Celebratio Mathematica

Michael H. Freedman

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

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 $$H_2(V)=0$$.)

The­or­em A: In this situ­ation, $$\Theta$$ 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], [e22] 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.

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

Re­main­ing smooth prob­lem: (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 $$S$$ 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 $$S\cap S^{\prime}$$ 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).

The­or­em C: (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.

The­or­em D: (Casson [e38], compare [e34].)  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.

The­or­em E: (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.

The­or­em F: Let $$(W;V,V^{\prime})$$ 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 [e33]) 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], [e33]) 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

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 $$\{x\in X\mid f(x)\neq x\}$$. 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

We will use two ver­sions $$B^2$$ 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:

1. $$(X, X\cap \partial_+H)$$ is a handle where $$\partial_+H$$ is the skin of the handle $$(H,\partial_-H)$$;
2. $$(\partial_+H,X\cap \partial_+H)$$ is (de­gree $$\pm 1$$) dif­feo­morph­ic to the White­head double $(B^2\times S^1,i(B^2\times S^1))$ il­lus­trated in Fig­ure 1;
3. 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.

Lemma 2.1: The tri­ad $$(H^4\smallsetminus \mathring{X};\partial_-H,\delta X)$$ 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 $$E_k$$. 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 $$\widehat{H}_\infty$$ 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.

The­or­em 2.2: (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 [e33]).

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 $$(H,\partial_-H)$$.

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 $$(H_{12}^{\prime},\partial_-H_{12}^{\prime})$$, such that
1. $$\partial_-H_{12}^{\prime}=\partial_-H_6$$.
2. $$H_{12}^{\prime}\smallsetminus \partial_-H_6\subset \operatorname{Int}H_6$$.
3. $$H_{12}^{\prime}\smallsetminus H_6^{\prime}$$ 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 [e33]) al­lows one to give a proof of The­or­em 2.3. However, it is slightly more de­tailed than the ana­logues in [1], [e33]. 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 $$(k,2k)$$, $$k > 6$$, 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:

Change of Nota­tion 2.4: From now on, we write $$H_k$$ 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.)
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 $$(H_\infty^{\prime},\partial_-H_\infty)\subset (H_\infty, \partial_-H_\infty)$$ sat­is­fy­ing the con­di­tions:
1. $$H_{k-1}^{\prime}=H_{k-1}$$ if $$k\geq 1$$.
2. $$\overline{H^{\prime}_\infty}\smallsetminus H_{k-1}^{\prime}\subset (\operatorname{Int} H_k)\smallsetminus H_{k-1}$$.
3. 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

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 $$B^2\times \mathbb{R}^2$$, 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
For each fi­nite se­quence $$(a_1,\ldots,a_k)$$ 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):
1. $$H_\infty=H_\infty(\emptyset)$$ (case $$k=0$$) as a presen­ted Cas­son handle.
2. $$H_\infty(a_1,\ldots,a_k,1)=H_\infty(a_1,\ldots,a_k)$$.
3. $$H_k(a_1,\ldots,a_k,0)=H_k(a_1,\ldots,a_k)$$ (re­call that $$H_k$$ are sets of 6-stages).
4. The clos­ure $\overline{H}_\infty(a_1,\ldots,a_k,0) \quad\text{in}\quad \widehat{H}_\infty$ is an en­d­point com­pac­ti­fic­a­tion of $$H_\infty(a_1,\ldots,a_k,0)$$.
5. $$\overline{H}_\infty(a_1,\ldots,a_k,0)\smallsetminus H_k(a_1,\ldots,a_k)\subset \mathring{H}_{k+1}(a_1,\ldots,a_k)\smallsetminus H_k(a_1,\ldots,a_k)$$.
6. $$i_k(a_1,\ldots,a_k,0)=i_k(a_1,\ldots,a_k)$$, so $$X_k(a_1,\ldots,a_k,0)=X_k(a_1,\ldots,a_k)$$.
7. 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 $$X_k(a_1,\ldots,a_k)$$.
8. (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.  ◻

Re­mark: If $$(a_1,a_2,\dots)$$ 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.

