \)__ and such
that __\( d(y_k,y_k^{\prime})\geq \epsilon \)__. By compactness of __\( X\times Y \)__, we can
arrange that the sequences __\( x_k \)__, __\( y_k \)__ and __\( y_k^{\prime} \)__ converge to __\( x \)__, __\( y \)__
and __\( y^{\prime} \)__, respectively. Then, __\( (x,y) \)__ and __\( (x,y^{\prime}) \)__ belong to compact __\( R_* \)__,
but __\( d(y,y^{\prime})\geq \epsilon \)__, which is a contradiction.
◻

Proposition 5.5 (with Addendum) will be used as a machine that swallows
the data __\( f \)__, __\( g \)__, __\( R \)__, __\( L \)__, __\( N \)__, __\( \epsilon \)__ and manufactures __\( f \)__, __\( g_* \)__,
__\( R_* \)__, __\( L_* \)__, __\( N_* \)__.

Let us continue constructing the homeomorphism __\( H \)__, assuming
Proposition 5.5. For __\( k\geq 1 \)__, the __\( k \)__-th step constructs a triangle
shown to the right (where __\( Z \)__ is a
copy of __\( Y \)__); a submanifold __\( L_k\subset Z \)__ and a neighbourhood __\( N_k \)__ of __\( R_k \)__ in __\( X\times
Y \)__ such that __\( f_k \)__, __\( g_k \)__, __\( R_k \)__, __\( L_k \)__, __\( N_k \)__ satisfy the conditions
imposed on __\( f \)__, __\( g \)__, __\( R \)__, __\( L \)__, __\( N \)__ in Proposition 5.5.
The first step is
already specified: Proposition 5.5 creates __\( f_1 \)__, __\( g_1 \)__, __\( R_1 \)__, __\( L_1 \)__,
__\( N_1 \)__ from __\( f \)__, __\( \operatorname{Id} \)__, __\( f \)__, __\( Y \)__, __\( N \)__, 1.

Suppose that the __\( k \)__-th triangle is constructed and we construct the
__\( (k+1) \)__-th triangle.

- If
__\( k \)__is odd, then Proposition 5.5 gives__\( g_{k+1} \)__,__\( f_{k+1} \)__,__\( R_{k+1}^{-1} \)__,__\( L_{k+1} \)__,__\( N_{k+1}^{-1} \)__from__\( g_k \)__,__\( f_k \)__,__\( R_{k}^{-1} \)__,__\( L_k \)__,__\( N_k^{-1} \)__,__\( 1/k \)__. In brief, we apply Proposition 5.5 to the reverse triangle shown to the right. - If
__\( k \)__is even, then it is same as the first step: Proposition 5.5 gives__\( f_{k+1} \)__,__\( g_{k+1} \)__,__\( R_{k+1} \)__,__\( L_{k+1} \)__,__\( N_{k+1} \)__from__\( f_k \)__,__\( g_k \)__,__\( R_k \)__,__\( L_k \)__,__\( N_k \)__,__\( 1/k \)__.

By induction, we have __\( N\supset N_1\supset N_2\supset \cdots \)__. We define
__\( H=\bigcap_k N_k \)__.
Then, __\( H \)__ is a homeomorphism since, for all __\( x \)__, we
have
__\[
\operatorname{diam} H(x)\leq \operatorname{diam} N_k(x)\leq 1/k,
\]__
for
all even __\( k \)__,
and
__\[
\operatorname{diam} H^{-1}(x)\leq \operatorname{diam}
N_k^{-1}(x)\leq 1/k,
\]__
for all odd __\( k \)__. This homeomorphism __\( H \)__ in the
neighbourhood __\( N \)__ of __\( f \)__ completes the proof of Theorem 5.1 assuming
Proposition 5.5.
◻

*Proof of Proposition 5.5*.
To explain the essential idea of
Freedman, the reader should read the proof with a view to (re)proving that a
surjection __\( f : B^n\to B^n \)__ such
that
__\[
S(f)=\{\text{a point}\}\subset \operatorname{Int} B^n
\]__
is approximable by homeomorphisms (for this, we set
__\( f=R \)__ and __\( g=\operatorname{Id} \)__). Then, it should be noted that as soon
as
__\[
S(f)=\{k\text{ points}\}\subset \operatorname{Int} B^n,
\]__
the same
argument leads us to approximate __\( f \)__ by relations which crush nothing,
but which blow up __\( k(k-1) \)__ points.

