# Celebratio Mathematica

## Robion C. Kirby

### Kirby’s work on topological manifolds

#### 1. Manifolds pre-Kirby

##### 1.1. Smooth and PL manifolds before 1969

Once upon a time, not so long ago, to­po­lo­gic­al man­i­folds were rather like uni­corns. Oth­er kinds of man­i­folds, like oth­er an­im­als, were plen­ti­ful and in­tens­ively stud­ied: nonsin­gu­lar al­geb­ra­ic vari­et­ies had been stud­ied for over a cen­tury; Poin­caré [e77] worked with PL man­i­folds; and by the early nine­teen fifties, many of the fun­da­ment­al res­ults on smooth man­i­folds had been de­veloped, in­clud­ing the the­or­em that smooth man­i­folds were PL. Still, no one had yet seen either a uni­corn or a to­po­lo­gic­al man­i­fold that was not smooth.

In­deed, by the mid-1950s we knew all to­po­lo­gic­al man­i­folds of di­men­sion less than 4 were smooth, where by to­po­lo­gic­al man­i­fold we mean a loc­ally Eu­c­lidean, para­com­pact, Haus­dorff to­po­lo­gic­al space. As for homeo­morph­isms, even on the real line there are homeo­morph­isms that are not dif­fer­en­ti­able. But any homeo­morph­ism between smooth man­i­folds of di­men­sion less than 4 is iso­top­ic to a dif­feo­morph­ism and the iso­topy can be chosen to move every point no more than a pre­de­ter­mined pos­it­ive dis­tance (usu­ally called an $$\epsilon$$-iso­topy).

Ori­gin­al cita­tions for these res­ults in di­men­sions 0 and 1 are un­known to this au­thor. T. Radó [e5] is usu­ally cred­ited with the fi­nal de­tails in di­men­sion 2 and Moise [e10] did di­men­sion 3 in a series of pa­pers around 1952.

Mil­nor’s [e14] 1956 ex­ample of a smooth man­i­fold which was homeo­morph­ic but not dif­feo­morph­ic to the 7-sphere ended the hope that all homeo­morph­isms might be iso­top­ic to dif­feo­morph­isms. By 1961 Smale [e25] had proved the h-cobor­d­ism the­or­em, which im­plied that smooth man­i­folds ho­mo­topy equi­val­ent to a sphere were PL homeo­morph­ic if the di­men­sion was at least 7 (later lowered to 5). This meant that sim­pli­cial com­plexes con­struc­ted by Mil­nor in 1956 were ac­tu­ally PL man­i­folds with no smooth struc­ture. Oth­er ex­amples fol­lowed. Some were by ex­pli­cit con­struc­tion but oth­ers needed more tech­nique. The Ker­vaire–Mil­nor [e29] clas­si­fic­a­tion of ho­mo­topy spheres in di­men­sions at least 5 used trans­vers­al­ity, sur­gery the­ory and the h-cobor­d­ism the­or­em. Handle­body the­ory was used to clas­si­fy cer­tain types of smooth man­i­folds.

Smooth trans­vers­al­ity due to Pontry­agin [e7] and Thom [e13] works in all di­men­sions, as does handle­body the­ory, ini­ti­ated by Morse [e6] and Bott [e15]. However, oth­er ba­sic tools such as the h-cobor­d­ism the­or­em and sur­gery the­ory de­pend on the Whit­ney trick and so only were known to work in di­men­sion at least 5.

In 1963 Mil­nor [e36] in­tro­duced mi­crobundles for all three kinds of man­i­folds and showed man­i­folds of each type had tan­gent bundles of each type. Smooth mi­crobundles are es­sen­tially vec­tor bundles. There are clas­si­fy­ing spaces and maps $B\mathrm{O}(n)\to B\mathrm{PL}(n) \to B\mathrm{TOP}(n) .$ A great deal was known about $$B\mathrm{O}(n)$$ but the oth­er two were mys­ter­i­ous.

Smale [e16] and Hirsch [e18] re­duced im­mer­sions of smooth man­i­folds to bundle the­ory and after mi­crobundles were avail­able Hae­fli­ger and Poen­aru [e37] pro­duced the ana­log­ous res­ults for loc­ally flat PL im­mer­sions. Much later, but just be­fore Kirby’s big break­through, Lees [e55] proved his im­mer­sion the­or­em which brought to­po­lo­gic­ally loc­ally flat im­mer­sion in­to line with the smooth and PL cases.

Dur­ing the six­ties the the­ory of PL man­i­folds caught up with the the­ory for smooth man­i­folds. In 1966 Wil­li­am­son [e46] proved a trans­vers­al­ity res­ult for loc­ally flat PL sub­man­i­folds. Handle­body the­ory, the s-cobor­d­ism the­or­em and sur­gery for PL man­i­folds in di­men­sions at least 5 were de­veloped by the com­bined work of many au­thors. So now even the­or­ems first proved in the smooth cat­egory be­came avail­able in the PL cat­egory with es­sen­tially the same proofs.

