Celebratio Mathematica

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 \( H_2(V)=0 \).)

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

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

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.

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.

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).

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.

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

The non­com­pact ver­sion of The­or­em B is also im­port­ant.

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

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

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

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.

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.

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.

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.

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

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

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

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

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

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

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.

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.

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.  ◻

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 \( (H_\infty(a_1,\ldots,a_k),\partial_-H_\infty) \).

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 \( I(a_1,\ldots,a_k) \).

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.

• \( T(a_1,\ldots,a_k)\equiv T_k(a_1,\ldots,a_k)=\partial_+X(a_1,\ldots,a_k) \), 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 \( (B^2\times D^2)_0\smallsetminus W_0 \).

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.

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.

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.

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.

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:

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

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 \( \pi_0(K) \).)

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 \( \mathcal{R}(X,A;\mathcal{D}) \).

Proof.  The proof is an edi­fy­ing ex­er­cise.  ◻

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 \[ (\mathbb{R}\times T,[0,1]\times T^{\prime}) . \]

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.

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

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

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.

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 \( W(\alpha)=\bigcup \mathcal{W}(\alpha) \).

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.  ◻

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.

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 \[ \lambda B^2\times \mu \partial D^2\subset \partial B_* .\]

This will res­ult in the fol­low­ing pro­pos­i­tion.

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 \( (B^2\times \mathring{D}^2,r(\partial B_*)) \).  ◻

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

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.

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 \( r(l(\alpha)) \).  ◻

Proof.  Ap­ply Pro­pos­i­tions 4.18, 4.16 and 4.5.  ◻

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 \( B^n \).

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.

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.  ◻

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

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 \( x\in X \).

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 \( \{(y,x)\in Y\times X\mid (x,y)\in R\} \).

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.

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 \[ R^{\prime}\subset R\subset X\times Y .\]

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).

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 \( X_2 \).  ◻

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.)  ◻