Consider the preimages __\( R^{-1}(y) \)__, __\( y\in Y \)__, of diameter __\( \geq \epsilon \)__,
that we want to eliminate. According to (a), (b) and (c), these sets
constitute the preimage by __\( f \)__ of the set __\( (S_\epsilon(f)\cap L)\subset
Z \)__,
where
__\[
S_\epsilon(f)=\{z\in Z\mid \operatorname{diam}f^{-1}(z)\geq
\epsilon\},
\]__
which will allow us to follow the case in __\( Z \)__. Note that
__\( S_\epsilon(f) \)__ is compact although, typically, __\( S(f) \)__ is not. For example,
__\( S_\epsilon(f) \)__ is finite in the case of interest to Freedman (see Section 4).

__(General position.)__In the interior of a compact topological manifold

__\( M \)__, let

__\( A \)__and

__\( B \)__be two countable sets and nowhere dense. Then there exists a small automorphism

__\( \theta \)__of

__\( M \)__fixing all points of

__\( \partial M \)__, such that

__\( \theta(A) \)__and

__\( B \)__are separated, that is, contained in disjoint open sets.

*Proof of Lemma 5.6*
Consider the space
__\( \operatorname{Aut}(M,\partial M) \)__ of automorphisms of
__\( M \)__ fixing __\( \partial M \)__, provided with the complete
metric
__\[
\operatorname{sup}(d(f,g),d(f^{-1},g^{-1}))
\]__
where __\( d \)__ is the
uniform convergence metric. In __\( \operatorname{Aut}(M,\partial
M) \)__, the set of automorphisms __\( \theta \)__, such that the first __\( k \)__
points __\( A_k \)__ of __\( A \)__ and __\( B_k \)__ of __\( B \)__
satisfying
__\[
\theta(A_k)\cap
\overline{B}=\emptyset=\theta(\bar{A})\cap B_k,
\]__
constitute an open
subset
__\[ U_k\subset \operatorname{Aut}(M,\partial M) \]__
everywhere dense in
__\( \operatorname{Aut}(M,\partial M) \)__, because __\( \bar{A} \)__ and __\( \overline{B} \)__
are closed, nowhere dense in __\( M \)__.

Then, the famous Baire category theorem asserts that the countable
intersection __\( \bigcap_k U_k \)__ is everywhere dense in __\( \operatorname{Aut}(M,
\partial M) \)__. Note that __\( \bigcap_k U_k \)__ is the set of __\( \theta \)__
in __\( \operatorname{Aut}(M,\partial M) \)__ such
that
__\[
\theta(A)\cap \overline{B}=\emptyset =\theta(\bar{A})\cap B.
\]__
But, for __\( X_1 \)__, __\( X_2 \)__ in a metrisable __\( M \)__, the condition that
__\[
X_1\cap \overline{X}_2=\emptyset =\overline{X}_1\cap X_2
\]__
leads to the separation of __\( X_1 \)__ and __\( X_2 \)__ in __\( M \)__. In
effect, seen in the open subset __\( M\smallsetminus (\overline{X}_1\cap
\overline{X}_2) \)__ of __\( M \)__, the
sets
__\[ \overline{X}_1\smallsetminus
(\overline{X}_1\cap \overline{X}_2)
\quad\text{and}\quad
(\overline{X}_1\cap \overline{X}_2) \]__
are always disjoint, closed and hence separated. The mentioned condition
ensures that they contain respectively __\( X_1 \)__ and __\( X_2 \)__.
◻

__(Trivial if__There exists a finite union

__\( S_\epsilon(f) \)__is finite.)__\( B_+ \)__of disjoint

__\( n \)__-balls in

__\( L \)__satisfying the following conditions:

__\( S_\epsilon(f)\cap L\subset \mathring{B}_+ \)__.__\( S(g)\cap B_+=\emptyset \)__.- Each connected component
__\( B_+^{\prime} \)__of__\( B_+ \)__is small in the sense that__\[ (f^{-1}(B_+^{\prime}))\times (g^{-1}(B_+^{\prime}))\subset N, \]__and standard in the sense that__\( Z\smallsetminus \operatorname{Int}B_+^{\prime} \)__is homeomorphic to__\( S^{n-1}\times [0,1] \)__.