The ques­tions of when a PL man­i­fold is smooth and when a PL homeo­morph­ism is iso­top­ic to a dif­feo­morph­ism were re­duced to lift­ing prob­lems for stable bundles and the ho­mo­topy groups of the fibra­tion $\mathrm{PL}/\mathrm{O}\to B\mathrm{O}\to B\mathrm{PL}$ were un­der­stood in terms of the groups of ho­mo­topy spheres.

The role of al­geb­ra­ic K-the­ory in the the­ory of man­i­folds be­came clear­er. Prob­lems whose solu­tions were un­ob­struc­ted in the simply con­nec­ted case be­came ob­struc­ted in the pres­ence of a non­trivi­al fun­da­ment­al group but ne­ces­sary and suf­fi­cient con­di­tions for a solu­tion could be giv­en us­ing al­geb­ra­ic K-the­ory, coupled with handle­body the­ory, trans­vers­al­ity and em­bed­ding the­ory. Smale’s h-cobor­d­ism the­or­em was ex­ten­ded to the s-cobor­d­ism the­or­em by Barden [e32], Mazur [e30] and Stallings [e40] us­ing J. H. C. White­head’s the­ory of tor­sion [e1]. Wall [e41] used the pro­ject­ive class group to study when a fi­nitely gen­er­ated, pro­ject­ive chain com­plex was chain ho­mo­topy equi­val­ent to a fi­nitely gen­er­ated, free one. Browder, Lev­ine and Livesay [e44] showed that a non­com­pact man­i­fold with fi­nitely gen­er­ated ho­mo­logy which was simply con­nec­ted at in­fin­ity was the in­teri­or of a com­pact man­i­fold with bound­ary, and Sieben­mann’s thes­is [e45] gave ne­ces­sary and suf­fi­cient con­di­tions for this to be true in gen­er­al us­ing the pro­ject­ive class group. Browder and Lev­ine [e47] showed that a con­nec­ted man­i­fold with fun­da­ment­al group $$\mathbb Z$$ fibers over a circle if and only if an ob­vi­ously ne­ces­sary fi­nite­ness con­di­tion holds. Far­rell’s thes­is [e62] gives a ne­ces­sary and suf­fi­cient con­di­tion for this to hold in gen­er­al in terms of the White­head group of the fun­da­ment­al group.

Sur­gery the­ory as in­tro­duced by Wal­lace [e21], Mil­nor [e22] and Ker­vaire–Mil­nor [e29] is ob­struc­ted even in the simply con­nec­ted case. Browder [e76](ori­gin­ally pub­lished in [e26]) and Novikov [e27] in­tro­duced the idea of sur­gery on a nor­mal map.

Clas­si­fic­a­tion of com­pact man­i­folds in a fixed ho­mo­topy type of di­men­sion at least 5 was be­com­ing a prob­lem that could be solved. Sul­li­van [e50] fo­cused at­ten­tion on the struc­ture set of a space $$X$$, $$S^{\mathrm{PL}}_n(X)$$. It con­sists of simple ho­mo­topy equi­val­ences $$f: M^n \to X$$ mod­ulo the re­la­tion $$f_1: M_1^n \to X$$ is equi­val­ent to $$f_2: M_2^n \to X$$ if and only if there ex­ists a PL homeo­morph­ism $$h: M_1 \to M_2$$ with $$f_2\circ h$$ ho­mo­top­ic to $$f_1$$. There are also $$\mathrm{TOP}$$ and smooth ver­sions and a ver­sion for man­i­folds with bound­ary.

For $$S^{\mathrm{PL}}_n(X)$$ to be nonempty re­stricts $$X$$: it must sat­is­fy Poin­caré du­al­ity since $$M$$ does. Us­ing Spivak’s con­struc­tion [e51] of a nor­mal spher­ic­al fibra­tion for Poin­caré spaces and Atiyah’s unique­ness res­ult [e23] for them, Sul­li­van con­struc­ted a “dif­fer­en­tial” $\mathfrak d: S^{\mathrm{PL}}_n(X) \to [X,G/\mathrm{PL}]$ provided the Spivak nor­mal fibra­tion, $$X\to BG$$ lifts to $$B\mathrm{PL}$$, in ana­logy with the $$\mathrm{PL}$$ to $$\mathrm{O}$$ smooth­ing the­ory. Moreover the ho­mo­topy fiber of $$B\mathrm{PL} \to BG$$ is $$G/\mathrm{PL}$$. From the de­scrip­tion of $$G/\mathrm{PL}$$ as this ho­mo­topy fiber, one can see it is a ho­mo­topy as­so­ci­at­ive, ho­mo­topy com­mut­at­ive H-space.

For a simply con­nec­ted Poin­caré du­al­ity space $$X^n$$, Sul­li­van defined a map $[X,G/\mathrm{PL}] \xrightarrow{\alpha\,} L_n(\mathbb Z)$ and for $$n\geqslant 5$$, fit them in­to the “ex­act se­quence” be­low. The map $$\beta$$ be­low is just a ver­sion of $$\alpha$$ for $$X\times [0,1]$$ rel $$X\times\{0, 1\}$$.