1. 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.
2. 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 $$g_0 : G_0^4\to B^2\times D^2$$
Let $$G_0^4$$ 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 $$g_0(G_0^4)\subset B^2\times D^2$$
Some nota­tion again (see Fig­ure 13).
• $$T(a_1,\ldots,a_k)\equiv T_k(a_1,\ldots,a_k)=\partial_+X(a_1,\ldots,a_k)$$, 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
The com­mut­at­ive dia­gram to the right gives an over­view of the con­struc­tion to come. The ele­ments will be con­struc­ted in the or­der $$\mathcal{W}_0$$, $$g_1$$, $$\mathcal{D}^{\prime}$$, $$g_2$$, $$\mathcal{D}$$, $$g_3$$, $$\mathcal{D}_+$$, $$f$$. 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 $$g_2$$
Let $$\mathcal{D}^{\prime}$$ 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 $$\mathcal{D}_+$$
We set $W=W_0\cap (B^2\times \mathring{D}^2)=W_0\smallsetminus (B^2\times \partial D^2).$ 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

1. the bound­ary $$\partial E(\alpha)$$; and
2. 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)
Ac­cept­ing from Sec­tion 4 that $$(B^2\times \mathring{D}^2)/\mathcal{D}_+$$ 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
where the in­clu­sion $$\operatorname{Int} M\subset S^4$$ 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.

Re­mark: $$\overline{S(f_*)}\subset S^4$$ 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

We need to show that the space $$B^2\times \mathring{D}^2/\mathcal{D}_+$$ 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.

The­or­em 4.1: (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:

Re­mark: In The­or­em 4.1, if $$h$$ re­spects (or fixes) a closed set $$A\subset X$$, then $$f$$ is ap­prox­im­able by homeo­morph­isms that send $$A$$ on $$f(A)$$ (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 [e19], $$\mathcal{D}_1$$ 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 [e21], [e30]. However, upon re­flec­tion, the com­plic­ated ar­gu­ment of [e21] 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 $\operatorname{Cantor}\times [0,1]=2^{\mathbb{N}}\times [0,1] ,$ and $$f=g\times \operatorname{Id}_{[0,1]}$$, where $g(1,a_2,a_3,\dots)=(a_2,a_3,\dots), \quad g(0,a_2,a_3,\dots)=(0,0,0,\dots).$
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
1. $$g^{-1}(V)=f^{-1}(V)$$,
2. $$g_V : g^{-1}(V)\to V$$ is a homeo­morph­ism, and
3. $$g= f$$ 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 $$\pi_0(K)$$.)

The­or­em 4.4: (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 $$\mathcal{R}(X,A;\mathcal{D})$$.

Ob­ser­va­tion: For every neigh­bour­hood $$U$$ 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.
Pro­pos­i­tion 4.5: If $$\mathcal{D}$$ 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.  ◻

Pro­pos­i­tion 4.6: $$\mathcal{R}(X,A)$$ 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 $(\mathbb{R}\times T,[0,1]\times T^{\prime}) .$

Lemma 4.7: For $$\epsilon > 0$$, 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.

The­or­em 4.8: (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 [e32], 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.

Lemma 4.9: With this data, we can find in $$\mathbb{R}\times T$$ 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.

Pro­pos­i­tion 4.10: The de­com­pos­i­tion $$\mathcal{W}$$ 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 $$W(\alpha)=\bigcup \mathcal{W}(\alpha)$$.

Pro­pos­i­tion 4.11: $$\mathcal{W}(\alpha)$$ 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

1. each in­ter­sec­tion $$B_i\cap D(\alpha)$$ is a dia­metral 2-disc and not knot­ted in $$B_i$$, and
2. $$\mathring{B}_1\cup\cdots\cup \mathring{B}_s$$ 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.  ◻

Pro­pos­i­tion 4.12: $$q(D(\alpha))=E(\alpha)$$ 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.

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 $r : B^2\times\mathring{D}^2\to B^2\times \mathring{D}^2$ 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 $\lambda B^2\times \mu \partial D^2\subset \partial B_* .$

Pro­pos­i­tion 4.14: $$r(\partial B_*)$$ 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.

Pro­pos­i­tion 4.15: The quo­tient of $$\partial B_*$$ 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 $$(B^2\times \mathring{D}^2,r(\partial B_*))$$.  ◻

Pro­pos­i­tion 4.16: $$\mathcal{R}(B^2\times \mathring{D}^2, r(B_*);\mathcal{P})$$ 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 $$\mathcal{R}(B^2\times \mathring{D}^2,B_*;\mathcal{P})$$.  ◻

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 $$\epsilon > 0$$, 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.

Pro­pos­i­tion 4.18: $$\mathcal{R}(B^2\times \mathring{D}^2,r(T_*^{\prime}(\alpha));\mathcal{P})$$ 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 $$r(l(\alpha))$$.  ◻

Pro­pos­i­tion 4.19: $$p_2$$ 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

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 $S(f)=\{y\in Y\mid f^{-1}(y)\neq \text{a point}\}$ 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 [e20], [e18], [e30]. 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 $$\operatorname{Int} M$$ 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.

