Celebratio Mathematica

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 $V^{'}$ be two such manifolds. Suppose that there is an isomorphism $Θ : H_{2} (V) \to H_{2} (V^{'})$ which preserves the intersection forms. (Note that $V$ is a homotopy 4-sphere if and only if $H_{2} (V) = 0$ .)

Theorem A: In this situation,

Θ

is realised by a homeomorphism

V \to V^{'}

Proof. It is not difficult to realise $Θ$ by a homotopy equivalence $g : V \to V^{'}$ [e25]. Surgery theory [e14], [e21] gives a compact 5-manifold $W$ with boundary $\partial W = V ⊔ - V^{'}$ such that the inclusions $V \to W$ and $V^{'} \to W$ are homotopy equivalences and such that the restriction $r |_{V} : V \to V^{'}$ of the retraction $r : W \to V^{'}$ is homotopic to $g$ . The compact triad $(W; V, V^{'})$ is called an $h$ -cobordism. Smale’s theory of handles tries to improve a Morse function $f : (W; V, V^{'}) \to ([0, 1]; 0, 1)$ 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.

Theorem B: Every smooth, compact, simply connected, 5-dimensional

h

-cobordism

(W; V, V^{'})

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.

Remaining smooth problem: (Unresolved in February 1982.) Let

S = S_{1} ⊔ \dots ⊔ S_{k} and S^{'} = S_{1}^{'} ⊔ \dots ⊔ S_{k}^{'}

be two families of disjointly embedded 2-spheres in a simply connected 4-manifold

M

(in fact

f^{- 1} (a point)

) in such a way that the homological intersection number

S_{i} \cdot S_{j}^{'} = \pm δ_{i, j}

. Can one reduce

S \cap S^{'}

k

points of intersection (smooth and transverse) by a smooth isotopy of

S

~~in $M$ ?~~

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.

Remaining topological problem: (Resolved here.) With the data of the smooth problem, reduce

S \cap S^{'}

k

points by a topological isotopy of

S

M

, that is given by an ambient isotopy

h_{t},

1 \leq t \leq 1

, of

Id |_{M}

fixing a neighbourhood of

k

-points of

S \cap S^{'}

Whitney introduced a natural method for solving these problems. In the model $(R^{2}; A, A^{'})$ , (this is a straight line $A$ cutting a parabola $A^{'}$ in two points), we can disengage $A$ from $A^{'}$ 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, $(R^{4}; A_{+}^{}, A_{+}^{'}) = (R^{2} \times R^{2}; A \times 0 \times R, A^{'} \times R \times 0),$ there is an isotopy with compact support that makes the plane $A_{+}^{}$ disjoint from the plane $A_{+}^{'}$ , deleting the two transverse intersection points between $A_{+}^{}$ and $A_{+}^{'}$ .

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

Theorem C: (Casson–Freedman.) In this context, after a preliminary smooth isotopy of

S

in M

, (adding intersection points with

S^{'}

by finger moves, far from

S \cap S^{'}

), 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 $R^{2} ∖ A \cup A^{'}$ . The product $B \times R^{2}$ is an open, embedded 2-handle (as a closed submanifold) in the Whitney model, and disjoint from $A_{+}^{} \cup A_{+}^{'}$ . In $B \times R^{2}$ , Casson constructed certain open sets $H = B \times R^{2} ∖ Ω$ with boundary $\partial H = \partial B \times R^{2}$ , 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 $B \times R^{2}$ or not. Replacing $B \times R^{2}$ by $H \subset B \times R^{2}$ in this Whitney model $(R^{4}; A_{+}^{}, A_{+}^{'})$ we have an open set $(R^{4} ∖ Ω; A_{+}^{}, A_{+}^{'})$ , that we call the Whitney–Casson model. By a remarkable infinite process, Casson proved the following.

Theorem D: (Casson [e38], compare [e33].) After a preliminary smooth isotopy of

S

M

, one can find in

(M; S, S^{'})

smoothly embedded, disjoint Whitney–Casson models so that the models contain all the points of

S \cap S^{'}

except the

k

intersection points.

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

Theorem E: (Freedman, 1981.) Every open Casson handle is homeomorphic to

B^{2} \times R^{2}

. Therefore, the Whitney model

(R^{4}; A_{+}^{}, A_{+}^{'})

is homeomorphic to

(R^{4} ∖ Ω; A_{+}, A_{+}^{'})

The noncompact version of Theorem B is also important.

Theorem F: Let

(W; V, V^{'})

be a simply connected, proper smooth 5-dimensional

h

-cobordism with a finite number of ends and a trivial

π_{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 $s -cobordism$ 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 $Z$ is realised, as well as every $x \in Z_{2}$ , except that for even forms, $x \in σ / 8 \in Z_{2}$ . Every topological 4-manifold $V$ which is homotopy equivalent to $S^{4}$ is in this class, because $V ∖ {point}$ is contractible and thus $V ∖ {point}$ can be immersed into $R^{4}$ (compare [e29]).

It also follows (see [1], [e34]) that every smooth homology 3-sphere $V$ (that is, $H_{*} (V) ≅ H_{*} (S^{3})$ ) 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 $D (α)$ of Section 3.9) deviates, probably for reasons of taste. I am indebted to A. Marin for his brotherly and insightful comments.

1. Terminology

This terminology is used from now on except when otherwise indicated. All spaces admit a metric, denoted generally by

d

. Maps are all continuous. The support of a map

f : X \to X

is the closure of

{x \in X ∣ 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 ∣ f_{t} (x) \neq x for some t \in [0, 1]} .

For a subset

A

, define the closure

\bar{A}

, the interior

\overset{˚}{A}

and the frontier

δ A

, always with respect to the understood ambient space (the largest involved). If

A

is a manifold, it is often necessary to distinguish

\overset{˚}{A}

from its formal interior

Int A

and

δ A

from the formal boundary

\partial A

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

The set of connected components of a space $X$ is denoted by $π_{0} (A)$ . If $A$ is compact, $π_{0} (A)$ is at the same time a decomposition of $A$ for which the quotient $A / π_{0} (A)$ is a compact set of dimension 0 (totally discontinuous), that is identified with $π_{0} (A)$ as a set. If $A \subset X$ , $π_{0} (A)$ gives a decomposition of $X$ whose quotient space is denoted by $X / π_{0} (A)$ . 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, $R^{n}$ is the Euclidean space with the metric $d (x, y) = | x - y |$ ; $B^{n} = {x \in R^{n} ∣ | x | \leq 1}$ ; $I = [0, 1]$ . A multidisc is a disjoint union of finitely many discs (each are diffeomorphic to $B^{2}$ ). Similarly, for multihandle, etc. The symbols $≅$ , $\approx$ and $≃$ indicate a diffeomorphism, a homeomorphism and a homotopy equivalence, respectively.

2. Casson tower and Freedman’s mitosis

We will use two versions

B^{2}

and

D^{2}

of the standard smooth 2-disc

{(x, y) \in R^{2} ∣ 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_{-}

\partial B^{2} \times D^{2}

; its skin

\partial_{+}

B^{2} \times \partial D^{2}

, its core is

B^{2} \times 0

. A 2-handle is a pair

(H^{4}, \partial_{-} H) 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 \overset{˚}{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

\overset{˚}{D^{2}}

where we ought strictly to write

Int D^{2}

A defect $X$ in a handle $(H^{4}, \partial_{-} H)$ is a compact submanifold $X$ of $H^{4} ∖ \partial_{-} H$ 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 $Int B^{2}$ . A multi-defect $X$ in a handle $(H^{4}, \partial H)$ is a finite, disjoint union $⨆_{i} X (i) = X$ of $\geq 1$ defects $X (i)$ , that, for a suitable identification $(H^{4}, \partial H) ≅ (B^{2}, \partial B^{2}) \times D^{2},$ 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.