Wall [e49] used quad­rat­ic al­geb­ra­ic K-the­ory, also known as L-the­ory, to ex­tend sur­gery on nor­mal maps to the nonsimply con­nec­ted case. Just as in the simply con­nec­ted case, these groups are 4-fold peri­od­ic, es­sen­tially by their con­struc­tion. Wall gen­er­al­ized Sul­li­van’s maps $$\alpha$$ and $$\beta$$ be­low to the nonsimply con­nec­ted case. They are defined in all di­men­sions. Sul­li­van’s $$\mathfrak d$$ already worked with no $$\pi_1$$ or di­men­sion as­sump­tions. The se­quence is easi­est to ex­plain when $$X$$ starts out as a PL man­i­fold $$M$$. The sense in which the se­quence is ex­act needs some ex­plan­a­tion. Here is where $$n\geqslant 5$$ comes in.

The struc­ture set $$S^{\mathrm{PL}}_n(M)$$ is only a based set with the iden­tity as the base point and $$\alpha$$ need not be a ho­mo­morph­ism. Nev­er­the­less, $$\alpha^{-1}(0)$$ is the im­age of $$\mathfrak d$$. The map $$\scriptstyle\bullet$$ comes from a group ac­tion on a set: $L_{n+1}(\mathbb Z[\pi_1(M)])\times S^{\mathrm{PL}}_n(M)\to S^{\mathrm{PL}}_n(M) .$ The map $$\beta$$ is a ho­mo­morph­ism and the quo­tient group $L_{n+1}(\mathbb Z[\pi_1(M)])/\operatorname{image}(\beta)$ acts freely on $$S^{\mathrm{PL}}_n(M)$$ with or­bit space $$\alpha^{-1}(0)$$. \begin{equation*} [\Sigma M, G/\mathrm{PL}] \xrightarrow{\beta\,} L_{n+1}(\mathbb Z[\pi_1(M)]) \xrightarrow{\bullet\,} S^{\mathrm{PL}}_n(M)\xrightarrow{\mathfrak d} [M,G/\mathrm{PL}] \xrightarrow{\alpha\,} L_n(\mathbb Z[\pi_1(M)]). \end{equation*}

To com­pute the struc­ture set of a man­i­fold $$M^n$$, $$n\geqslant5$$, one needs to com­pute two of the four Wall groups. One also needs to com­pute the group of nor­mal in­vari­ants, $$[M,G/\mathrm{PL}]$$ and $$[\Sigma M, G/\mathrm{PL}]$$. Sul­li­van gave a de­tailed ana­lys­is of the ho­mo­topy type of $$G/\mathrm{PL}$$ but for Kirby’s needs, only the ho­mo­topy groups are needed. For low di­men­sions they can be com­puted from the ho­mo­topy ex­act se­quence of the fibra­tion and the fact that $$B\mathrm{O}\to B\mathrm{PL}$$ is an iso­morph­ism on ho­mo­topy groups for $$n\leqslant 6$$. For $$n\geqslant5$$ the Sul­li­van ex­act se­quence com­putes the groups in di­men­sions $$\geqslant5$$ since all ho­mo­topy spheres in these di­men­sions are PL-stand­ard.

Of par­tic­u­lar rel­ev­ance to Kirby’s work, Shaneson [e57] and Wall [e60] used Far­rell’s thes­is to com­pute the Wall groups of free abeli­an groups BassHellerSwan [e38] com­puted the White­head, pro­ject­ive class and Nil groups of free abeli­an groups.

##### 1.2. Topological manifolds before 1969
Whilst non-PL man­i­folds were un­known be­fore 1969, em­bed­dings and homeo­morph­isms which were not smooth or PL or even piece­wise-dif­fer­en­ti­able were well known. These can be con­struc­ted by pro­du­cing a se­quence of smooth/PL func­tions, em­bed­dings, etc., and by ar­guing that the lim­it of the se­quence has whatever prop­er­ties are de­sired.

Al­ex­an­der [e4] con­struc­ted his fam­ous horned sphere in 1924 as an ex­ample of an em­bed­ding of the 3-ball in the 3-sphere which could not be iso­top­ic to a smooth em­bed­ding. In the 1950’s, Bing [e12] and oth­ers con­struc­ted many strange ob­jects, pres­ci­ently to­po­lo­gic­al spaces which were not man­i­folds but which be­came man­i­folds after cross­ing with some Eu­c­lidean space.

Res­ults about man­i­folds without as­sum­ing a smooth or PL struc­ture be­fore 1950 were very rare. Al­geb­ra­ic to­po­logy res­ults such as Poin­caré du­al­ity, the Jor­don–Brouwer sep­ar­a­tion the­or­em and Brouwer’s in­vari­ance of do­main res­ult were known.