*Proof of Claim 5.7*.
Identify __\( Z \)__ with __\( B^n\subset \mathbb{R}^n \)__
to give __\( L \)__ an affine linear structure. Let __\( K \)__ be a compact neighbourhood
of the compact set __\( S_\epsilon(f)\cap L \)__ which is a subpolyhedra of __\( L \)__
and disjoint from __\( S(g) \)__, see Proposition 5.5(c).
We subdivide __\( K \)__ into a simplicial complex of which each simplex
__\( \mathcal{L} \)__ is linear in __\( L \)__ and so small such that
__\[
f^{-1}(\Delta)\cap g^{-1}(\Delta)\subset N.
\]__
Then (compare, the proof of Lemma 5.6), by a small
perturbation (a translation if we want) of __\( K \)__ in __\( L \)__, we disengage the
__\( (n{-}1) \)__-skeleton __\( K^{(n-1)} \)__ from the compact countable __\( S_\epsilon(f) \)__,
without harming the properties of __\( K \)__ already established. Finally,
__\( B_+ \)__ is defined as __\( K \)__ minus a small __\( \delta \)__ open neighbourhood of
__\( K^{(n-1)} \)__ in __\( \mathbb{R}^n \)__. Each component __\( B_+^{\prime} \)__ of __\( B_+ \)__ is convex
and in __\( \operatorname{Int} Z=\mathring{B}^n \)__; therefore __\( Z\smallsetminus
\mathring{B}_+^{\prime} \)__ is homeomorphic to __\( S^{n-1}\times [0,1] \)__, by an elementary
argument.
◻

In __\( \operatorname{Int} B_+ \)__, we choose now a union __\( B \)__ of balls (one in each
connected component of __\( B_+ \)__), which still satisfies (1), (2), (3) and also

(4) __\( S(f)\cap \partial B=\emptyset \)__.

We set __\( L_*=L\smallsetminus B \)__. For each connected component __\( B_+^{\prime} \)__
of __\( B_+ \)__, we are now modifying __\( g \)__ and __\( R \)__ above __\( B_+^{\prime} \)__ to define __\( g_* \)__
and __\( R_* \)__. These changes for the various connected components __\( B_+^{\prime} \)__ are
disjoint and independent. Therefore, it is enough to specify one. Moreover,
in order to simplify the notation, we allow ourselves to specify this
change only in the case that __\( B_+ \)__ is connected.

Let __\( c : Z\to B_+ \)__ be a homeomorphism, called the compression, which
fixes all points of __\( B \)__. (We remember that __\( Z\smallsetminus \mathring{B} \)__
is homeomorphic to
__\[ S^{n-1}\times [0,1] \]__
and __\( B_+\smallsetminus
\mathring{B} \)__.) We should modify __\( c \)__ by composing with a homeomorphism
__\( \theta \)__ of __\( B_+\smallsetminus \mathring{B} \)__ fixing
__\[ \partial B_+\cup
\partial B \]__
given by Lemma 5.6, to assure that __\( S(f) \)__ and __\( c(S(f)) \)__ are
separated on the open __\( \mathring{B}_+\smallsetminus B \)__.

Since __\( g^{-1}(B_+) \)__ is a ball in __\( Y \)__ (in fact, __\( g \)__ is a homeomorphism over
__\( B_+ \)__), we can also choose __\( i \)__ so that __\( i|_{\partial X} \)__ is
__\[ (g^{-1}\circ
c\circ f)|_{\partial X} .\]__
We set
__\[g_*=\begin{cases}g&\mbox{on }g^{-1}(Z\smallsetminus \mathring{B}_+),\\
c\circ f\circ i^{-1}&\mbox{on }g^{-1}(B_+).\end{cases}\]__
On __\( g^{-1}(\partial B_+) \)__, __\( g_* \)__ is well-defined since
__\[ g=c\circ f\circ
(g^{-1}\circ c\circ f)^{-1} \]__
on __\( g^{-1}(\partial B_+) \)__. We set
__\[R_*=\begin{cases}R&\mbox{on }f^{-1}(Z\smallsetminus \mathring{B}_+),\\
g^{-1}\circ f=(i\circ f^{-1}\circ c^{-1})\circ f&\mbox{on
}f^{-1}(B_+).\end{cases}\]__
More precisely, on __\( f^{-1}(B_+) \)__, we specify
__\[R_*=\begin{cases}(i\circ f^{-1}\circ c^{-1})\circ f &\mbox{on
}f^{-1}(B_+-\mathring{B}),\\
i&\mbox{on }f^{-1}(B).
\end{cases}\]__
On __\( f^{-1}(\partial B) \)__, __\( R_* \)__ is well-defined since __\( c \)__ fixes all points
of __\( \partial B \)__, and
__\[ S(f)\cap \partial B=\emptyset .\]__

