Celebratio Mathematica

An­drew Ran­icki ded­ic­ated the largest part of his math­em­at­ic­al life to sur­gery the­ory, a branch of to­po­logy that deals with clas­si­fic­a­tion prob­lems of man­i­folds in­side a giv­en ho­mo­topy type. The total sur­gery ob­struc­tion (TSO) can be seen as the cul­min­a­tion of his work in this area. It provides in some sense a defin­it­ive an­swer to the fol­low­ing ex­ist­ence ques­tion: Giv­en a space $X$, when is it ho­mo­topy equi­val­ent to a closed $n$-di­men­sion­al to­po­lo­gic­al man­i­fold? There is a closely re­lated unique­ness ques­tion of such a man­i­fold, and the TSO provides an an­swer here as well.

We will now make an in­form­al for­mu­la­tion of the main the­or­em about the TSO and then we dis­cuss the con­text, some ap­plic­a­tions and some ideas that go in­to the proofs. The prime sources are the pa­per [1] from 1979, and also the book [4] from 1992, which provided a re­for­mu­la­tion and vari­ous im­prove­ments of the ori­gin­al treat­ment.

Giv­en a fi­nite sim­pli­cial com­plex $X$, and an in­teger $n\in \mathbb{Z}$, Ran­icki defined an abeli­an group ${\mathbb{S}}_n(X)$, as a cer­tain cobor­d­ism group of chain com­plexes para­met­rized over sim­plices of $X$ and equipped with an $(n-1)$-di­men­sion­al quad­rat­ic struc­ture sat­is­fy­ing vari­ous con­di­tions de­scribed in more de­tail be­low. For $X$ a fi­nite sim­pli­cial com­plex, which sat­is­fies a suit­able form of $n$-di­men­sion­al Poin­caré du­al­ity, he defined an ele­ment $$s(X)\in {\mathbb{S}}_n(X),$$ called the TSO, and proved that, if $n\ge 5$, then $s(X)=0$ if and only if $X$ is ho­mo­topy equi­val­ent to a closed $n$-di­men­sion­al to­po­lo­gic­al man­i­fold. Moreover, for $X$ a closed $n$-di­men­sion­al man­i­fold, and a ho­mo­topy equi­val­ence $h:M\to X$ of such man­i­folds, he defined an ele­ment $$s(h)\in {\mathbb{S}}_{n+1}(X),$$ called the struc­ture in­vari­ant of $h$, and proved that, if $n\ge 5$, then $s(h)=0$ if and only if $h$ is ho­mo­top­ic to a homeo­morph­ism.

It is im­port­ant that the groups ${\mathbb{S}}_n(X)$ and ${\mathbb{S}}_{n+1}(X)$ above are in many cases cal­cul­able in a prac­tic­al sense, as they are a part of the al­geb­ra­ic sur­gery ex­act se­quence, which was in­ven­ted by Ran­icki to­geth­er with the TSO. This se­quence con­tains the so-called as­sembly map, which has been ex­tens­ively stud­ied and about which many res­ults have been proven. This know­ledge can then be used to ob­tain in­form­a­tion about the TSO via the al­geb­ra­ic sur­gery ex­act se­quence.

To il­lus­trate this meth­od, we re­call an ap­plic­a­tion to the Borel con­jec­ture, which says that any ho­mo­topy equi­val­ence between two closed as­pher­ic­al man­i­folds is ho­mo­top­ic to a homeo­morph­ism. Giv­en a closed $n$-di­men­sion­al as­pher­ic­al man­i­fold $X$ with ${\pi }_1(X)=G$, such that the so-called Far­rell–Jones con­jec­ture about the as­sembly maps in both $K$-the­ory and $L$-the­ory holds for $G$, the the­ory around the TSO can be used to show that ${\mathbb{S}}_{n+1}(X)=0$. Hence any ho­mo­topy equi­val­ence $h:M\to X$ from any closed $n$-di­men­sion­al man­i­fold $M$ has $s(h)=0$ and is there­fore ho­mo­top­ic to a homeo­morph­ism. So the Borel con­jec­ture holds for $X\simeq BG$ in this case. There is an ex­ist­ence ver­sion of this con­jec­ture and the TSO can be used to say a great deal about it too, but the state­ment is a bit more tech­nic­al, so we re­leg­ate that dis­cus­sion for later.

Let us now put the TSO in­to broad­er his­tor­ic­al con­text. In 1956 Mil­nor found the first ex­ample of an exot­ic sphere, a 7-di­men­sion­al man­i­fold ${\mathrm{\Sigma }}^7$ ho­mo­topy equi­val­ent, but not dif­feo­morph­ic to the stand­ard sphere $S^7$. This raised the ques­tion of clas­si­fic­a­tion of ho­mo­topy spheres, which means smooth man­i­folds ho­mo­topy equi­val­ent to the stand­ard sphere, up to dif­feo­morph­ism. Sur­gery the­ory was born in 1963, when Ker­vaire and Mil­nor provided such a clas­si­fic­a­tion in their pa­per [e3], in terms of ho­mo­topy the­ory and quad­rat­ic forms over $\mathbb{Z}$. The pro­cess of sur­gery was used when at­tempt­ing to modi­fy cobor­d­isms to $h$-cobor­d­isms, a task which may be im­possible due to sur­gery ob­struc­tions.