Han­ner [e9] proved to­po­lo­gic­al man­i­folds were ANR’s and hence, by a res­ult of White­head’s [e8], the ho­mo­topy type of CW com­plexes. But even com­pact to­po­lo­gic­al man­i­folds were not known to al­ways have the ho­mo­topy type of fi­nite CW com­plexes. Wall’s fi­nite­ness ob­struc­tion [e41] in the pro­ject­ive class group of the fun­da­ment­al group was known to be defined, but not known to be zero.

Mazur [e17], Morse [e19] and Brown [e20] proved the to­po­lo­gic­al Schoen­flies the­or­em in all di­men­sions which says that a loc­ally flat em­bed­ding of $$\mathbb{S}^{n-1}$$ in $$\mathbb{S}^n$$ di­vides the sphere in­to two disks, each with bound­ary the em­bed­ded $$\mathbb{S}^{n-1}$$. Mazur’s proof re­quired an ad­di­tion­al hy­po­thes­is that the em­bed­ding was “nice” at one point, a con­di­tion which was re­moved by Morse. Brown gave a self-con­tained proof of the full res­ult.

The next most com­plic­ated res­ult along these lines was to con­sider a loc­ally flat em­bed­ding of $$\mathbb{S}^{n-1}$$ in­to the in­teri­or of an $$n$$-disk. The sphere di­vides the disk in­to two pieces: one piece is a smal­ler $$n$$-disk by the Schoen­flies the­or­em. The oth­er piece was con­jec­tured to be an an­nu­lus, a space homeo­morph­ic to $$\mathbb{S}^{n-1}\times[0,1]$$. Mil­nor’s in­clu­sion of the an­nu­lus con­jec­ture on his 1963 prob­lem list from the Seattle con­fer­ence [e42] plus its in­her­ent sim­pli­city to state made it an at­tract­ive prob­lem which gen­er­ated some in­terest. Cantrell [e43], LaBach [e52] and even Kirby [1] proved ver­sions of the an­nu­lus con­jec­ture with “small ad­di­tion­al hy­po­thes­is” but a proof of the full con­jec­ture re­mained elu­sive.

In a pos­it­ive dir­ec­tion, M. New­man [e48] was able to ex­tend Stallings’s the­ory of en­gulf­ing [e28] to to­po­lo­gic­al man­i­folds and man­aged to prove, amongst oth­er things, that any to­po­lo­gic­al man­i­fold ho­mo­topy equi­val­ent to a sphere was homeo­morph­ic to a sphere.

In a dif­fer­ent, seem­ingly un­re­lated dir­ec­tion, people began to study homeo­morph­isms of $$\mathbb R^n$$ and em­bed­dings of $$\mathbb R^n$$ in $$\mathbb R^n$$. Kister [e39] proved that mi­crobundles are fiber bundles so that Mil­nor’s mys­ter­i­ous clas­si­fy­ing spaces, $$B\mathrm{TOP}(n)$$ are the clas­si­fy­ing spaces of the group of homeo­morph­isms, $$\mathbb R^n \to \mathbb R^n$$. These however were also very mys­ter­i­ous. Be­fore 1969 the num­ber of path com­pon­ents of $$\operatorname{Homeo}(\mathbb R^n)$$ was un­known. Two homeo­morph­ism are in the same path com­pon­ent if and only if they are iso­top­ic. As a first step, people tried to de­term­ine if a homeo­morph­ism is iso­top­ic to the iden­tity. Clearly such a homeo­morph­ism must pre­serve ori­ent­a­tion.

In the oth­er dir­ec­tion, it is not hard to show that a homeo­morph­ism which is the iden­tity in a neigh­bor­hood of a point is iso­top­ic to the iden­tity. Define a stable homeo­morph­ism of $$\mathbb R^n$$ to be one that is a com­pos­ite of a fi­nite num­ber of homeo­morph­isms, each of which is the iden­tity in a neigh­bor­hood of some point. All ori­ent­a­tion-pre­serving homeo­morph­isms of $$\mathbb R^n$$ which are dif­fer­en­ti­able or PL in a neigh­bor­hood of a point are stable. The stable homeo­morph­ism con­jec­ture con­jec­tures that all ori­ent­a­tion-pre­serving homeo­morph­isms of $$\mathbb R^n$$ are stable.

Brown and Gluck [e33], [e34], [e35] ex­ten­ded these ideas. They defined a no­tion of stable in a neigh­bor­hood of a point and showed that a homeo­morph­ism of $$\mathbb R^n$$ was stable if and only if it was stable in a neigh­bor­hood of any one point. This gives a no­tion of stable between dif­fer­ent open sub­sets of $$\mathbb R^n$$ and from there to the no­tion of a stable at­las for a man­i­fold and hence the no­tion of a stable man­i­fold. There is also the no­tion of a stable im­mer­sion between stable man­i­folds.

It fur­ther fol­lows that all ori­ent­able smooth and ori­ent­able PL man­i­folds are stable as are all smooth or PL ori­ent­a­tion-pre­serving im­mer­sions.