The­or­em 5.3: (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.  ◻

Ob­ser­va­tion: The case of The­or­em 5.3, where we know in ad­vance that $$\Sigma$$ 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 [e31] 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 $$R$$ 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 $$x\in X$$.

Re­mark: The nat­ur­al func­tion $$[0,1)\to \mathbb{R}/\mathbb{Z}$$ 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

1. the im­age $$R(A)=\{y\in Y\mid \text{there exists }x\in A\text{ such that }(x,y)\in R\}$$,
2. the re­stric­tion $$R|_{A} : A\to Y$$ is the closed sub­set $$R\cap A\times Y$$ in $$A\times Y$$,
3. the in­verse $$R^{-1} : Y\to X$$ such that $$\{(y,x)\in Y\times X\mid (x,y)\in R\}$$.
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 $$D\subset Y\smallsetminus\overline{S(f)}$$ from $$Y$$ and re­mov­ing its preim­age $$f^{-1}(D)$$ from $$X$$, we see that it is per­miss­ible to ad­opt the fol­low­ing.

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 $S(f)=\{y\in Y\mid f^{-1}(y)\neq \text{a point}\}$ 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:

1. $$R\subset X\times Y$$ pro­jects onto $$X$$ and onto $$Y$$.
2. $$R(x)$$ 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 $R^{\prime}\subset R\subset X\times Y .$

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 $$N$$ of $$R$$ in $$X\times Y$$; and $$L\subset Z$$ an open sub­set (called the gap). We im­pose the fol­low­ing con­di­tions:
1. $$R\subset (f^{-1}(\overline{L})\times g^{-1}(\overline{L}))\cup (f^{-1}(Z\smallsetminus L)\times g^{-1}(Z\smallsetminus L))$$; it is in­ev­it­able if the tri­angle switches.
2. $$R=g^{-1}\circ f$$ on $$f^{-1}(\overline{L})$$.
3. $$R$$ is giv­en by the in­ter­sec­tion graph of a homeo­morph­ism $f^{-1}(Z\smallsetminus L)\to g^{-1}(Z\smallsetminus L) .$
4. 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 $$y\in Y$$, $$\operatorname{diam} R_*^{-1}(y) < \epsilon$$.

Ad­dendum: There ex­ists a neigh­bour­hood $$N_*\subset N$$ 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.

1. 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.
2. 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).

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 $$\theta$$ 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 $$X_2$$.  ◻

Claim 5.7: (Trivial if $$S_\epsilon(f)$$ is finite.)  There ex­ists a fi­nite uni­on $$B_+$$ of dis­joint $$n$$-balls in $$L$$ sat­is­fy­ing the fol­low­ing con­di­tions:
1. $$S_\epsilon(f)\cap L\subset \mathring{B}_+$$.
2. $$S(g)\cap B_+=\emptyset$$.
3. 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.)  ◻

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 $$\varphi$$ 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 $$g_*^{-1}\circ f$$.
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 $$f_\infty$$ 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].
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 $$f^{\prime} : X\to Y$$ 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 $$\mathbb{Z}\cup \{\infty, -\infty\}$$.

Works

[1] article M. H. Freed­man: “A fake $$S^{3}\times \mathbf{R}$$,” Ann. of Math. (2) 110 : 1 (1979), pp. 177–​201. MR 0541336 Zbl 0442.​57014

[2]F. An­cel: “Nowhere dense tame 0-di­men­sion­al de­com­pos­i­tions of $${S}^4$$ (an ex­pos­i­tion of a the­or­em of Mike Freed­man)”. Un­pub­lished manuscript, 1981. This work was ori­gi­nally (in­cor­rec­tly) cited by Sie­ben­mann in his list of re­fe­ren­ces as “No­where dense tame 0-di­men­sio­nal de­com­po­si­tions of $$S^4$$; an ex­po­si­tion of a the­orem of Mike Freed­man”. Ric Ancel notes the cor­rect title (email to the editor, dated 28 May 2020) and has made the PDF avail­able to rea­ders of CM. techreport

[3]L. Sieben­mann: “La con­jec­ture de Poin­caré to­po­lo­gique en di­men­sion 4 (d’après M. H. Freed­man),” pp. 219–​248 in Bourbaki Sem­in­ar, vol. 1981/1982. Astérisque 92. Soc. Math. France, Par­is, 1982. MR 689532 incollection