If one is in­ter­ested in clas­si­fic­a­tion of to­po­lo­gic­al man­i­folds, then the gen­er­al­ized Poin­caré con­jec­ture says that there is up to homeo­morph­ism only one to­po­lo­gic­al man­i­fold ho­mo­topy equi­val­ent to the stand­ard sphere. This con­jec­ture was fam­ously settled by Smale [e1], [e2] for PL-man­i­folds in di­men­sions $\ge 5$ and later by New­man [e4] for to­po­lo­gic­al man­i­folds, and even later by oth­ers in all di­men­sions. However, even be­fore these res­ults there were known pairs of man­i­folds that were ho­mo­topy equi­val­ent, but not homeo­morph­ic, for ex­ample, the lens spaces; see [e13]. So the quest for non­trivi­al clas­si­fic­a­tion res­ults of to­po­lo­gic­al man­i­folds up to homeo­morph­ism was cer­tainly also in­ter­est­ing.

This re­quired fur­ther work. Wall de­veloped sur­gery the­ory for nonsimply con­nec­ted man­i­folds in [e9]. Kirby and Sieben­mann ex­ten­ded sur­gery the­ory to to­po­lo­gic­al man­i­folds in [e6]. More ideas led to con­struct­ing what is now known as the Browder–Novikov–Sul­li­van–Wall sur­gery ex­act se­quence, stated be­low as $\text{(1)}$, re­du­cing the clas­si­fic­a­tion of man­i­folds ho­mo­topy equi­val­ent to $X$ to ho­mo­topy the­ory and quad­rat­ic forms over $\mathbb{Z}[{\pi }_{1}X]$. It has a ver­sion for both smooth and to­po­lo­gic­al man­i­folds, but the res­ults hap­pen to be more cal­cul­able in the to­po­lo­gic­al case. This is due to the work of Sul­li­van, who de­scribed the ho­mo­topy type of the cru­cial clas­si­fy­ing space $\mathrm{G}/\text{TOP}$; see [e7]. We will dis­cuss this space in more de­tail be­low; now we want to point out that the work of Sul­li­van starts from the above­men­tioned Poin­caré con­jec­ture, which from the clas­si­fic­a­tion point of view says that “the base case” of $S^n$ has no oth­er to­po­lo­gic­al man­i­fold in its ho­mo­topy type, while in the smooth case we may have many ho­mo­topy spheres.

This his­tor­ic­al dis­cus­sion has fo­cused on the unique­ness part of the clas­si­fic­a­tion ques­tion so far, but it is im­port­ant to men­tion that the ex­ist­ence part also played a role. The spher­ic­al space form prob­lem, con­cern­ing the ques­tion of which fi­nite groups ad­mit free ac­tions on spheres, and an­oth­er ques­tion of which fi­nite $H$-spaces are ho­mo­topy equi­val­ent to man­i­folds were con­sidered in the 1960s. These ques­tions mo­tiv­ated the de­vel­op­ment of the the­ory, which would ex­plain when is a to­po­lo­gic­al space ho­mo­topy equi­val­ent to a closed man­i­fold. Moreover, the ex­ist­ence and unique­ness ques­tions are closely linked. For a ho­mo­topy equi­val­ence $$h:\partial M_0\to \partial M_1$$ where $M_0$ and $M_1$ are com­pact $(n+1)$-di­men­sion­al man­i­folds with bound­ar­ies $\partial M_0$ and $\partial M_1$, one can con­sider the to­po­lo­gic­al space $$X:=M_0{\cup }_h{M}_1.$$ If $h$ is ho­mo­top­ic to a homeo­morph­ism, say $h^{\prime }$, then the space $X$ is ho­mo­topy equi­val­ent to a closed $(n+1)$-di­men­sion­al man­i­fold, say $M=M_0{\cup }_{h^{\prime }}{M}_1$. So if meth­ods can be de­veloped to ad­dress this ex­ist­ence ques­tion, one can say something about the unique­ness ques­tion too. This is in the spir­it of the well-known slo­gan in to­po­logy that “unique­ness is the re­l­at­ive form of ex­ist­ence”, which turned out to be ful­filled in sur­gery the­ory.