Lemma 2.1: The triad

(H^{4} ∖ \overset{˚}{X}; \partial_{-} H, δ X)

determines

H^{4}

and

X

in the following sense. If

X^{'}

is a multidefect in a handle

(H^{'}, \partial_{-} H^{'})

and

θ : (H ∖ \overset{˚}{X}; \partial_{-} H, δ X) \to (H^{'} ∖ \overset{˚}{X^{'}}; \partial_{-} H^{'}, δ X^{'})

is a diffeomorphism, there exists a diffeomorphism

Θ : H \to H^{'}

extending

θ

Sketch of proof (see [e38]). If we attach a multihandle $(X^{'}, \partial_{-} X^{'})$ to $H ∖ \overset{˚}{X}$ along the frontier $δ X$ , in such a way that there exists no extension of $θ$ to a diffeomorphism $Θ : H \to (H ∖ \overset{˚}{X}) \cup X^{'} = H^{'},$ we claim that $(\partial H^{'}, \partial_{-} H)$ is diffeomorphic to $(S^{3}, solid torus)$ where the solid torus is tied in a nontrivial knot — in fact, a connected sum of $k$ nontrivial twist knots, $1 \leq k \leq | π_{0} (X) |$ . ◻

A residual defect $Ω$ in a handle $(H^{4}, \partial_{-} H^{4})$ is the intersection of a sequence $X_{1} \supset \overset{˚}{X_{1}} \supset X_{2} \supset \overset{˚}{X_{2}} \supset X_{3} \supset \dots$ of compact submanifolds of $H^{4} ∖ \partial_{-} H^{4}$ such that, for all $k$ , $(X_{k}, δ X_{k})$ is a multihandle in which $X_{k + 1}$ is a multidefect. The sequence $X_{1} \supset X_{2} \supset \dots$ 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 $Ω \subset H$ and an open smooth embedding $i_{\infty} : H_{\infty} \to H$ with image $H ∖ Ω$ , which induces a diffeomorphism $i_{\infty} | : \partial_{-} H_{\infty} \to \partial_{-} H .$ In other words, $(H_{\infty}, \partial_{-} H_{\infty})$ is diffeomorphic to $(H ∖ Ω, \partial_{-} H)$ .

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 $H_{k} = i_{\infty}^{- 1} (H ∖ \overset{˚}{X_{k}}) and \partial_{-} H_{k} = \partial_{-} H_{\infty} .$ Then, $H_{\infty} = ⋃_{k} H_{k}$ . The manifold $H_{k}$ is called a tower of height $k$ , its stages are $E_{j} = i_{\infty}^{- 1} (X_{j - 1} ∖ X_{j})$ 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 $E_{k}$ .~~ Similarly $\partial_{+} X_{k} = X_{k} \cap \partial_{+} H .$

A Casson handle $(H_{\infty}, \partial_{-} H_{\infty})$ is never compact; we will often encounter the endpoint compactification ${\hat{H}}_{\infty}$ of $H_{\infty}$ . Recall that the endpoint compactification $\hat{M}$ of a connected, locally connected and locally compact space $M$ is the Freudenthal compactification that adds to $M$ the compact 0-dimensional space $Ends (M)$ which is the (projective) limit of an inverse system ${π_{0} (M ∖ K) ∣ K \subset M such that K is compact} .$

By $i_{\infty}$ , ${\hat{H}}_{\infty}$ is identified with the quotient of $H^{4}$ obtained by crushing each connected component of $Ω$ to a point. (To verify this, note that $π_{0} (Ω)$ with the compact topology is the (projective) limit of an inverse system ${π_{0} (U) ∣ U$ is an open subset of $H$ containing $Ω}$ .)

We remark that ${\hat{H}}_{\infty}$ is the Alexandroff compactification by a point, exactly when $Ω \subset H$ 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.

${\hat{H}}_{\infty}$ 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 $Ω$ is connected, by definition, $\partial H_{\infty}$ (which is homeomorphic to $\partial H ∖ \partial_{+} Ω$ ) is one of the contractible 3-manifolds of J. H. C. Whitehead [e1], [e2], with a nontrivial $π_{1}$ -system at infinity. $\partial_{+} Ω \subset \partial H ≅ S^{3}$ is a Whitehead compactum. In the general case, $\partial_{+} Ω$ is called a ramified Whitehead compactum. Thus, $(\hat{H ∖ Ω}, \partial_{-} H)$ has no chance of being a topological handle. On the other hand, $H ∖ (\partial_{+} H \cup Ω)$ is homeomorphic to $B^{2} \times R^{2}$ ; this will be the central result of this paper.

Theorem 2.2: (Freedman, 1981.) Every open Casson handle

M

is homeomorphic to

B^{2} \times 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 $S^{3} \times R$ that is however the image of a proper map of degree $\pm 1$ , $S^{3} \times R \to M$ (see [1] and [e34]).

A Casson tower of height $k$ , or more briefly $C_{k}$ , is a pair diffeomorphic to $(H ∖ \overset{˚}{X_{k}}, \partial_{-} H)$ where $X_{1} \supset X_{2} \supset \dots$ is a Russian doll of defects in a handle $(H, \partial_{-} H)$ .

Theorem 2.3: (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}^{'}, \partial_{-} H_{12}^{'})

, such that

$\partial_{-} H_{12}^{'} = \partial_{-} H_{6}$ .
$H_{12}^{'} ∖ \partial_{-} H_{6} \subset Int H_{6}$ .
$H_{12}^{'} ∖ H_{6}^{'}$ is contained in a disjoint union of balls in $Int H_{6}$ , 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).

Remark: Every pair

(k, 2 k)

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:

Change of Notation 2.4: From now on, we write

H_{k}

and

X_{k}

in place of

H_{6 k + 6}

and

X_{6 k + 6}

k = 0, 1, 2, \dots

. (Also the meaning of

E_{k} = H_{k} ∖ H_{k - 1}

i_{k}

, etc. is changed.)

Theorem 2.5: (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}^{'}, \partial_{-} H_{\infty}) \subset (H_{\infty}, \partial_{-} H_{\infty})

satisfying the conditions:

$H_{k - 1}^{'} = H_{k - 1}$ if $k \geq 1$ .
$\overset{―}{H_{\infty}^{'}} ∖ H_{k - 1}^{'} \subset (Int H_{k}) ∖ H_{k - 1}$ .
The closure $\overset{―}{H_{\infty}^{'}}$ of $H_{\infty}^{'}$ in $H_{\infty}$ is the endpoint compactification of $H_{\infty}^{'}$ .

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

The ambitious construction to come applies the mitosis theorem (Theorem 2.5) and elementary geometry, to convert Theorem 2.2, that every open Casson handle is homeomorphic to

B^{2} \times 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 $N ∖ \partial_{+} N$ where $(N, \partial_{-} N)$ is a Casson handle (not open). Let $\hat{N}$ be the endpoint compactification $of N$ . Subtracting from $N$ the (topological) interior of a collar neighbourhood of $\partial_{+} N$ in $N$ , very pinched towards the ends $of N,$ we obtain a Casson handle $(H_{\infty}, \partial_{-} H_{\infty}) \subset (M, \partial M) \subset (N, \partial_{-} N)$ whose closure in $\hat{N}$ is the endpoint compactification ${\hat{H}}_{\infty}$ 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

For each finite sequence

(a_{1}, \dots, a_{k})

{0, 1}

(finite dyadic sequence), we can define a presented Casson handle

(H_{\infty} (a_{1}, \dots, a_{k}), \partial_{-} H_{\infty})

contained in

(H_{\infty}, \partial_{-} H_{\infty})

, whose presentation consists of an embedding

i_{\infty} (a_{1}, \dots, a_{k}) : H_{\infty} (a_{1}, \dots, a_{k}) \to B^{2} \times D^{2},