We have now
specified the modification __\( L_* \)__, __\( g_* \)__, __\( R_* \)__ of __\( L \)__, __\( g \)__, __\( R \)__
claimed by Proposition 5.5. (We remark that if __\( B_+ \)__ is a union of __\( k \)__ balls,
rather than one ball, the modification is done in __\( k \)__ disjoint and independent
steps, each similar to the one just specified for connected __\( B_+ \)__.)

Verifying the claimed properties for __\( L_* \)__, __\( g_* \)__, __\( R_* \)__ is direct. (There
are already manuscripts
[1],
[2]
which offer more details.)
◻

__\( g_* \)__and

__\( R_* \)__, hides the geometry. We now try to reveal it by looking

__\( f \)__and

__\( g \)__respectively as fibrations

__\( \varphi \)__and

__\( \gamma \)__, of base

__\( Z \)__, and variable fibre, which allows us to use the notion of fibre restriction. Let

__\[ \gamma_0=\gamma-(\gamma|_{\mathring{B}_+}) .\]__We form

__\( \gamma_* \)__of

__\( \varphi\sqcup \gamma_0 \)__by identifying, via

__\( c|_{\partial Z} \)__, the subfibres (whose fibres are points)

__\( \varphi|_{\partial Z} \)__and

__\( \gamma_0|_{\partial B_+} \)__. Then, we can identify the total space and the base of

__\[ \gamma_*=\varphi\cup \gamma_0 \]__to those of

__\( \gamma \)__by an extension of

__\( \gamma_0\to \gamma \)__. More precisely, we use

__\[ (\operatorname{Id}|_{Z\smallsetminus \mathring{B}_+})\cup c \]__between bases, which is the identity on

__\( B\subset B_+ \)__. Then,

__\[ \varphi \quad\text{and}\quad \gamma_* : Y\xrightarrow{g_*}Z \]__are fibrations over

__\( Z \)__naturally isomorphic on

__\( B\cup L \)__, which defines a relation

__\( R_* \)__finer than the simple correspondence of fibres

__\( g_*^{-1}\circ f \)__.

__\( f_n \)__and

__\( g_n \)__converge towards

__\( f_\infty \)__and

__\( g_\infty \)__, and that

__\[ g_\infty\circ H=f_\infty .\]__Thus, as fibres,

__\( f_\infty \)__and

__\( g_\infty \)__are isomorphic. Moreover, each fibre of

__\( f_\infty \)__or

__\( g_\infty \)__is homeomorphic to a fibre of

__\( f \)__. I point out that

__\( f_\infty \)__and

__\( g_\infty \)__remind me of the two infinite products of Mazur

__[e7]__and that

__\( H \)__reminds me of the famous

__Eilenberg–Mazur swindle__, which completes the proof of Theorem 5.3

__(__weakened version

__)__given in

__[e7]__.

__(Following Remark 2.)__If we want to avoid unnecessary complications in the structure of

__\( f_\infty \)__and

__\( g_\infty \)__, it should be noted that in the definition of

__\( g_* \)__above, we have the right to replace the map

__\( f \)__which occurs by any good map

__\( f^{\prime} : X\to Y \)__such that

__\( f=f^{\prime} \)__on

__\( f^{-1}(B) \)__. Then, for every use of Proposition 5.5 in the proof of Theorem 5.1, we find that one always has the possibility of choosing for an alternative

__\( f^{\prime} \)__a map isomorphic to a map

__\( f \)__given in Theorem 5.1. With this little refinement, the proof of Theorem 5.1 in the case

__\( S(f)=\{2\text{ points}\} \)__is close to the argument of

__[e7]__. In particular,

__\( \overline{S(f_\infty)} \)__and

__\( \overline{S(g_\infty)} \)__can be homeomorphic to

__\( \mathbb{Z}\cup \{\infty, -\infty\} \)__.