At this point it is ne­ces­sary to men­tion some tech­nic­al is­sues. In the above for­mu­la­tion of the ex­ist­ence ques­tion there are some ob­vi­ous con­di­tions, which the space $X$ has to ful­fill. Namely, it must be ho­mo­topy equi­val­ent to a fi­nite CW-com­plex sat­is­fy­ing Poin­caré du­al­ity. Such com­plexes were provided in the con­text of the spher­ic­al space form prob­lem, the fi­nite $H$-spaces ques­tion, and the com­plex above ob­tained by glu­ing along a ho­mo­topy equi­val­ence ob­vi­ously also sat­is­fies these con­di­tions. There­fore a good tech­nic­al for­mu­la­tion of the ex­ist­ence ques­tion is: Giv­en a fi­nite CW-com­plex sat­is­fy­ing Poin­caré du­al­ity, is there a man­i­fold ho­mo­topy equi­val­ent to it?

To­po­lo­gic­al sur­gery pro­duces the stand­ard an­swer in a two-stage ob­struc­tion pro­cess. As we have seen, the TSO pro­duces a single ob­struc­tion, which can be of ad­vant­age. We also em­phas­ize again that the abeli­an group ${\mathbb{S}}_n(X)$ in­ven­ted by Ran­icki is cal­cul­able in a prac­tic­al sense. An­drew used to joke that if we did not take this re­quire­ment in­to ac­count, then there would a very easy way of solv­ing the two-stage de­fi­ciency. Namely, we could simply take the group $\mathbb{Z}/2$ and say that $s(X)=0$, if the an­swer is yes, and $s(X)=1$, if not. Of course, the prob­lem would be that we would not have any real­ist­ic way of de­term­in­ing which case hap­pens for a giv­en $X$.

Let us now briefly re­call the clas­sic­al sur­gery ap­proach to the ex­ist­ence ques­tion. Giv­en a fi­nite $n$-di­men­sion­al Poin­caré com­plex $X$, the clas­sic­al sur­gery the­ory asks first wheth­er there ex­ists a de­gree one nor­mal map $$(f,\bar{f}):M\to X,$$ from some closed $n$-di­men­sion­al man­i­fold $M$ to $X$. In­tu­it­ively, if we want a ho­mo­topy equi­val­ence, then we should ob­vi­ously have a de­gree one map, pos­sibly equipped with some bundle data, and this is per­haps easi­er to find. The bundle data are a mi­crobundle map from the stable nor­mal mi­crobundle ${\nu }_M$ to some mi­crobundle $\xi$ over $X$.1 This setup is so made, that the as­so­ci­ated spher­ic­al fibra­tion $S(\xi )$ of $\xi$ will al­ways be fiber ho­mo­topy equi­val­ent to the so-called Spivak nor­mal fibra­tion ${\nu }_X$ of $X$. The Spivak nor­mal fibra­tion (SNF) is for Poin­caré com­plexes an ana­logue of the stable nor­mal mi­crobundle of a man­i­fold, and it is a unique spher­ic­al fibra­tion in a spe­cif­ic sense. If a de­gree one nor­mal map ex­ists, then this spher­ic­al fibra­tion comes from a to­po­lo­gic­al mi­crobundle. Con­versely, if it comes from a to­po­lo­gic­al mi­crobundle, then by the to­po­lo­gic­al ver­sion of the Pon­trja­gin–Thom con­struc­tion one can con­struct a de­gree one nor­mal map.

So the first ob­struc­tion is the ob­struc­tion to the ex­ist­ence of the to­po­lo­gic­al mi­crobundle re­duc­tion of the SNF, which can be for­mu­lated in terms of the clas­si­fy­ing spaces of stable spher­ic­al fibra­tions $\mathrm{BG}$, and of stable to­po­lo­gic­al mi­crobundles $\mathrm{BTOP}$, and the ca­non­ic­al map $J:\mathrm{BTOP}\to \mathrm{BG}$, as fol­lows. It is the ob­struc­tion to the ex­ist­ence of a lift $\xi$ of the clas­si­fy­ing map ${\nu }_X$ of the SNF of $X$ in the dia­gram shown to the right.

If this first ob­struc­tion van­ishes, i.e., if such a $\xi$ ex­ists, then the second ob­struc­tion is the sur­gery ob­struc­tion. Giv­en a de­gree one nor­mal map $(f,\bar{f})$, this is a well-defined ele­ment $${\sigma }_n(f,\bar{f})\in L_n(\mathbb{Z}[{\pi }_1(X)])$$ in a cer­tain abeli­an group $L_n(\mathbb{Z}[{\pi }_1(X)])$ defined us­ing quad­rat­ic forms or form­a­tions, or chain com­plexes. The ele­ment it­self is defined, when $n=2k$ and $f$ is $k$-con­nec­ted, by us­ing the middle-di­men­sion­al in­ter­sec­tion form on the ker­nel of the map in­duced by $f$ and its quad­rat­ic re­fine­ment, which has to do with self-in­ter­sec­tions of spheres rep­res­ent­ing ele­ments in the ker­nel. In the odd-di­men­sion­al case the defin­i­tion is more com­plic­ated, it in­volves the no­tion of a form­a­tion, which is an al­geb­ra­ic struc­ture that can be ob­tained by study­ing auto­morph­isms of quad­rat­ic forms, and we re­frain from go­ing in­to a more de­tailed dis­cus­sion here.