and a Russian doll of defects

X_{i} (a_{1}, \dots, 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.
$H_{\infty} (a_{1}, \dots, a_{k}, 1) = H_{\infty} (a_{1}, \dots, a_{k})$ .
$H_{k} (a_{1}, \dots, a_{k}, 0) = H_{k} (a_{1}, \dots, a_{k})$ (recall that $H_{k}$ are sets of 6-stages).
The closure ${\overset{―}{H}}_{\infty} (a_{1}, \dots, a_{k}, 0) in {\hat{H}}_{\infty}$ is an endpoint compactification of $H_{\infty} (a_{1}, \dots, a_{k}, 0)$ .
${\overset{―}{H}}_{\infty} (a_{1}, \dots, a_{k}, 0) ∖ H_{k} (a_{1}, \dots, a_{k}) \subset {\overset{˚}{H}}_{k + 1} (a_{1}, \dots, a_{k}) ∖ H_{k} (a_{1}, \dots, a_{k})$ .
$i_{k} (a_{1}, \dots, a_{k}, 0) = i_{k} (a_{1}, \dots, a_{k})$ , so $X_{k} (a_{1}, \dots, a_{k}, 0) = X_{k} (a_{1}, \dots, a_{k})$ .
The intersection of $X_{k + 1} (a_{1}, \dots, a_{k}, 0)$ and $X_{k + 1} (a_{1}, \dots, a_{k})$ is empty, and their union is a multiple defect in $X_{k} (a_{1}, \dots, a_{k})$ .
(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 $X_{k} = ⋃ X_{k} (a_{1}, \dots, a_{k})$ . 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 $(a_{1}, \dots, a_{k}, 1)$ by (2). Next, we define $H_{\infty} (a_{1}, \dots, a_{k}, 0)$ 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 $(H_{\infty} (a_{1}, \dots, a_{n}, 0), \partial_{-} H_{\infty})$ in such a fashion that the two last conditions (6) and (7) are satisfied. To define $i_{\infty} (a_{1}, \dots, a_{k}, 0)$ , it is convenient to graft, onto $i_{k} (a_{1}, \dots, a_{k})$ , a presentation the near part of the Casson handle $(H_{\infty} (a_{1}, \dots, a_{k}, 0), \partial_{-} H_{\infty}),$ to know the Casson multihandle $(H_{\infty} (a_{1}, \dots, a_{k}, 0) ∖ {\overset{˚}{H}}_{k} (a_{1}, \dots, a_{k}, 0), δ H_{k} (a_{1}, \dots, a_{k}, 0)),$ where exceptionally $\overset{˚}{}$ and $δ$ denote the interior and the frontier in $H_{\infty} (a_{1}, \dots, a_{k}, 0)$ rather than in $\hat{N}$ . The grafting is done with the help of Lemma 2.1. The last condition (7) is assured afterwards by an isotopy in ${\overset{˚}{X}}_{k} (a_{1}, \dots, a_{k})$ . Having (1) to (7), the reader will know how to arrange that (8) is also satisfied. ◻

Remark: If

(a_{1}, a_{2}, \dots)

is an infinite sequence in

{0, 1}

, the union

H_{\infty} (a_{1}, a_{2}, \dots) = ⋃_{k} H_{\infty} (a_{1}, a_{2}, \dots, a_{k})

gives a Casson handle with an obvious presentation. Moreover, the closure

{\overset{―}{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 $(H_{\infty} (a_{1}, \dots, a_{k}), \partial_{-} H_{\infty}),$ we especially use their skins $\partial_{+} H_{\infty} (a_{1}, \dots, a_{k})$ . The union $P^{3} = ⋃ \partial_{+} H_{\infty} (a_{1}, \dots, a_{k})$ of the skins is what one calls a branched manifold in $N^{4}$ , since near every point $P^{3} ∖ \partial_{-} H_{\infty}$ , 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 $R^{2}$ with the model of branching $(R^{2}, Y^{1})$ where $Y^{1}$ is the union of two smooth curves (isomorphic to $R^{1}$ ), properly embedded in $R^{2}$ and which have in common exactly one closed half-line. One observes without difficulty that the closure $\overset{―}{P}$ of $P$ in $\hat{N}$ is the endpoint compactification of $P$ .

The branched manifold $P$ splits along the singular points into compact manifolds: $\begin{aligned} P_{k} (a_{1}, \dots, a_{k}) & = \partial_{+} E_{k} (a_{1}, \dots, a_{k}) \\ = E_{k} (a_{1}, \dots, a_{k}) \cap \partial_{+} H_{\infty} (a_{1}, \dots, a_{k}) . \end{aligned}$ Thus, $P_{k} (a_{1}, \dots, a_{k})$ is the skin of the $k$ -th stage of $(H_{\infty} (a_{1}, \dots, a_{k}), \partial_{-} H_{\infty})$ .

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 $I$ of $G^{4}$ into disjoint intervals, satisfying the following.

For every interval $I_{α}$ of $I$ , the intersection $I_{α} \cap \partial_{-} N$ is $I_{α}$ or the empty set. A neighbourhood of $I_{α}$ in $(G^{4}, P^{3}; I)$ is isomorphic to the product of $R^{2}$ with an open 2-dimensional model $(G^{2}, P^{1}; I^{'})$ as in Figure 12.
The closure $\overset{―}{G}$ of $G$ in $\hat{N}$ is its endpoint compactification, and hence coincides with $G \cup \overset{―}{P}$ .

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 $N ∖ {\overset{˚}{H}}_{\infty}$ of $\partial_{+} N$ .

The design $(G^{4}, I)$ decomposed into intervals splits in a canonical fashion (along the 3-manifold formed by the exceptional intervals of $I$ having interior points on $\partial G^{4}$ ) into genuine trivial $I$ -bundles $I (a_{1}, \dots, a_{k}) \times P_{k} (a_{1}, \dots, a_{k}),$ where $I (a_{1}, \dots, a_{k})$ is a 1-simplex and $(its centre) \times P_{k} (a_{1}, \dots, a_{k}) \subset G^{4}$ is nearly the natural inclusion $P_{k} (a_{1}, \dots, a_{k}) \subset G^{4}$ . More precisely, the two embeddings are isotopic in $G^{4}$ by an isotopy which moves only a collar of the boundary of $P_{k} (a_{1}, \dots, a_{k})$ . 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 $I (a_{1}, \dots, a_{k})$ .

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 $I (a_{1}, \dots, a_{k}) \subset (0, 1]$ 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 $I (a_{1}, \dots, a_{k}, 0)$ and $I (a_{1}, \dots, a_{k}, 1)$ respectively on the initial third and the final third of the interval $I (a_{1}, \dots, a_{k}) \subset (0, 1]$ .

The central third of $I (a_{1}, \dots, a_{k})$ is a closed interval that we may call $J (a_{1}, \dots, a_{k})$ . The complement in $I (\emptyset)$ of all the open intervals $\overset{˚}{J} (a_{1}, \dots, a_{k})$ is then a compact Cantor set in $(0, 1]$ .

On the other hand, we claim that the embeddings $i_{k} (a_{1}, \dots, a_{k}) | : \partial_{+} H_{k} (a_{1}, \dots, a_{k}) \to B^{2} \times \partial D^{2}$ define together a smooth map $i : P \to B^{2} \times \partial D^{2}$ . Let $φ : (0, 1] \times B^{2} \times \partial D^{2} \to B^{2} \times D^{2}$ be the embedding $(t, x, y) \mapsto (x, t y)$ . We will have the tendency to identify domain and codomain by $φ$ .

We define $g : G^{4} \to B^{2} \times D^{2}$ on $I (a_{1}, \dots, a_{k}) \times P_{k} (a_{1}, \dots, a_{k})$ by the rule that $(t, x) \mapsto φ (t, i (x))$ . For that definition to make sense, we have to first adjust, by isotopy, the trivialisation given by the $I$ -fibres $I (a_{1}, \dots, a_{k}) \times P_{k} (a_{1}, \dots, a_{k})$ in $(G^{4}, I)$ , a routine task that is left to the reader.

3.4. Construction of $g_{0} : G_{0}^{4} \to B^{2} \times D^{2}$

Let

G_{0}^{4}

be the union of

G^{4}

and a small collar neighbourhood

C^{4}

\partial_{-} N

N

that respects

δ 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} ∖ {\overset{˚}{G}}^{4}

(B^{2} ∖ λ B^{2}) \times μ D^{2},

where

λ \in (0, 1]

is near to 1 and

μ

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

\overset{―}{G_{0}^{4}}

G_{0}^{4}

{\hat{N}}^{4}

3.5. The image $g_{0} (G_{0}^{4}) \subset B^{2} \times D^{2}$

Some notation again (see Figure 13).

$T (a_{1}, \dots, a_{k}) \equiv T_{k} (a_{1}, \dots, a_{k}) = \partial_{+} X (a_{1}, \dots, a_{k})$ , a multisolid torus $\subset B^{2} \times \partial D^{2}$ .
$T_{*} (a_{1}, \dots, a_{k}) = φ (J (a_{1}, \dots, a_{k}) \times T (a_{1}, \dots, a_{k})) \subset B^{2} \times {\overset{˚}{D}}^{2}$ , a radially thickened copy of $T (a_{1}, \dots, a_{k})$ , called a hole.
$B_{*} = λ B^{2} \times μ D^{2}$ (see definition of $g_{0}$ ), called the central hole.
$F_{k} = ⋃ {φ (I (a_{1}, \dots, a_{k - 1}) \times T (a_{1}, \dots, a_{k})) ∣ k fixed}$ ; the frontiers $δ F_{k}$ , $k \geq 2$ , are indicated in dashed lines in the right-hand figure below.
$(B^{2} \times D^{2})_{0} = (B^{2} \times D^{2} ∖ {\overset{˚}{B}}_{*}) ∖ ⋃ {{\overset{˚}{T}}_{*} (a_{1}, \dots, a_{k})}$ , called the holed standard handle.
$W_{0} = ⋂_{k} F_{k}$ , a compactum in $(B^{2} \times D^{2})_{0}$ .