Us­ing this circle of ideas, Brown and Gluck were able to show that the stable homeo­morph­ism con­jec­ture in di­men­sion $$n$$ im­plies the an­nu­lus con­jec­ture in di­men­sion $$n$$.

Slightly earli­er, E. Con­nell [e31] proved that stable homeo­morph­isms of $$\mathbb R^n$$ could be ap­prox­im­ated by PL homeo­morph­isms if $$n\geqslant 7$$. In the midst of the proof is a lemma that bounded homeo­morph­isms of $$\mathbb R^n$$ are stable, $$n\geqslant 7$$. In 1967, Bing [e54] re­duced 7 to 5.

#### 2. Kirby’s breakthrough

Kirby was fa­mil­i­ar with Con­nell’s res­ults since he had used them in his thes­is [2]. Lemma 5 of Con­nell’s pa­per proves that bounded homeo­morph­isms of $$\mathbb R^n$$ are stable, $$n\geqslant 5$$. The loc­al nature of stable im­plies that any cov­er­ing space of a stable man­i­fold is stable and if $$f: N \to M$$ is a homeo­morph­ism between stable man­i­folds and if $$\tilde{f}: \widetilde{N} \to \widetilde{M}$$ is the homeo­morph­ism in­duced between cov­er­ing spaces, $$f$$ is stable if and only if $$\tilde{f}$$ is stable. Giv­en any homeo­morph­ism $$f$$ of $$\mathbb{T}^n$$, the in­duced map on the uni­ver­sal cov­er is bounded and hence stable ($$n\geqslant 5$$) by Con­nell. Hence all homeo­morph­isms $$\mathbb{T}^n \to \mathbb{T}^n$$ are stable ($$n\geqslant 5$$).

Kirby says his big break­through came one night in Au­gust 1968 when he real­ized that giv­en a homeo­morph­ism $$h: \mathbb R^n \to\mathbb R^n$$, he could con­struct a homeo­morph­ism $$\hat{h}$$ from a PL man­i­fold, $$K^n$$, to $$\mathbb{T}^n$$, $$n$$ large. The homeo­morph­ism $$h$$ would be stable if and only if $$\hat{h}$$ were stable.

This is not quite enough since $$\hat{h}$$ may not be a self-homeo­morph­ism of $$\mathbb{T}^n$$. If there is a PL homeo­morph­ism $$g: \mathbb{T}^n\to K^n$$ then $$\hat{h}$$ is stable if and only if $$\hat{h}\circ g: \mathbb{T}^n\to \mathbb{T}^n$$ is stable.

There is an old con­jec­ture, the Hauptver­mu­tung, due to Stein­itz [e3] and Tiet­ze [e2] in 1908, which con­jec­tures that if two sim­pli­cial com­plexes are homeo­morph­ic then they are PL homeo­morph­ic. Mil­nor [e24] dis­proved this for com­plexes but the con­jec­ture was still open for PL man­i­folds in 1968. At this point Kirby had a proof that the Hauptver­mu­tung for $$\mathbb{T}^n$$ im­plies the stable homeo­morph­ism con­jec­ture for $$\mathbb R^n$$ for $$n\geqslant 5$$.

In the fall of ’68 ex­perts would have prob­ably bet that the Hauptver­mu­tung for $$\mathbb{T}^n$$ was false. Nev­er­the­less dur­ing that fall and the spring of ’69, Kirby, to­geth­er with Sieben­mann, tried to make this idea work. The Hauptver­mu­tung can be at­tacked via the Sul­li­van–Wall se­quence. Sev­er­al phe­nom­ena can res­ult in non­trivi­al ele­ments in $$S^{\mathrm{PL}}(M^n)$$. There might be a pair $$(N, f)$$ with $$N$$ not homeo­morph­ic to $$M$$. The Hauptver­mu­tung makes no pre­dic­tion in this case. There might be $$(N, f)$$ with $$f$$ a homeo­morph­ism but $$N$$ not PL homeo­morph­ic to $$M$$. In this case the Hauptver­mu­tung is false. There might be $$(N, f)$$ with $$N$$ PL homeo­morph­ic to $$M$$ but $$f$$ not ho­mo­top­ic to a PL homeo­morph­ism. The Hauptver­mu­tung still holds in this case. In gen­er­al it can be dif­fi­cult to sort out which ele­ments are which, even in 2021, but if $$S^{\mathrm{PL}}_n(M)$$, $$n\geqslant 5$$, is a single point the Hauptver­mu­tung holds for $$M$$.