The sur­gery ob­struc­tion has the prop­erty that ${\sigma }_n(f,\bar{f})=0$ if and only if $(f,\bar{f})$ is nor­mally cobord­ant to a ho­mo­topy equi­val­ence, provided $n\ge 5$. Hence, if both ob­struc­tions van­ish, then we ob­tain a pos­it­ive an­swer to our ques­tion. However, ob­serve that if the first ob­struc­tion van­ishes, then we have to take in­to ac­count all the pos­sible lifts $\xi$ of ${\nu }_X$ and check the sur­gery ob­struc­tion of de­gree one nor­mal maps as­so­ci­ated to all of them to con­clude a neg­at­ive an­swer for a giv­en $X$. This may be quite hard to do in prac­tice, which is the stand­ard de­fi­ciency in a two-stage ob­struc­tion the­ory.

Both of the above ob­struc­tions can be loc­ated in the to­po­lo­gic­al sur­gery ex­act se­quence. It con­tains the struc­ture set ${\mathcal{S}}^{\text{TOP}}(X)$, which con­sists of ho­mo­topy equi­val­ences $f:M\to X$, where $M$ is some $n$-di­men­sion­al closed man­i­fold, up to homeo­morph­ism in the source. The pairs $(M,f)$ are called man­i­fold struc­tures on $X$. It is clear that the clas­si­fic­a­tion ques­tions we have been dis­cuss­ing so far can be for­mu­lated as fol­lows:

• Ex­ist­ence ques­tion: Is ${\mathcal{S}}^{\text{TOP}}(X)$ nonempty?
• Unique­ness ques­tion: If it is nonempty, what is it?

The next term ap­pear­ing in the sur­gery ex­act se­quence is that of the nor­mal in­vari­ants ${\mathcal{N}}^{\text{TOP}}(X)$, a cobor­d­ism group of de­gree one nor­mal maps $(f,\bar{f}):M\to X$, as dis­cussed above. The third term is the $L$-groups. Here it is also ap­pro­pri­ate to men­tion that Ran­icki cla­ri­fied the ex­ist­ing the­ory of $L$-groups them­selves by provid­ing a defin­i­tion as cer­tain cobor­d­ism groups of fi­nite chain com­plexes equipped with what he called a quad­rat­ic struc­ture, which should be viewed as gen­er­al­iz­ing the no­tion of a quad­rat­ic form on a mod­ule to chain com­plexes. Among oth­er ad­vant­ages this gives a uni­fied defin­i­tion of $L$-groups in­de­pend­ent of the par­ity of $n$; see the found­a­tion­al pa­pers [2], [3].

Ab­bre­vi­at­ing $\pi ={\pi }_1(X)$, we can then give the to­po­lo­gic­al sur­gery ex­act se­quence as $$\begin{array}{}\text{(1)}& \cdots \stackrel{{\sigma }_{n+1}}{\to }{L}_{n+1}(\mathbb{Z}[\pi ])\stackrel{{\partial }_{n+1}}{\to }{\mathcal{S}}^{\text{TOP}}(X)\stackrel{{\eta }_n}{\to }{\mathcal{N}}^{\text{TOP}}(X)\stackrel{{\sigma }_n}{\to }{L}_n(\mathbb{Z}[\pi ]).\end{array}$$ The first ob­struc­tion in the above dis­cus­sion van­ishes if and only if the nor­mal in­vari­ants ${\mathcal{N}}^{\text{TOP}}(X)$ are nonempty; the second ob­struc­tion van­ishes if and only if 0 is in the im­age of ${\sigma }_n$.

This ex­act se­quence is a great res­ult that can be and was used in ap­plic­a­tions, but it also has con­cep­tu­al prob­lems. The $L$-groups are co­v­ari­ant, the nor­mal in­vari­ants are con­trav­ari­ant, and the struc­ture set has no ob­vi­ous in­vari­ance at all. Moreover, there is a group struc­ture on ${\mathcal{N}}^{\text{TOP}}(X)$ giv­en by the Whit­ney sum op­er­a­tion, but the map ${\sigma }_n$ is not a ho­mo­morph­ism with re­spect to this struc­ture, so it is only an ex­act se­quence of poin­ted sets in a spe­cif­ic sense.

For­tu­nately, these prob­lems can be over­come by the the­ory around Ran­icki’s TSO. We first de­scribe some ideas that lead to the re­for­mu­la­tion of the part of the to­po­lo­gic­al sur­gery that are re­lated to the unique­ness ques­tion, and then re­turn to the ex­ist­ence.