With this notation, we claim that the image $g_{0} (G_{0}^{4})$ is $(B^{2} \times D^{2})_{0} ∖ W_{0}$ .

3.6. The main diagram

The commutative diagram to the right gives an overview of the construction to come. The elements will be constructed in the order

W_{0}

g_{1}

D^{'}

g_{2}

D

g_{3}

D_{+}

f

. The proof that

(B^{2} \times {\overset{˚}{D}}^{2}) / D_{+}

is homeomorphic to

B^{2} \times {\overset{˚}{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 $W_{0}$ and $g_{1}$

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 W_{0}

is a Whitehead compactum in a single level

φ (t \times B^{2} \times \partial D^{2})

. We check naively that the inclusion

(B^{2} \times D^{2})_{0} ∖ W_{0} \to (B^{2} \times D^{2})_{0} / W_{0}

induces a homeomorphism

((B^{2} \times D^{2})_{0} ∖ W_{0})^{\land} \to (B^{2} \times D^{2})_{0} / W_{0} .

We already know that

{\hat{G}}_{0}

is identified with

{\overset{―}{G}}_{0} \subset \hat{N}

. We define the homeomorphism

g_{1}

as a composition of homeomorphisms:

g_{1} : {\overset{―}{G}}_{0} \to {\hat{G}}_{0} \overset{{\hat{g}}_{0}}{\to} ((B^{2} \times D^{2})_{0} ∖ W_{0})^{\land} \to (B^{2} \times D^{2})_{0} / W_{0} .

3.8. Construction of $D^{'}$ and $g_{2}$

Let

D^{'}

be the decomposition of

B^{2} \times D^{2}

given by the

B_{*}

T_{*} (α)

(

α

can be any finite dyadic sequence), and the elements of

W

which are disjoint from them. To define

g_{2} : \hat{N} \to (B^{2} \times D^{2}) / D^{'},

we must extend

q_{1} g_{1} : {\overset{―}{G}}_{0} \to (B^{2} \times D^{2}) / D^{'}

to each connected component

Y

\hat{N} ∖ {\overset{―}{G}}_{0}

. Its frontier

δ Y

is identified by

g_{1}

to the quotient in

(B^{2} \times D^{2})_{0} / W_{0},

either of

\partial B_{*}

, or of a boundary of a connected component of a hole

T_{*} (a_{1}, \dots, a_{k})

. By definition,

g_{2} (Y)

is the image in

(B^{2} \times D^{2}) / D

of this boundary. It is easy to check the continuity of

g_{2}

Next, $g_{3}$ and $D$ in the main diagram are defined by restriction. The design $G^{4}$ has led us inexorably to define $g_{3} : M^{4} \to B^{2} \times {\overset{˚}{D}}^{2} / D,$ which compares the open Casson handle $M^{4}$ with a very explicit quotient of the open handle $B^{2} \times {\overset{˚}{D}}^{2}$ .

The decomposition $D$ which specifies this quotient has noncellular elements, that is, the holes $T_{*} (a_{1}, \dots, a_{k})$ , each of which has the homotopy type of a circle. Therefore the quotient map $B^{2} \times {\overset{˚}{D}}^{2} \to B^{2} \times {\overset{˚}{D}}^{2} / D$ is certainly not approximable by homeomorphisms. One can also check that the Čech cohomology ${\overset{ˇ}{H}}^{2}$ of the quotient is of infinite type.

The construction of $D_{+}$ below repairs this terrible defect; it will be constructed by hand; $D_{+}$ will be less fine than $D$ , which will enable us to define $f = q_{3} \circ g_{3}$ without effort.

3.9. Construction of $D_{+}$

We set

W = W_{0} \cap (B^{2} \times {\overset{˚}{D}}^{2}) = W_{0} ∖ (B^{2} \times \partial D^{2}) .

Its connected components define a decomposition

W

B^{2} \times {\overset{˚}{D}}^{2}

. We have known since the 1950s how to show that

B^{2} \times {\overset{˚}{D}}^{2} / W

is homeomorphic to

B^{2} \times {\overset{˚}{D}}^{2}

, see Section 4.

For the requirements of the next paragraph, the quotient $(B^{2} \times {\overset{˚}{D}}^{2}) / D_{+}$ must be a quotient of $B^{2} \times {\overset{˚}{D}}^{2} / W$ by a decomposition whose elements are the connected components of $⋃ {q (T_{*} (α)) \cup E (α) ∣ α a finite dyadic sequence} .$ Here ${E (α)}$ is a collection of disjoint, topologically flat multi-2-discs such that for each finite dyadic sequence $α$ , the intersection $E (α) \cap (⋃_{α^{'}} q (T_{*} (α^{'})))$ is

the boundary $\partial E (α)$ ; and
a multilongitude of $\partial T_{*} (α)$ far from $W$ (each connected component of $q (T_{*} (α)) \cup E (α)$ is then contractible).

Moreover, we want that the diameter of the connected components of $E (a_{1}, \dots, a_{k})$ tends towards 0 (on each compact set) as $k \to \infty$ . Section 4 does not demand any more than this and visibly, ${E (α)}$ specifies $D_{+}$ .

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

We have $T_{k} = ⋃_{α} T_{k} (α)$ ; conditions (6) and (7) of Construction 3.1 assure that $T_{k}$ is a multisolid torus of which certain connected components constitute $T_{k} (α)$ . We have $⋂_{k} T_{k} = p (W),$ 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 $D^{'} (α)$ which will be the projection $p (D (α)) = D^{'} (α)$ . We assume easily the two properties (a) and (b), where (b) uses (8) of Construction 3.1.

(a) $D^{'} (a_{1}, \dots, a_{k})$ is a disjoint union of immersed discs in $T_{k - 1}$ , with as their only singularities, an arc of double points for each, above $T_{k} (a_{1}, \dots, a_{k}) .$ The boundary $\partial D^{'} (a_{1}, \dots, a_{k})$ is formed from one longitude of each connected component of $\partial T_{k} (a_{1}, \dots, a_{k})$ . The double points of $D^{'} (a_{1}, \dots, a_{k})$ are outside ${\overset{˚}{T}}_{k} (a_{1}, \dots, a_{k})$ .

(b) For each $l \geq k$ , the intersection ${\overset{˚}{D}}^{'} (a_{1}, \dots, a_{k}) \cap T_{l}$ is a multidisc (embedded in $T_{k} (a_{1}, \dots, a_{k})$ ) 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 $D^{'} (α)$ , 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 $ρ (α) : D (α) \to (0, 1)$ .

We will embed a single $D (α)$ at a time (following some chosen order). We embed first $D (a_{1}, \dots, a_{k})$ closer and closer (by a secondary induction). Some notation: $\begin{aligned} T_{*}^{+} (a_{1}^{'}, \dots, a_{l}^{'}) & = J (a_{1}^{'}, \dots, a_{l}^{'}) \times T (a_{1}^{'}, \dots, a_{l - 1}^{'}), \\ F_{l}^{*} = p^{- 1} (p (F_{l})) & = (0, 1) \times T_{l}, \\ W^{+} = p^{- 1} (p (W)) & = (0, 1) \times (⋂_{k} T_{k}) . \end{aligned}$

One can easily check that, for $D (a_{1}, \dots, a_{k}),$ the properties (c) and (d) for $l > k$ , of which (d) for $l$ is only provisional.

(c) $D (a_{1}, \dots, a_{k})$ is embedded, is contained in $I (a_{1}, \dots, a_{k - 1}) \times T (a_{1}, \dots, a_{k - 1}),$ and is disjoint from $B_{*}$ and from $⋃ {T_{*}^{+} (α^{'}) ∣ α^{'} \neq (a_{1}, \dots, a_{k})} .$ The boundary $\partial D (a_{1}, \dots, a_{k})$ is in a single level $t \times B^{2} \times \partial D^{2}$ , where $t \in \overset{˚}{J} (a_{1}, \dots, a_{k})$ .