Un­for­tu­nately, $$S^{\mathrm{PL}}(\mathbb{T}^n)$$ is nev­er a single point for $$n\geqslant 5$$ but it can be com­puted from the Sul­li­van–Wall se­quence. For­tu­it­ously the cal­cu­la­tions of $$L_k(\mathbb Z[\pi_1(\mathbb{T}^n)])$$, $$[\mathbb{T}^n, G/\mathrm{PL}]$$ and $$[\Sigma \mathbb{T}^n, G/\mathrm{PL}]$$ can be done in a sim­il­ar fash­ion, which en­ables an in­duct­ive cal­cu­la­tion of $$\alpha$$ and $$\beta$$. It turns out that $$\alpha$$ and $$\beta$$ are in­ject­ive and $$\operatorname{coker}(\beta)$$ is nat­ur­ally iso­morph­ic to $$H^3(\mathbb{T}^n;\mathbb Z/2\mathbb Z)$$, so $$S^{\mathrm{PL}}(\mathbb{T}^n)$$ and $$H^3(\mathbb{T}^n;\mathbb Z/2\mathbb Z)$$ have the same num­ber of ele­ments. The $$\mathbb Z/2\mathbb Z$$’s come from the fact that PL sur­gery prob­lems over $$\mathbb{T}^3\times [0,1]$$ rel $$\mathbb{T}^3\times\{0,1\}$$ have sig­na­ture di­vis­ible by 16 by Rohlin’s the­or­em [e11], where­as the sur­gery group real­izes all sig­na­tures di­vis­ible by 8. Both Wall [e56] and HsiangShaneson [e58] have proofs.

The stand­ard $$r^n$$-fold cov­er of $$\mathbb{T}^n$$ is the product of $$n$$ cop­ies of the de­gree-$$r$$ cov­er on $$\mathbb{S}^1$$. The ex­pli­cit form of the iden­ti­fic­a­tion of $$S^{\mathrm{PL}}(\mathbb{T}^n)$$ with $$H^3(\mathbb{T}^n;\mathbb Z/2\mathbb Z)$$ shows that the map $$S^{\mathrm{PL}}(\mathbb{T}^n) \to S^{\mathrm{PL}}(\mathbb{T}^n)$$ in­duced by the stand­ard $$r^n$$-fold cov­er is the iden­tity if $$r$$ is odd and maps all ele­ments to the iden­tity homeo­morph­ism if $$n$$ is even.

To go back to that night in Au­gust, $$h$$ is stable if and only if $$\hat{h}$$ is stable if and only if $$\widetilde{\hat{h}}$$ is stable where tilde de­notes the stand­ard $$2^n$$-fold cov­er. But $$\widetilde{\hat{h}}$$ is a homeo­morph­ism from a man­i­fold $$K^n \to \mathbb{T}^n$$, where $$K^n$$ is PL homeo­morph­ic to $$\mathbb{T}^n$$ and hence $$\widetilde{\hat{h}}$$ is stable.

All ori­ent­a­tion-pre­serving homeo­morph­isms of $$\mathbb R^n$$ with $$n\geqslant 5$$ are stable.

Here is the tor­us trick. Start with an ori­ent­a­tion-pre­serving homeo­morph­ism $$h: \mathbb R^n \to \mathbb R^n$$ and pick a PL im­mer­sion of $$g: \mathbb{T}^n \to \mathbb R^n$$. The map $$h\circ g: \mathbb{T}^n \to \mathbb R^n$$ is a to­po­lo­gic­al im­mer­sion so there is a, po­ten­tially dif­fer­ent, PL struc­ture on $$\mathbb{T}^n$$, say $$K^n$$, such that $$h\circ g$$ is a PL im­mer­sion and hence stable. There is a map $$\hat{h}: K^n \to \mathbb{T}^n$$ which is the iden­tity on the un­der­ly­ing to­po­lo­gic­al man­i­folds and hence a homeo­morph­ism. Moreover $$h$$ is stable if and only if $$\hat{h}$$ is stable.

The only prob­lem is that there is no such im­mer­sion. However, the tan­gent bundle of $$\mathbb{T}^n$$ is trivi­al and if $$t_0$$ is a point in $$\mathbb{T}^n$$, $$\mathcal T=\mathbb{T}^n - t_0$$ is a non­com­pact man­i­fold. Hirsch [e18] sup­plies a smooth im­mer­sion $$g: \mathcal T\to \mathbb R^n$$. Let $$\tau^n$$ be the smooth­ing of $$\mathcal T$$ for which $$h\circ g$$ is smooth and let $$h_1: \tau \to \mathcal T$$ be the in­duced homeo­morph­ism as above.