To pre­pare, note that Quinn [e5] has defined an $\mathrm{\Omega }$-spec­trum ${\mathcal{L}}_{\bullet }$, such that ${\pi }_n({\mathcal{L}}_{\bullet })=L_n(\mathbb{Z})$, and it can be shown that the space $\mathrm{G}/\text{TOP}$, defined as the ho­mo­topy fiber of the map $J:\mathrm{BTOP}\to \mathrm{BG}$, is the 0th space $\mathrm{G}/\text{TOP}\simeq {\mathcal{L}}_0$. Ran­icki him­self provided an al­geb­ra­ic ver­sion of these $L$-spec­tra ${\mathbf{L}}_{\bullet }⟨1⟩$ us­ing his quad­rat­ic chain com­plexes the­ory and so one ob­tains $${\mathcal{N}}^{\text{TOP}}(X)\cong [X;\mathrm{G}/\text{TOP}]\cong H^0(X;{\mathcal{L}}_{\bullet })\cong H^0(X;{\mathbf{L}}_{\bullet }⟨1⟩).$$ Now, one may won­der wheth­er there ex­ists a kind of Poin­caré du­al­ity with re­spect to the spec­trum ${\mathbf{L}}_{\bullet }⟨1⟩$. Ran­icki showed that for a Poin­caré com­plex $X$ with ${\mathcal{N}}^{\text{TOP}}(X)$ nonempty, there are bijec­tions $${\mathcal{N}}^{\text{TOP}}(X)\cong [X;\mathrm{G}/\text{TOP}]\cong H^0(X;{\mathbf{L}}_{\bullet }⟨1⟩)\cong H_n(X;{\mathbf{L}}_{\bullet }⟨1⟩),$$ mak­ing the nor­mal in­vari­ants a co­v­ari­ant func­tor of $X$.

Next he defined the so-called as­sembly maps $\text{asmb}:H_n(X;{\mathbf{L}}_{\bullet }⟨1⟩)\to L_n(\mathbb{Z}[\pi ])$, so that the com­pos­i­tion of the above Poin­caré du­al­ity with this map pro­duces the sur­gery ob­struc­tion map ${\sigma }_n$. At this stage it was pos­sible to make the first defin­i­tion of ${\mathbb{S}}_n(X)$ and the TSO in the pa­per [1].

However, Ran­icki kept work­ing on the top­ic and a much more de­veloped ver­sion ap­pears in his book [4]. A cru­cial in­gredi­ent is the chain com­plex ver­sion of the or­din­ary sur­gery which was already men­tioned. Here the $L$-groups of a ring are cobor­d­ism groups of quad­rat­ic chain com­plexes and the sur­gery ob­struc­tion is defined at the chain com­plex level without the need for first mak­ing the de­gree one nor­mal map highly con­nec­ted. This leads to bet­ter func­tori­al prop­er­ties that are used later.

His next key idea is that he was able to define $L$-groups in the set­ting of ad­dit­ive cat­egor­ies with chain du­al­ity and later al­geb­ra­ic bor­d­ism cat­egor­ies. An ex­ample of an ad­dit­ive cat­egory with chain du­al­ity is the cat­egory of mod­ules para­met­rized by sim­plices of a fi­nite sim­pli­cial com­plex $X$. A chain du­al­ity is a no­tion that al­lows one to con­sider a quad­rat­ic com­plex in such a cat­egory as a col­lec­tion of quad­rat­ic com­plexes para­met­rized by the sim­plices with suit­able com­pat­ib­il­ity con­di­tion. The al­geb­ra­ic bor­d­ism cat­egor­ies are a setup that en­ables de­fin­ing vari­ous nonde­gen­er­acy con­di­tions in this set­ting: one can ask for the com­plex to be Poin­caré loc­ally, mean­ing the com­plex over each sim­plex is Poin­caré, or to be Poin­caré glob­ally, mean­ing the single com­plex ob­tained by as­sem­bling the com­plexes over all sim­plices is Poin­caré. One can also spe­cify a sub­class of com­plexes that are to be con­sidered, and an im­port­ant class here is that of com­plexes that are loc­ally Poin­caré and glob­ally con­tract­ible.