(d) Each connected component of the multidisc $F_{l}^{+} \cap D (a_{1}, \dots, a_{k})$ is in a single level $t \times B^{2} \times \partial D^{2}$ ; this level is disjoint from each box $T_{*} (α^{'})$ , and does not contain any other connected component of $F_{l}^{+} \cap D (a_{1}, \dots, a_{k})$ .

For $l = k$ and $k + 1$ , here are the illustrations of the graph of $ρ$ in a simple case.

We observed that in pushing $D (a_{1}, \dots, a_{k})$ vertically, as small as we want, and only on ${\overset{˚}{F}}_{l}^{+} \cap D (a_{1}, \dots, a_{k}),$ 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 $F_{l}^{+} \cap D (a_{1}, \dots, a_{k})$ project onto as many disjoint intervals of radius in $(0, 1)$ .

This condition assures that, for all $W \in W$ , the intersection $W \cap D (a_{1}, \dots, a_{k})$ is an intersection of discs (and so cellular). Therefore $q (D (a_{1}, \dots, a_{k}))$ is certainly a disc (compare Theorem 4.4). In Section 4, we will prove by hand that it is a flat disc. If, before $D (a_{1}, \dots, a_{k})$ , we have already defined (for the main induction) a finite collection of discs $D (α_{1}), \dots, D (α_{n}),$ we follow the same construction as above, always staying in a neighbourhood of $T_{*}^{+} (a_{1}, \dots, a_{k})$ (guaranteed by (c)), disjoint from $D (α_{1}) \cup \dots \cup D (α_{n})$ and for all elements of $W$ that touch $D (α_{1}) \cup \dots \cup D (α_{n})$ .

Thus the family ${D (α)}$ of disjoint 2-discs is defined by a double induction and satisfies the properties (a), (b), (c) and (e) with $p (D (α)) = D^{'} (α) .$ Next ${D (α)}$ defines $D_{+}$ as already indicated. One easily checks all the properties wanted for $q (D (α)) = E (α) in (B^{2} \times {\overset{˚}{D}}^{2}) / W,$ except local flatness of $E (α)$ which is postponed to Section 4.

3.10. End of the proof that $M$ is homeomorphic to $B^{2} \times {\overset{˚}{D}}^{2}$ (modulo Sections 4 and 5)

Accepting from Section 4 that

(B^{2} \times {\overset{˚}{D}}^{2}) / D_{+}

is homeomorphic to

B^{2} \times {\overset{˚}{D}}^{2}

, we show modulo Section 5 the approximability by homeomorphisms of

f : M^{4} \to (B^{2} \times {\overset{˚}{D}}^{2}) / D_{+}

in the following fashion. We form the commutative diagram

where the inclusion

Int M \subset S^{4}

exists since

M

embeds in

B^{2} \times D^{2}

(the experts also know that

Int M

is diffeomorphic to

R^{4}

[e38]), and where

f_{*} (S^{4} ∖ Int M^{4}) = \infty .

Therefore,

S (f_{*}) = {y \in S^{4} ∣ f_{*}^{- 1} (y) \neq a point}

is visibly a contractible set.

Also $S (f_{*})$ is nowhere dense. [Here is a proof. The restriction $f_{*} |$ is the same as $q_{3} \circ q_{1} \circ g_{1} | : M \cap {\overset{―}{G}}_{0}^{4} \to (B^{2} \times {\overset{˚}{D}}^{2}) / D_{+},$ which is already surjective and $f_{*}^{- 1} (S (f_{*}))$ is contained in the nowhere dense set of $M \cap {\overset{―}{G}}_{0}^{4}$ given by $(\partial G_{0}) \cup ($ ends of $G_{0}^{4}) \cup g_{1}^{- 1} (⋃_{α} E (α))$ .]

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

Remark:

\overset{―}{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

M^{4} ∖ G_{0}^{4}

. For reasons of cohomology,

\dim \overset{―}{S (f_{*})} \geq 1

. Therefore it is a compactum of dimension exactly 1.

4. Bing shrinking

We need to show that the space

B^{2} \times {\overset{˚}{D}}^{2} / D_{+}

defined in Section 3 is homeomorphic to

B^{2} \times {\overset{˚}{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 $D = {f^{- 1} (y) ∣ y \in Y}$ be the decomposition associated with $f$ . When is $f$ (strongly) approximable by homeomorphisms, in the sense that for all open coverings $V$ of $Y$ , the $V$ -neighbourhood $N (f, V) = {g : X \to Y ∣ for all x \in X, there exists V \in V such that f (x), g (x) \in V}$ contains a homeomorphism?

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

Theorem 4.1: (Bing shrinking criterion.)

f

is approximable by homeomorphism if and only if, for every covering

U

X

and

V

Y

, there exists a homeomorphism

h : X \to X

such that

h (D) < U

, and for all compact

D \in D

D

and

h (D)

are

f^{- 1} (V)

-near in the sense that there exists an

f^{- 1} (V) \in f^{- 1} (V)

that contains

D \cup h (D)

We then say that $D$ 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 $D$ ). The proof also gives:

Remark: In Theorem 4.1, if

h

respects (or fixes) a closed set

A \subset X

, then

f

is approximable by homeomorphisms that send

A

f (A)

(or which coincide on

A

with

f

), and reciprocally.

Proposition 4.2: (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],

D_{1}

of [e23]), then, for each open set

V

Y

, the restriction

f_{V} : f^{- 1} (V) \to V

f

is approximable by homeomorphisms.

Proof (indication). To approximate $f_{V}$ , we combine (by the principle $D_{1}$ ) 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. ◻

Counterexample: This principle is false if

X

and

Y

are

Cantor \times [0, 1] = 2^{N} \times [0, 1],

and

f = g \times {Id}_{[0, 1]}

, where

g (1, a_{2}, a_{3}, \dots) = (a_{2}, a_{3}, \dots), g (0, a_{2}, a_{3}, \dots) = (0, 0, 0, \dots) .

Proposition 4.3: (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 $X ∖ f^{- 1} (V)$ .

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

Theorem 4.4: (Criterion.) Under these conditions,

D

is shrinkable if for all

ε > 0

and for all open

D

-saturated

U

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

< ε

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

For all $ε > 0$ , one can consider $D_{ε} = {D \in D ∣ diam D \geq ε} .$ We say that $⋃_{D \in D_{ε}} D$ is a closed subset of $X$ . Here is a remarkable but disturbing example where $D$ is null, $D_{ε}$ is shrinkable for any $ε > 0$ , but $D$ is not shrinkable. The elements of $D$ are the connected components of a compact set $X = ⋂_{n} F_{n}$ 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 $D \in D$ is clearly cellular, hence $D_{ε}$ is shrinkable by Lemma 5.2. But, with the help of cyclic covers, one can check that $D$ 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 $R (X, A)$ : for each $ε > 0$ , for every null decomposition $D$ 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 $D \in D$ , $diam f (D) \leq max (diam D, ε) .$ If $D$ is fixed in advance, we call the (weaker) property $R (X, A; D)$ .

Observation: For every neighbourhood

U

A

, we have

R (X, A)

is equivalent to

R (U, A)

. Moreover,

R (X, A)

is independent of the metric.

Proposition 4.5: If

D

is null, and

R (X, D; D)

is satisfied for all

D \in D

, then

D

is shrinkable.

Proof. The proof is an edifying exercise. ◻

Proposition 4.6:

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 $R (R^{n}, B^{k})$ for $k \leq n$ . The proof of $R (R^{2}, B^{1})$ which is indicated by Figure 16.