There re­mains the prob­lem of “filling in the hole”. By Browder–Livesay, $$n\geqslant 6$$, and Wall [e53] $$n=5$$, there is a smooth em­bed­ding of $$e: \mathbb{S}^{n-1}\times \mathbb R \to \tau$$ so that $$\tau - e(\mathbb{S}^{n-1}\times \mathbb R)$$ is com­pact. Let $$K=\tau \cup \mathbb R^n$$, us­ing $$e$$ to identi­fy $$\mathbb R^n - \vec{0}=\mathbb{S}^{n-1}\times \mathbb R$$, where $$\vec{0}$$ de­notes the ori­gin in $$\mathbb R^n$$, with the im­age of $$e$$ in $$\tau$$. Ex­tend $$h_1: \tau^n \to \mathcal{T}^n$$ to $$\hat{h}$$ by let­ting $$\hat{h}(\vec{0})=t_0$$. Clearly $$\hat{h}: K^n \to \mathbb{T}^n$$ is a bijec­tion. If $$U\subset \mathbb{T}^n-t_0$$ is open, $$\hat{h}^{-1}(U)=h_1^{-1}(U)$$ is open in $$\tau$$ and hence open in $$K^n$$. If $$U \subset \mathbb{T}^n-t_0$$ is closed, it is com­pact. Hence $$\hat{h}^{-1}(U)=h_1^{-1}(U)$$ is com­pact in $$\tau$$ and hence closed in $$K^n$$. If $$U\subset \mathbb{T}^n$$ is open and $$t_0\in U$$, then $$\mathbb{T}^n-U$$ is closed so $$\hat{h}^{-1}(\mathbb{T}^n-U)\subset K$$ is closed and, since $$\hat{h}$$ is a bijec­tion, $\hat{h}^{-1}(U) = K - \hat{h}^{-1}(X-U)$ is open, so $$\hat{h}$$ is con­tinu­ous and there­fore is a homeo­morph­ism.

At some point dur­ing aca­dem­ic ’68–69, it was no­ticed that by cross­ing everything with the $$n$$-ball, $$B^n$$, the tor­us trick be­came a solu­tion to the handle-straight­en­ing prob­lem. After that most of the res­ults needed to put to­po­lo­gic­al man­i­folds on the same foot­ing as smooth and PL man­i­folds fol­low.

The most fam­ous was the res­ult that iso­topy classes of PL-tri­an­gu­la­tions of a to­po­lo­gic­al man­i­fold $$M^n$$, $$n\geqslant 5$$, were giv­en by lifts of the tan­gent bundle $M \xrightarrow{\tau_M} B\mathrm{TOP}$ to $$B\mathrm{PL}$$. The ho­mo­topy groups of $$\mathrm{TOP}/\mathrm{PL}$$ were de­term­ined. First it was shown that $$\mathrm{TOP}/\mathrm{PL}$$ was either a point or $$K(\mathbb Z/2\mathbb Z,3)$$ and then in the spring, they showed it was $$K(\mathbb Z/2\mathbb Z,3)$$. It fol­lows from the Serre spec­tral se­quence that there is a class $\kappa\in H^4(B\mathrm{TOP};\mathbb Z/2\mathbb Z)$ such that a to­po­lo­gic­al man­i­fold $$M^n$$, $$n\geqslant 5$$, is PL if and only if $$\kappa(M)=\tau_M^\ast(\kappa)=0$$. If $$M$$ is PL, define $$S^{\mathrm{TOP}/\mathrm{PL}}(M)$$ to be homeo­morph­isms $$f: N \to M$$ with $$N$$ PL mod­ulo the re­la­tion $$f_1: N_1 \to M$$ is equi­val­ent to $$f_2: N_2 \to M$$ if and only if there ex­ists a PL homeo­morph­ism $$h: N_1\to N_2$$ with $$f_2\circ h$$ iso­top­ic to $$f_1$$. Then $$S^{\mathrm{TOP}/\mathrm{PL}}(M)$$ and $$H^3(M;\mathbb Z/2\mathbb Z)$$ are biject­ive. To see $\mathrm{TOP}/\mathrm{PL}=K(\mathbb Z/2\mathbb Z,3) ,$ they con­struc­ted a non­trivi­al ele­ment in $$S^{\mathrm{TOP}/\mathrm{PL}}(\mathbb{T}^n)$$. For the Arbeit­sta­gung that sum­mer, Sieben­mann de­scribed a to­po­lo­gic­al man­i­fold with no PL struc­ture: see [e61] or page 309 of [3].

Many long-stand­ing prob­lems with to­po­lo­gic­al man­i­folds now had solu­tions. Since the total space of a nor­mal bundle to $$M$$ is PL, com­pact man­i­folds of any di­men­sion have the ho­mo­topy type of fi­nite com­plexes. This also al­lows for the as­sign­ment of a White­head tor­sion for ho­mo­topy equi­val­ences between man­i­folds.

Every map $$f: N \to M$$ is ho­mo­top­ic to a map $$g$$ trans­verse to a sub­man­i­fold $$P\subset M$$ with nor­mal mi­crobundle provided none of $$g^{-1}(P)$$, $$N$$ and $$M$$ have di­men­sion 4. Man­i­folds of di­men­sion $$\geqslant 6$$ have handle­body struc­tures. Hence sur­gery and the s-cobor­d­ism the­or­em fol­low by their usu­al proofs and there is a Sul­li­van–Wall se­quence for to­po­lo­gic­al man­i­folds, $$n\geqslant5$$. The ho­mo­topy groups of the Thom spec­tra as­so­ci­ated to $$B\mathrm{TOP}$$ and the ori­ented ver­sion $$B\mathrm{S}\mathrm{TOP}$$ give the bor­d­ism groups of to­po­lo­gic­al man­i­folds and ori­ented to­po­lo­gic­al man­i­folds re­spect­ively, ex­cept maybe in di­men­sion 4.