Giv­en a de­gree one nor­mal map $(f,\bar{f}):M\to X$ such that $X$ is a tri­an­gu­lated man­i­fold (this as­sump­tion may be weakened, but we do not want to go in­to such de­tails here), one can make this map trans­verse to dual cells $D(\sigma ,X)$ and take the preim­ages to ob­tain for each sim­plex $\sigma \in X$ a de­gree one nor­mal map of man­i­folds with bound­ary $$(f(\sigma ),\bar{f}(\sigma )):(M(\sigma ),\partial M(\sigma ))\to (D(\sigma ,X),\partial D(\sigma ,X)).$$ The col­lec­tion of the quad­rat­ic chain com­plexes para­met­rized by the sim­plices $\sigma \in X$ ob­tained from these de­gree one nor­mal maps by tak­ing the chain com­plex ver­sion of the sur­gery ob­struc­tion pro­duces a quad­rat­ic com­plex which is loc­ally Poin­caré. As­sem­bling the col­lec­tion of these com­plexes to a single com­plex pro­duces the chain com­plex ver­sion of the usu­al sur­gery ob­struc­tion. In case $f$ is a ho­mo­topy equi­val­ence one ob­tains a com­plex that is glob­ally con­tract­ible, since its un­der­ly­ing chain com­plex is the ho­mo­topy fiber of the ho­mo­topy equi­val­ence $f$, which is con­tract­ible. However, such a com­plex may be loc­ally not con­tract­ible — in fact we only know that it is loc­ally Poin­caré, since even if $f$ is a ho­mo­topy equi­val­ence, for a giv­en $\sigma$ the map $(f(\sigma ),\bar{f}(\sigma ))$ above may not be a ho­mo­topy equi­val­ence.

These ideas can be put to­geth­er in the al­geb­ra­ic sur­gery ex­act se­quence and in the proof of its iden­ti­fic­a­tion with the to­po­lo­gic­al sur­gery ex­act se­quence, which was already dis­cussed. The terms in the al­geb­ra­ic sur­gery ex­act se­quence are for a sim­pli­cial com­plex $X$ with ${\pi }_{1}X=\pi$ and for any $m\in \mathbb{Z}$ (sweep­ing some tech­nic­al con­nectiv­ity re­quire­ments un­der the rug):

The above in­dic­ated con­struc­tion of mak­ing a de­gree one nor­mal map trans­verse to dual cells yields a map from the to­po­lo­gic­al to the al­geb­ra­ic sur­gery ex­act se­quence: $$\begin{array}{ccccccccc}\cdots & \underset{}{\overset{}{\to }}& L_{n+1}(\mathbb{Z}[\pi ])& \underset{}{\overset{}{\to }}& {\mathcal{S}}^{\text{TOP}}(X)& \underset{}{\overset{}{\to }}& {\mathcal{N}}^{\text{TOP}}(X)& \underset{}{\overset{}{\to }}& L_n(\mathbb{Z}[\pi ])\\ & & \cong \downarrow & & \cong \downarrow s& & \cong \downarrow t& & \cong \downarrow & \\ \cdots & \underset{}{\overset{}{\to }}& L_{n+1}(\mathbb{Z}[\pi ])& \underset{}{\overset{}{\to }}& {\mathbb{S}}_{n+1}(X)& \underset{}{\overset{}{\to }}& H_n(X,{\mathbf{L}}_{\bullet }⟨1⟩)& \underset{\text{asmb}}{\overset{}{\to }}& L_n(\mathbb{Z}[\pi ])& \underset{}{\overset{}{\to }}& {\mathbb{S}}_n(X)\end{array}$$ which is shown to be a bijec­tion at all terms where it is defined.

Note that all the terms in the bot­tom row are $L$-groups of some al­geb­ra­ic bor­d­ism cat­egory and that the as­sembly map $\text{asmb}$ ap­pears. This map hap­pens to have a cer­tain uni­ver­sal prop­erty, which char­ac­ter­izes it, and it also has many in­carn­a­tions. In par­tic­u­lar, there is a con­trolled to­po­logy ver­sion of it, which ap­pears in the iso­morph­ism con­jec­tures of Far­rell–Jones. These have been ex­tens­ively stud­ied and many res­ults were achieved us­ing it, such as the proof of the Borel con­jec­ture for a vast class of groups, see for ex­ample [e12].

Fi­nally, we come to the ac­tu­al TSO and ad­dress­ing the ex­ist­ence ques­tion in Ran­icki’s set­ting. Note that in the above dia­gram both groups ${\mathbb{S}}_{n+1}(X)$ and ${\mathbb{S}}_n(X)$ ap­pear. In fact the al­geb­ra­ic sur­gery ex­act se­quence at the bot­tom is an ex­act se­quence of abeli­an groups and can be ex­ten­ded in­fin­itely also to the right. This is a ma­jor dif­fer­ence to the top se­quence, which ends with $L_n(\mathbb{Z}[\pi ])$. This is cru­cial from the point of view of the ex­ist­ence ques­tion since philo­soph­ic­ally, if the unique­ness ob­struc­tion lives in ${\mathbb{S}}_{n+1}(X)$, then the ex­ist­ence ob­struc­tion should live in its “de­loop­ing” ${\mathbb{S}}_n(X)$ and this de­loop­ing is now ex­ist­ing. (In fact the bot­tom se­quence is the long ex­act se­quence of ho­mo­topy groups of a cer­tain cofibra­tion se­quence of spec­tra.)