In (a), every element of $D$ that meets the big rectangle has already diameter $< ε / 4$ ; if $D \in D$ meets a gap between successive rectangles, it is disjoint from the rectangle after. We set $f (B^{1}) = 0$ , and $f = Id$ 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 $∖ B^{1}$ ). Moreover, $p \circ f = p$ where $p$ is the projection to the $y$ -axis (the $R^{n - k}$ normal to $B^{k}$ ). Finally, the size of the image of each of the vertical rectangle is $< ε / 4$ . ◻

We consider the Whitehead pair $(B^{2} \times S^{1}, j (B^{2} \times S^{1})) = (T, T^{'}),$ and the thickened pair $(R \times T, [0, 1] \times T^{'}) .$

Lemma 4.7: For

ε > 0

, there exists an isotopy

h_{t}

(t \in [0, 1])

Id |_{R \times T}

with compact support in

(- ε, 1 + ε) \times Int T

such that we have

diam (h_{1} (t \times T^{'})) < ε and h_{1} (t \times T^{'}) \subset [t - ε, t + ε] \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 $W \subset R^{3}$ be a Whitehead compactum and let $D = {t \in W ∣ t \in [0, 1], W \in W}$ be the decomposition $I \times W$ of $R \times R^{3} = R^{4}$ . Then $D$ is shrinkable by Lemma 4.7 applied to the solid tori $T$ , $T^{'}$ , $T^{''}$ , …whose intersection is $W$ . Therefore $R^{4} / D$ is homeomorphic to $R^{4}$ . Moreover, by Proposition 4.2 (localisation principle), we have that $(0, 1) \times R^{3} / W$ is homeomorphic to $(0, 1) \times R^{3}$ . Hence we have the following celebrated fact.

Theorem 4.8: (Celebrated fact [e16].)

R \times (R^{3} / W) = 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 $R \times (S^{3} ∖ W)$ is homeomorphic to $R^{4}$ , compare [e6]. Did the proof of Shapiro from the 50s disappear without a trace?

To establish the flatness of the discs ${E (α)}$ constructed in Section 3.9, we will also need a lemma that is easier than Lemma 4.7, treating again the Whitehead pair $(T, T^{'})$ . Let $D$ be a meridional disc of $T$ that cuts $T^{'}$ transversally in two discs.

Lemma 4.9: With this data, we can find in

R \times T

a topological 4-ball

B

, such that

Int B \supset [0, 1] \times T^{'}

and

B \cap (R \times D)

is an equatorial 3-ball of the form

(interval) \times D_{0} \subset 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 $(B^{2} \times {\overset{˚}{D}}^{2}) / D_{+}$ is homeomorphic to $B^{2} \times {\overset{˚}{D}}^{2}$ , we will now use the construction of Section 3.

Proposition 4.10: The decomposition

W

B^{2} \times {\overset{˚}{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 $W$ -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 ${\overset{˚}{I}}^{'} \times \overset{˚}{T} (a_{1}, \dots, a_{k}),$ where $I^{'}$ is an interval. ◻

Our next goal is the flatness of the discs $E (α) = q (D (α)) \subset (B^{2} \times {\overset{˚}{D}}^{2}) / W .$ Let $W (α) = {w \in W ∣ w \cap D (α) \neq \emptyset},$ and let $W (α) = ⋃ W (α)$ .

Proposition 4.11:

W (α)

is shrinkable respecting

D (α)

. Therefore, the quotient

q_{α} (D (α))

D (α)

is flat in

(B^{2} \times {\overset{˚}{D}}^{2}) / W (α) .

Proof of Proposition 4.11. We apply Lemma 4.9 and the relative criteria (Theorem 4.4). For every open $W_{α}$ -saturated $U$ of $B^{2} \times {\overset{˚}{D}}^{2}$ , the intersection $W_{α} \cap U$ is trivially contained in an open set which, for some integer $l$ , is a disjoint union of open sets of the form ${\overset{˚}{I}}^{'} \times {\overset{˚}{T}}^{'} \subset U,$ where $T^{'}$ is a connected component of multiple solid tori $T_{l} (b_{1}, \dots, b_{l})$ and $i^{'}$ is an interval.

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

$D (α) \cap (I^{'} \times T^{'})$ is a single 2-disc, which is projected onto a meridional disc $D$ of $T^{'}$ which is also a connected component of $D^{'} (α) \cap T^{'}$ ; see Section 3.9.

By condition (b) of Section 3.9 the meridional disc $D$ ideally chopped off $T_{l + 1 / 6} \cap T^{'}$ , so Lemma 4.9 gives us disjoint 4-balls $B_{1}, \dots, B_{s}$ in ${\overset{˚}{I}}^{'} \times {\overset{˚}{T}}^{'}$ , such that

each intersection $B_{i} \cap D (α)$ is a diametral 2-disc and not knotted in $B_{i}$ , and
${\overset{˚}{B}}_{1} \cup \dots \cup {\overset{˚}{B}}_{s}$ contains the compact set $W^{+} \cap ({\overset{˚}{I}}^{'} \times {\overset{˚}{T}}^{'}) \supset W_{α} \cap ({\overset{˚}{I}}^{'} \times \overset{˚}{T^{'}}) .$

For all compact $K$ in ${\overset{˚}{B}}_{i}$ and all $ε > 0$ , we can easily find a homeomorphism $h : B_{i} \to B_{i}$ with compact support which respects ${\overset{˚}{B}}_{i} \cap D (α)$ and such that $diam h (K) < ε$ . The criteria of Theorem 4.4 (respecting $D (α)$ ) is therefore satisfied. ◻

Proposition 4.12:

q (D (α)) = E (α)

is flat in

(B^{2} \times {\overset{˚}{D}}^{2}) / W

Proof of Proposition 4.12. The open set $U_{α} = (B^{2} \times {\overset{˚}{D}}^{2}) ∖ (W_{α} \cup D (α))$ is clearly homeomorphic to $(B^{2} \times {\overset{˚}{D}}^{2}) / W_{α} - q_{α} (D (α))$ by $q_{α}$ . Therefore, by Propositions 4.2 and 4.3, the quotient map $q_{α}^{'} : (B^{2} \times {\overset{˚}{D}}^{2}) / W_{α} \to (B^{2} \times {\overset{˚}{D}}^{2}) / W$ is approximable by homeomorphisms fixing $q_{α}^{'}$ on the flat disc $q_{α} (D (α))$ . Therefore, $q (D (α)) = q^{'} (α) \circ q (α) (D (α))$ is flat. ◻

We now propose to finish by showing that the quotient maps $\begin{aligned} B^{2} \times {\overset{˚}{D}}^{2} & \overset{\approx}{\to} (B^{2} \times {\overset{˚}{D}}^{2}) / W \\ \overset{p_{1}}{\to} ((B^{2} \times {\overset{˚}{D}}^{2}) / W) / {E (α)} \\ \overset{p_{2}}{\to} (B^{2} \times {\overset{˚}{D}}^{2}) / D_{+} \end{aligned}$ are approximable by homeomorphisms.

Proposition 4.13:

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 $r : B^{2} \times {\overset{˚}{D}}^{2} \to B^{2} \times {\overset{˚}{D}}^{2}$ inducing the same decomposition as the quotient map $((B^{2} \times {\overset{˚}{D}}^{2}) / W) / {E (α)};$ we can identify the domain of $p_{2}$ with $B^{2} \times {\overset{˚}{D}}^{2}$ by $r$ .

The decomposition $P$ constituted of the preimages $p_{2}^{- 1} (y) = {a point}$ is the countable collection of natural quotients of connected components of holes $T_{*} (α)$ and $B_{*}$ , which now identify $r (T_{*} (α))$ and $r (B_{*}) \subset B^{2} \times {\overset{˚}{D}}^{2} .$ We observe that $P$ is null. The quotient map $λ B^{2} \times μ D^{2} = B_{*} \to r B_{*}$ shrinks the Whitehead compactum $W (\partial B_{*}) = {w \in W ∣ w \subset B_{*}},$ and these compact sets lie in $λ B^{2} \times μ \partial D^{2} \subset \partial B_{*} .$

Proposition 4.14:

r (\partial B_{*})

has a bicollar neighbourhood

V

B^{2} \times {\overset{˚}{D}}^{2}

, that is,

(V, r (\partial B_{*}))

is homeomorphic to

(R \times r (\partial B_{*}), 0 \times r (\partial B_{*})) .

This will result in the following proposition.

Proposition 4.15: The quotient of

\partial B_{*}

(B^{2} \times {\overset{˚}{D}}^{2}) / W (\partial B_{*})

admits a bicollar neighbourhood.

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

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

Proposition 4.16:

R (B^{2} \times {\overset{˚}{D}}^{2}, r (B_{*}); P)

is satisfied.

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

Lemma 4.17: (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

ε > 0

, there exists a homeomorphism with compact support

φ : [0, \infty) \to [0, \infty)

such that

Φ = φ \times {Id}_{X}

satisfies that, for all

E \in l

such that

Φ (E) \cap (X \times [0, 1]) \neq \emptyset,

we have

diam (Φ (E)) < ε

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

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

Proposition 4.18:

R (B^{2} \times {\overset{˚}{D}}^{2}, r (T_{*}^{'} (α)); P)

is satisfied.