Sieben­mann [e59] also poin­ted out that there was ten­sion between Moise’s 3-man­i­fold res­ults and the high-di­men­sion­al res­ults above. A simple ex­ample is that 4-di­men­sion­al handle­bod­ies are smooth so if there were genu­ine to­po­lo­gic­al 4-man­i­folds they could not be handle­bod­ies.

The pic­ture was cla­ri­fied a dec­ade later when Freed­man [e74] proved that Cas­son handles could be straightened to­po­lo­gic­ally. Quinn [e75] then proved the stable homeo­morph­ism con­jec­ture in di­men­sion 4, so both it and the an­nu­lus con­jec­ture now are known in all di­men­sions. The di­men­sion re­stric­tions on trans­vers­al­ity were re­moved and 5-man­i­folds ac­quired handle­body de­com­pos­i­tions. The the­ory of to­po­lo­gic­al man­i­folds of di­men­sion $$\geqslant 5$$ is now on an equal foot­ing with dif­fer­en­ti­able and PL man­i­folds.

Fi­nally, the au­thor can­not res­ist show­ing a genu­ine to­po­lo­gic­al man­i­fold. Sieben­mann and Freed­man cer­tainly con­struct lots of them but they are not easy to visu­al­ize the way spheres or pro­ject­ive spaces and such are.

It is said that the pure of heart can some­times see a uni­corn where or­din­ary folk see only a horse. Let $$E8$$ be the res­ult of plumb­ing eight cop­ies of the co­tan­gent bundle of $$\mathbb{S}^2$$ ac­cord­ing to the Dynkin dia­gram for the ex­cep­tion­al Lie group $$E_8$$ and then con­ing the bound­ary, which re­call is the Poin­caré ho­mo­logy sphere. Define the Bing-uni­corn as the sim­pli­cial com­plex $E8\times \mathbb{S}^1.$

It is a uni­corn be­cause it is a very or­din­ary sim­pli­cial com­plex but only the pure of heart can see that it is ac­tu­ally a man­i­fold [e71].

#### 3. The torus trick post-Kirby

The tor­us trick went on to have a fine ca­reer.

Al­most im­me­di­ately after the in­tro­duc­tion of the tor­us trick, Sieben­mann [e63] used a ver­sion to prove that a CE map between man­i­folds of di­men­sion great­er than 5 is near a homeo­morph­ism.

A bit later, T. Chap­man used a handle-straight­en­ing the­or­em for $$Q$$-man­i­folds to prove the to­po­lo­gic­al in­vari­ance of White­head tor­sion for fi­nite CW com­plexes [e67] and that the homeo­morph­ism group of a com­pact $$Q$$-man­i­fold is loc­ally con­tract­ible [e65]. An ad­apt­a­tion of Sieben­mann’s ideas pro­duced the­or­ems about CE maps between $$Q$$-man­i­folds [e68].

The Ed­wards–West res­ult that com­pact ANR’s are $$Q$$-man­i­fold factors and Chap­man’s work defines a unique simple-ho­mo­topy type for com­pact ANR’s and homeo­morph­isms between ANR’s are simple. See [e70].

Dav­er­man [e64] used a the­or­em of Price and See­beck [e66] (which also uses handle-straight­en­ing) and a tor­us trick to show that a 1-ULC em­bed­ding of man­i­folds in codi­men­sion one is loc­ally flat.

Us­ing some hy­per­bol­ic man­i­folds con­struc­ted by De­ligne and Sul­li­van [e69], Sul­li­van [e72] used cov­ers of these in the same way Kirby used the tor­us to show that man­i­folds of di­men­sion at least 5 have Lipschitz and quasicon­form­al struc­tures.

As late as 2013, M. Hast­ings [e78] was us­ing the tor­us trick to clas­si­fy quantum phases.

By now much of the “trick­ery” in the tor­us trick has been sub­sumed and gen­er­al­ized un­der the rub­ric of “con­trolled sur­gery”. Quinn’s ends-of-maps pa­pers [e73] are one such sub­lim­a­tion.

### Works

[1]R. C. Kirby: “On the an­nu­lus con­jec­ture,” Proc. Amer. Math. Soc. 17 (1966), pp. 178–​185. MR 0192481 Zbl 0151.​32902

[2]R. C. Kirby: “Smooth­ing loc­ally flat im­bed­dings of dif­fer­en­ti­able man­i­folds,” To­po­logy 6 (1967), pp. 207–​220. MR 0211410 Zbl 0152.​22401

[3]R. C. Kirby and L. C. Sieben­mann: Found­a­tion­al es­says on to­po­lo­gic­al man­i­folds, smooth­ings, and tri­an­gu­la­tions. An­nals of Math­em­at­ics Stud­ies 88. Prin­ceton Uni­versity Press, 1977. With notes by John Mil­nor and Mi­chael Atiyah. MR 0645390 Zbl 0361.​57004 book