The TSO is an ele­ment in the group ${\mathbb{S}}_n(X)$, which is a cobor­d­ism group of $(n-1)$-di­men­sion­al loc­ally Poin­caré, glob­ally con­tract­ible quad­rat­ic com­plexes over $X$. How can we ob­tain such an ele­ment out of an $n$-di­men­sion­al fi­nite sim­pli­cial Poin­caré com­plex $X$? The key prop­erty of such com­plexes $X$ is that al­though they are Poin­caré, they may not be loc­ally Poin­caré, mean­ing that for a giv­en sim­plex $\sigma \in X$ the dual cell $(D(\sigma ,X),\partial D(\sigma ,X))$ may not be a Poin­caré pair. Ob­serve that if $X$ is a tri­an­gu­lated to­po­lo­gic­al man­i­fold, then these pairs are Poin­caré, since they are in fact man­i­folds with bound­ary. Start­ing with an $n$-di­men­sion­al fi­nite sim­pli­cial Poin­caré com­plex $X$, Ran­icki first con­siders an $n$-di­men­sion­al glob­ally Poin­caré sym­met­ric com­plex $(C,\phi )$ para­met­rized by sim­plices of $\sigma \in X$, where the chain com­plex over each $\sigma$ is $$C(\sigma ):=C(D(\sigma ,X),\partial D(\sigma ,X)),$$ and there are cer­tain com­pat­ib­il­it­ies and cer­tain du­al­ity maps (which are not chain equi­val­ences in gen­er­al) $$\begin{array}{}\text{(2)}& C^{n-|\sigma |-\ast }(D(\sigma ,X))\to C(D(\sigma ,X),\partial D(\sigma ,X)),\end{array}$$ in­her­ited from the glob­al Poin­caré du­al­ity $$\begin{array}{}\text{(3)}& -\cap [X]:C^{n-\ast }(\tilde{X})\to C(\tilde{X}),\end{array}$$ which is a $\mathbb{Z}[\pi ]$-chain ho­mo­topy equi­val­ence since $X$ is Poin­caré.

Ran­icki next in­tro­duced the no­tion of al­geb­ra­ic bound­ary $\partial (E,\phi )$ of an $n$-di­men­sion­al glob­ally Poin­caré sym­met­ric com­plex $(E,\phi )$ para­met­rized by sim­plices of $X$, which is an $(n-1)$-di­men­sion­al quad­rat­ic com­plex $\partial (E,\phi )=(\partial E,\partial \phi )$ again para­met­rized by sim­plices of $X$, and which in some sense meas­ures the fail­ure of the ori­gin­al com­plex to be loc­ally Poin­caré. This $(n-1)$-di­men­sion­al quad­rat­ic com­plex al­ways turns out to be loc­ally Poin­caré and it is also glob­ally con­tract­ible since its un­der­ly­ing chain com­plex is the ho­mo­topy fiber of the glob­al Poin­caré du­al­ity map $\text{(3)}$, which is a chain ho­mo­topy equi­val­ence by as­sump­tion.

The TSO of $X$, that means $s(X)\in {\mathbb{S}}_n(X)$, is now defined by tak­ing first the $n$-di­men­sion­al glob­ally Poin­caré sym­met­ric com­plex $(C,\phi )$ para­met­rized by sim­plices of $\sigma \in X$ de­scribed above, and then tak­ing its al­geb­ra­ic bound­ary $\partial (C,\phi )=(\partial C,\partial \phi )$, which is now an $(n-1)$-di­men­sion­al quad­rat­ic com­plex again para­met­rized by sim­plices of $X$, and which is loc­ally Poin­caré and glob­ally con­tract­ible and thus rep­res­ents an ele­ment in ${\mathbb{S}}_n(X)$. The un­der­ly­ing chain com­plex $\partial C(\sigma )$ over each sim­plex $\sigma \in X$ turns out to be the ho­mo­topy fiber of the chain map from $\text{(2)}$. Fur­ther tech­nic­al de­tails of the con­struc­tion are bey­ond the scope of this art­icle, but we hope that the above de­scrip­tion of the un­der­ly­ing chain com­plex provides some in­tu­ition.

In sum­mary, we see that the TSO in some sense meas­ures the fail­ure of loc­al Poin­caré du­al­ity of a geo­met­ric Poin­caré com­plex. The idea that fail­ure of such a loc­al Poin­caré du­al­ity ob­structs the pos­it­ive an­swer to the ex­ist­ence ques­tion is quite nat­ur­al. What was hard here was find­ing the setup (that means the right kind of $L$-groups), such that van­ish­ing of the ele­ment ob­tained us­ing this idea guar­an­tees the ex­ist­ence of a man­i­fold in the ho­mo­topy type of $X$. The proof is very long, and it in­volves in par­tic­u­lar also all of the clas­sic­al sur­gery. We refer the read­er to [1], [4], [e14].