Proof of Proposition 4.18 (indications). The quotient of $T_{*}^{'} (α) = J (α) \times T^{'} (α),$ by the longitude $l (α)$ that is in $D (α)$ , is a cone whose centre is the quotient of $l (α)$ , and the base is a solid torus. $δ r (T_{*}^{'} (α)) - r (l (α))$ has a bicollar neighbourhood in $B^{2} \times {\overset{˚}{D}}^{2}$ , compare Proposition 4.13. The accumulation points of elements $P \neq a point$ of $P$ are the centre $r (l (α))$ and a compact set $r (W \cap \partial T_{*}^{'} (α))$ far from $r (l (α))$ . ◻

Proposition 4.19:

p_{2}

is approximable by homeomorphisms and hence

B^{2} \times {\overset{˚}{D}}^{2} / D_{+} \approx B^{2} \times {\overset{˚}{D}}^{2} .

Proof. Apply Propositions 4.18, 4.16 and 4.5. ◻

5. Freedman’s approximation theorem

Theorem 5.1: (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 ∣ f^{- 1} (y) \neq a point}

is nowhere dense and at most countable. Then,

f

can be approximated by homeomorphisms.

Remark: For all dimensions

\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 $B^{n}$ .

Lemma 5.2: Let

A

be a compact, cellular set in the interior

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 $X ∖ A$ is homeomorphic to $R^{n}$ . 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.

Theorem 5.3: (Schönflies theorem.) Let

Σ^{n - 1}

be a topologically embedded

(n - 1)

-sphere in

S^{n}

such that there is a bicollar neighbourhood

N

Σ

S^{n}

, that is,

(N, Σ)

is homeomorphic to

(Σ \times [- 1, 1], Σ \times 0) .

Then the closure of each of the two components of

S^{n} - Σ

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 $S^{n} ∖ \overset{˚}{N}$ and $W_{1}$ and $W_{2}$ be the closures of connected components of $S^{n} ∖ Σ^{n - 1}$ 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 $f : S^{n} \to S^{n} / {X_{1}, X_{2}} \approx (Σ \times [- 1, 1]) / {Σ \times 0, Σ \times 1} \approx S^{n}$ 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 $X_{i} \subset {\overset{˚}{W}}_{i}$ , we deduce that $W_{i} \to W_{i} / X_{i} \approx Σ \times [0, 1] / {Σ \times 1} \approx B^{n}$ is approximable by homeomorphisms. ◻

Observation: The case of Theorem 5.3, where we know in advance that

Σ

bounds an

n

-ball in

S^{n}

, already arises from the case of Theorem 5.1 where

S (f) = {1 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.

Definition: A closed relation

R : X \to Y

between metrisable spaces

X

and

Y

is a closed subset

R

X \times Y

. If

S : Y \to Z

is a closed relation, the composition

S \circ R : X \to Z

S \circ R = {(x, z) \in X \times Z ∣ there is y \in Y such that (x, y) \in R 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 ${(x, f (x)) ∣ x \in X}$ (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 $x \in X$ .

Remark: The natural function

[0, 1) \to R / 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 $R (A) = {y \in Y ∣ there exists x \in A such that (x, y) \in R}$ ,
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 ${(y, x) \in Y \times X ∣ (x, y) \in R}$ .

Remark:

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 $D \subset Y ∖ \overset{―}{S (f)}$ from $Y$ and removing its preimage $f^{- 1} (D)$ from $X$ , we see that it is permissible to adopt the following.

Theorem 5.4: (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 ∣ f^{- 1} (y) \neq a point}

is nowhere dense and at most countable and

S (f) \subset 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 $Σ^{n - 1}$ 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 $Int X$ . The same holds for $R^{- 1}$ .

It is said that a good relation $R^{'} : X \to Y$ is finer than $R$ if $R^{'} \subset R \subset X \times Y .$

Proposition 5.5: Given the triangle of good relations (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

R

X \times Y

; and

L \subset Z

an open subset (called the gap). We impose the following conditions:

$R \subset (f^{- 1} (\overset{―}{L}) \times g^{- 1} (\overset{―}{L})) \cup (f^{- 1} (Z ∖ L) \times g^{- 1} (Z ∖ L))$ ; it is inevitable if the triangle switches.
$R = g^{- 1} \circ f$ on $f^{- 1} (\overset{―}{L})$ .
$R$ is given by the intersection graph of a homeomorphism $f^{- 1} (Z ∖ L) \to g^{- 1} (Z ∖ L) .$
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 $ε > 0$ , 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 $f^{- 1} (Z ∖ L)$ , $L_{*} \subset L$ , and for all $y \in Y$ , $diam R_{*}^{- 1} (y) < ε$ .

Addendum: There exists a neighbourhood

N_{*} \subset N

R_{*}

X \times Y

such that

diam (N_{*}^{- 1} (y)) < ε

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 $(x_{k}, y_{k})$ , $(x_{k}, y_{k}^{'})$ , $k = 1, 2, 3, \dots$ , which converge in compact $R_{*}$ and such that $d (y_{k}, y_{k}^{'}) \geq ε$ . By compactness of $X \times Y$ , we can arrange that the sequences $x_{k}$ , $y_{k}$ and $y_{k}^{'}$ converge to $x$ , $y$ and $y^{'}$ , respectively. Then, $(x, y)$ and $(x, y^{'})$ belong to compact $R_{*}$ , but $d (y, y^{'}) \geq ε$ , which is a contradiction. ◻

Proposition 5.5 (with Addendum) will be used as a machine that swallows the data $f$ , $g$ , $R$ , $L$ , $N$ , $ε$ 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 $f$ , $Id$ , $f$ , $Y$ , $N$ , 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 $N \supset N_{1} \supset N_{2} \supset \dots$ . We define $H = ⋂_{k} N_{k}$ . Then, $H$ is a homeomorphism since, for all $x$ , we have $diam H (x) \leq diam N_{k} (x) \leq 1 / k,$ for all even $k$ , and $diam H^{- 1} (x) \leq diam N_{k}^{- 1} (x) \leq 1 / k,$ 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 $S (f) = {a point} \subset Int B^{n}$ is approximable by homeomorphisms (for this, we set $f = R$ and $g = Id$ ). Then, it should be noted that as soon as $S (f) = {k points} \subset Int B^{n},$ 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 $\geq ε$ , that we want to eliminate. According to (a), (b) and (c), these sets constitute the preimage by $f$ of the set $(S_{ε} (f) \cap L) \subset Z$ , where $S_{ε} (f) = {z \in Z ∣ diam f^{- 1} (z) \geq ε},$ which will allow us to follow the case in $Z$ . Note that $S_{ε} (f)$ is compact although, typically, $S (f)$ is not. For example, $S_{ε} (f)$ is finite in the case of interest to Freedman (see Section 4).

Lemma 5.6: (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

θ

M

fixing all points of

\partial M

, such that

θ (A)

and

B

are separated, that is, contained in disjoint open sets.

Proof of Lemma 5.6 Consider the space $Aut (M, \partial M)$ of automorphisms of $M$ fixing $\partial M$ , provided with the complete metric $\sup (d (f, g), d (f^{- 1}, g^{- 1}))$ where $d$ is the uniform convergence metric. In $Aut (M, \partial M)$ , the set of automorphisms $θ$ , such that the first $k$ points $A_{k}$ of $A$ and $B_{k}$ of $B$ satisfying $θ (A_{k}) \cap \overset{―}{B} = \emptyset = θ (\bar{A}) \cap B_{k},$ constitute an open subset $U_{k} \subset Aut (M, \partial M)$ everywhere dense in $Aut (M, \partial M)$ , because $\bar{A}$ and $\overset{―}{B}$ are closed, nowhere dense in $M$ .