We prom­ised an ap­plic­a­tion to the ex­ist­ence ver­sion of the Borel con­jec­ture. It is as fol­lows. Giv­en a tor­sion­free group $G$, such that $BG$ is an $n$-di­men­sion­al fi­nite sim­pli­cial com­plex, which sat­is­fies Poin­caré du­al­ity, and the Far­rell–Jones con­jec­ture in $K$-the­ory and $L$-the­ory holds for $G$, then the the­ory around the TSO can be used to identi­fy the TSO with the so-called Quinn ob­struc­tion, which is $\mathbb{Z}$-val­ued. If this ob­struc­tion van­ishes, then we ob­tain that $BG$ is ho­mo­topy equi­val­ent to a closed $n$-di­men­sion­al to­po­lo­gic­al man­i­fold (see, for ex­ample, [e10]).

Fur­ther ap­plic­a­tions can be found in part II of the book [4], in­clud­ing re­cov­er­ing the pre­vi­ously known ex­amples of Poin­caré com­plexes that are not ho­mo­topy equi­val­ent to man­i­folds, such as the ex­ample of a 4-di­men­sion­al com­plex found in the 1960s by Wall in Ex­ample 22.28 of [4], and a dis­cus­sion of oth­er ver­sions of the TSO, not­ably a ver­sion for ANR ho­mo­logy man­i­folds.

We would like to men­tion that the proof of the iden­ti­fic­a­tion of the geo­met­ric and al­geb­ra­ic sur­gery ex­act se­quence is done in par­al­lel with the proof of the main the­or­em about the TSO. It has also the fol­low­ing co­rol­lary. The struc­ture in­vari­ant provides us with a bijec­tion $$s:{\mathcal{S}}^{\text{TOP}}(X)\stackrel{\cong }{\to }{\mathbb{S}}_{n+1}(X),$$ whose tar­get is an abeli­an group. As a res­ult we ob­tain an abeli­an group struc­ture on ${\mathcal{S}}^{\text{TOP}}(X)$ as well, but it is un­clear what is its geo­met­ric in­ter­pret­a­tion. This re­mains an in­ter­est­ing open prob­lem.

The the­ory and ap­plic­a­tions of TSO form an im­press­ive col­lec­tion of ideas that en­com­passes much of the area of clas­si­fic­a­tion of man­i­folds in to­po­logy. Many con­tri­bu­tions of An­drew Ran­icki start­ing from his thes­is were in the spir­it of try­ing to find con­cep­tu­al treat­ments and al­geb­ra­ic ex­plan­a­tions of vari­ous beau­ti­ful geo­met­ric phe­nom­ena that oc­cur here. It of­ten means that one needs to grasp a huge body of ma­ter­i­al, but one is re­war­ded by a deep un­der­stand­ing. The TSO is a prime ex­ample where all this was achieved.

Let us also not for­get that the TSO is still a top­ic of con­tem­por­ary re­search. For ex­ample, in 2011 Lurie pos­ted on­line lec­ture notes [e11], where he offered his own per­spect­ive on al­geb­ra­ic sur­gery and the TSO by provid­ing a treat­ment in the setup of $\mathrm{\infty }$-cat­egor­ies. This was taken up re­cently by oth­ers and a spec­tac­u­lar col­lab­or­a­tion ap­peared start­ing with [e15], where vari­ous as­pects of al­geb­ra­ic sur­gery were ap­plied to the study of the re­la­tion­ship between her­mitian $K$-the­ory and $L$-the­ory. Al­though this work has not yet dir­ectly ad­dressed the TSO, it is closely re­lated and might bring new in­sights in the fu­ture.

Be­sides provid­ing the to­po­logy com­munity with the vast body of in­ter­est­ing math­em­at­ic­al lit­er­at­ure An­drew Ran­icki was a cent­ral fig­ure in this area and had a very dis­tinct­ive pres­ence wherever he ap­peared. Much about his kind per­son­al­ity has been said at the on­line con­fer­ence in his hon­or in 2021. Just like many oth­er to­po­lo­gists the au­thor of this note be­nefited greatly from the in­ter­ac­tions with An­drew and from his gen­er­os­ity and hos­pit­al­ity. I es­pe­cially re­call the fall semester in 2008, when he gave a semester-long lec­ture course about al­geb­ra­ic sur­gery in Münster. Al­though I had known some of his the­ory be­fore, it was great to build on that, and also to study it with a large group of people in­clud­ing his stu­dents. It was just one mani­fest­a­tion of his last­ing pos­it­ive in­flu­ence on how the to­po­logy com­munity func­tions.

Thank you everything, An­drew!

Tibor Macko is an As­so­ci­ate Pro­fess­or at the Fac­ulty of Math­em­at­ics, Phys­ics and In­form­at­ics of the Comeni­us Uni­versity and a Re­search Fel­low at the In­sti­tute of Math­em­at­ics of the Slov­ak Academy of Sci­ences, both in Brat­is­lava, Slov­akia.