Then, the famous Baire category theorem asserts that the countable intersection $⋂_{k} U_{k}$ is everywhere dense in $Aut (M, \partial M)$ . Note that $⋂_{k} U_{k}$ is the set of $θ$ in $Aut (M, \partial M)$ such that $θ (A) \cap \overset{―}{B} = \emptyset = θ (\bar{A}) \cap B .$ But, for $X_{1}$ , $X_{2}$ in a metrisable $M$ , the condition that $X_{1} \cap {\overset{―}{X}}_{2} = \emptyset = {\overset{―}{X}}_{1} \cap X_{2}$ leads to the separation of $X_{1}$ and $X_{2}$ in $M$ . In effect, seen in the open subset $M ∖ ({\overset{―}{X}}_{1} \cap {\overset{―}{X}}_{2})$ of $M$ , the sets ${\overset{―}{X}}_{1} ∖ ({\overset{―}{X}}_{1} \cap {\overset{―}{X}}_{2}) and ({\overset{―}{X}}_{1} \cap {\overset{―}{X}}_{2})$ are always disjoint, closed and hence separated. The mentioned condition ensures that they contain respectively $X_{1}$ and $X_{2}$ . ◻

Claim 5.7: (Trivial if

S_{ε} (f)

is finite.) There exists a finite union

B_{+}

of disjoint

n

-balls in

L

satisfying the following conditions:

$S_{ε} (f) \cap L \subset {\overset{˚}{B}}_{+}$ .
$S (g) \cap B_{+} = \emptyset$ .
Each connected component $B_{+}^{'}$ of $B_{+}$ is small in the sense that $(f^{- 1} (B_{+}^{'})) \times (g^{- 1} (B_{+}^{'})) \subset N,$ and standard in the sense that $Z ∖ Int B_{+}^{'}$ is homeomorphic to $S^{n - 1} \times [0, 1]$ .

Proof of Claim 5.7. Identify $Z$ with $B^{n} \subset R^{n}$ to give $L$ an affine linear structure. Let $K$ be a compact neighbourhood of the compact set $S_{ε} (f) \cap L$ 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 $L$ is linear in $L$ and so small such that $f^{- 1} (Δ) \cap g^{- 1} (Δ) \subset N .$ 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 $S_{ε} (f)$ , without harming the properties of $K$ already established. Finally, $B_{+}$ is defined as $K$ minus a small $δ$ open neighbourhood of $K^{(n - 1)}$ in $R^{n}$ . Each component $B_{+}^{'}$ of $B_{+}$ is convex and in $Int Z = {\overset{˚}{B}}^{n}$ ; therefore $Z ∖ {\overset{˚}{B}}_{+}^{'}$ is homeomorphic to $S^{n - 1} \times [0, 1]$ , by an elementary argument. ◻

In $Int B_{+}$ , 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 $L_{*} = L ∖ B$ . For each connected component $B_{+}^{'}$ of $B_{+}$ , we are now modifying $g$ and $R$ above $B_{+}^{'}$ to define $g_{*}$ and $R_{*}$ . These changes for the various connected components $B_{+}^{'}$ 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 $Z ∖ \overset{˚}{B}$ is homeomorphic to $S^{n - 1} \times [0, 1]$ and $B_{+} ∖ \overset{˚}{B}$ .) We should modify $c$ by composing with a homeomorphism $θ$ of $B_{+} ∖ \overset{˚}{B}$ 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 ${\overset{˚}{B}}_{+} ∖ B$ .

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 $g_{*} = {\begin{cases} g & on g^{- 1} (Z ∖ {\overset{˚}{B}}_{+}), \\ c \circ f \circ i^{- 1} & on g^{- 1} (B_{+}) . \end{cases}$ 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 $R_{*} = {\begin{cases} R & on f^{- 1} (Z ∖ {\overset{˚}{B}}_{+}), \\ g^{- 1} \circ f = (i \circ f^{- 1} \circ c^{- 1}) \circ f & on f^{- 1} (B_{+}) . \end{cases}$ More precisely, on $f^{- 1} (B_{+})$ , we specify $R_{*} = {\begin{cases} (i \circ f^{- 1} \circ c^{- 1}) \circ f & on f^{- 1} (B_{+} - \overset{˚}{B}), \\ i & on f^{- 1} (B) . \end{cases}$ 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.) ◻

Remark 1: The system of the above formula, specifying

g_{*}

and

R_{*}

, hides the geometry. We now try to reveal it by looking

f

and

g

respectively as fibrations

φ

and

γ

, of base

Z

, and variable fibre, which allows us to use the notion of fibre restriction. Let

γ_{0} = γ - (γ |_{{\overset{˚}{B}}_{+}}) .

We form

γ_{*}

φ ⊔ γ_{0}

by identifying, via

c |_{\partial Z}

, the subfibres (whose fibres are points)

φ |_{\partial Z}

and

γ_{0} |_{\partial B_{+}}

. Then, we can identify the total space and the base of

γ_{*} = φ \cup γ_{0}

to those of

γ

by an extension of

γ_{0} \to γ

. More precisely, we use

(Id |_{Z ∖ {\overset{˚}{B}}_{+}}) \cup c

between bases, which is the identity on

B \subset B_{+}

. Then,

φ and γ_{*} : Y \overset{g_{*}}{\to} 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

Remark 2: In the proof of Theorem 5.1, we can easily ensure that

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}

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].

Remark 3: (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^{'} : X \to Y

such that

f = f^{'}

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^{'}

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 points}

is close to the argument of [e7]. In particular,

\overset{―}{S (f_{\infty})}

and

\overset{―}{S (g_{\infty})}

can be homeomorphic to

Z \cup {\infty, - \infty}

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This paper describes a construction which can be exploited to recover the results of higher dimensional surgery ($ \dim \geq 5 $) for a wide variety of noncompact four dimensional surgery problems. These include any countable union over boundary components of compact problems with simply connected targets. Here we will concentrate on a special case: the construction of a 4-manifold $ W $ proper-homotopy-equivalent to $ \mathbb{S}^3\times\mathbb{R} $ which has a cross section diffeomorphic to the Poincaré sphere $ \Sigma^3 $ and thus cannot be diffeomorphic to $ \mathbb{S}^3\times\mathbb{R} $. Although the ideas of surgery are in the background, our arguments do not rely heavily on them. A sequel will appear to develop a version of the surgery exact sequence for noncompact 4-manifolds.

Michael H. Freedman

The topological Poincaré conjecture in dimension 4 (the work of M. H. Freedman)

by Laurent Siebenmann

French to English translation by Min Hoon Kim and Mark Powell

Introduction

Report

1. Terminology

2. Casson tower and Freedman’s mitosis

3. Architecture of topological coordinates

3.1. Construction

3.2. Construction of the design $G^{4}$ (see Figure 11)

3.3. Construction of $g : G^{4} \to B^{2} \times D^{2}$

3.4. Construction of $g_{0} : G_{0}^{4} \to B^{2} \times D^{2}$

3.5. The image $g_{0} (G_{0}^{4}) \subset B^{2} \times D^{2}$

3.6. The main diagram

3.7. Construction of $W_{0}$ and $g_{1}$

3.8. Construction of $D^{'}$ and $g_{2}$

3.9. Construction of $D_{+}$

3.10. End of the proof that $M$ is homeomorphic to $B^{2} \times {\overset{˚}{D}}^{2}$ (modulo Sections 4 and 5)

4. Bing shrinking

5. Freedman’s approximation theorem

Works

The topological Poincaré conjecture in dimension 4 (the work of M. H. Freedman)

by Laurent SiebenmannFrench to English translation by Min Hoon Kim and Mark Powell

Introduction

Report

1. Terminology

2. Casson tower and Freedman’s mitosis

3. Architecture of topological coordinates

3.1. Construction

3.2. Construction of the design G4 (see Figure 11)

3.3. Construction of g:G4→B2×D2

3.4. Construction of g0:G04→B2×D2

3.5. The image g0(G04)⊂B2×D2

3.6. The main diagram

3.7. Construction of W0 and g1

3.8. Construction of D′ and g2

3.9. Construction of D+

3.10. End of the proof that M is homeomorphic to B2×D˚2 (modulo Sections 4 and 5)

4. Bing shrinking

5. Freedman’s approximation theorem

Works

by Laurent Siebenmann

French to English translation by Min Hoon Kim and Mark Powell

3.2. Construction of the design $G^{4}$ (see Figure 11)

3.3. Construction of $g : G^{4} \to B^{2} \times D^{2}$

3.4. Construction of $g_{0} : G_{0}^{4} \to B^{2} \times D^{2}$

3.5. The image $g_{0} (G_{0}^{4}) \subset B^{2} \times D^{2}$

3.7. Construction of $W_{0}$ and $g_{1}$

3.8. Construction of $D^{'}$ and $g_{2}$

3.9. Construction of $D_{+}$

3.10. End of the proof that $M$ is homeomorphic to $B^{2} \times {\overset{˚}{D}}^{2}$ (modulo Sections 4 and 5)