Celebratio Mathematica

Andrew A. Ranicki

Apprenticeship with an algebraic surgeon

by Fabian Hebestreit and Markus Land

Early interactions

We both be­came dis­ciples of An­drew Ran­icki dur­ing our early edu­ca­tion on sur­gery the­ory: in the case of the first au­thor dir­ectly through lec­tures on the top­ic by An­drew, and in the case of the second au­thor mostly by proxy Wolfgang Lück, mo­tiv­ated by the re­la­tion between sur­gery the­or­et­ic and in­dex the­or­et­ic ap­proaches to the Novikov con­jec­ture. As usu­al, our first in­ter­ac­tions with An­drew were little chats at con­fer­ences and sum­mer schools; however, our com­mu­nic­a­tion in­tens­i­fied over time. We both had many email ex­changes and video calls with him, seek­ing in­put on our work, try­ing to de­cipher some writ­ing of his or look­ing for a ref­er­ence from the golden days of geo­met­ric to­po­logy. These in­ter­ac­tions gave us many fond memor­ies: for a good couple of months when the second au­thor was a postdoc in Re­gens­burg, the first thing to do in the morn­ing was to check Skype and have a brief video call with An­drew about math­em­at­ics and also life in gen­er­al. At al­most all times of day, his replies to any and all ap­proaches would be al­most im­me­di­ate, ir­re­spect­ive of wheth­er he was sup­posed to be oth­er­wise en­gaged. For ex­ample, he would oc­ca­sion­ally an­swer, com­plete with an ac­com­pa­ny­ing im­age or video snip­pet he thought amus­ing, while in the audi­ence of sem­inars or con­fer­ence talks. In ad­di­tion to an­swer­ing on the spot, An­drew’s replies were al­ways ex­tremely friendly and in­vit­ing, some­times to his own det­ri­ment. Here is an amus­ing in­stance of this. Once, in the middle of the night, we asked him (amongst oth­er things) via email about a sup­posedly miss­ing fi­nite­ness con­di­tion in the defin­i­tion of geo­met­ric nor­mal spaces in one of his books. Hav­ing missed the an­swer to our ques­tion in his al­most in­stant­an­eous reply, we un­re­lent­ingly re­peated the ques­tion sev­er­al times, ad­mit­tedly after a light ale or two, in ever lar­ger fonts. In­stead of call­ing out our mis­de­mean­our, An­drew po­litely en­gaged in this “ex­change” for the bet­ter part of an hour be­fore fi­nally giv­ing up on us for the first (and to the best of our memory only) time. Shortly after, ap­par­ently not fed up with our antics, he in­vited the two of us to vis­it him and his wife Ida at their home in Ed­in­burgh in Feb­ru­ary 2018, a meet­ing about which we shall say more later.

An­drew’s use of Poin­caré chain com­plexes in sur­gery the­ory provided what im­me­di­ately seemed to us the most nat­ur­al way to cap­ture sig­na­ture-like in­vari­ants in to­po­logy, such as the sur­gery ob­struc­tion of a de­gree one nor­mal map and An­drew’s sym­met­ric sig­na­ture of a closed ori­ented man­i­fold, were it not for the long and in­tim­id­at­ing ex­pli­cit for­mu­lae that An­drew was so fond of. We urge the in­cred­u­lous read­er to pick five pages from his book Ex­act Se­quences in the Al­geb­ra­ic The­ory of Sur­gery at ran­dom to see why it earned the af­fec­tion­ate nick­name “the phone book”, yel­low cov­er aside. While study­ing his work, many times the last re­sort was to write him an email with a call for aid, which in his char­ac­ter­ist­ic man­ner he would of­ten enough grant al­most im­me­di­ately, as al­luded to above. Full of en­ergy he would ex­plain the mean­ing un­der­ly­ing his for­mu­lae only to then res­ist any at­tempt at re­cast­ing them in more con­cep­tu­al terms.

With time it be­came pos­sible for us to give our own proofs of sev­er­al of his res­ults, in a frame­work of L-the­ory first sug­ges­ted and stud­ied by Jac­ob Lurie, which al­lows for much smooth­er treat­ments in many places (at least to our minds). Upon ex­tract­ing ex­pli­cit for­mu­lae from said frame­work (and neg­lect­ing signs) one would of course find ex­actly what An­drew had writ­ten down dec­ades earli­er. His abil­ity to guess and in­ter­pret ex­pli­cit chain-level con­struc­tions a pri­ori, however, re­mains a mys­tery to this day. His guid­ing hand and his eager­ness to pass on his in­tu­ition in this sub­ject is sorely missed. To this day, we find res­ults in his writ­ing that seem closely re­lated to ques­tions we have, or to res­ults we have ob­tained, if only we could eas­ily peer through his for­mu­lae.

How his work influenced ours

Prob­ably one of An­drew’s most prom­in­ent res­ults is the iden­ti­fic­a­tion of the geo­met­ric sur­gery ex­act se­quence for to­po­lo­gic­al man­i­folds with his al­geb­ra­ic sur­gery ex­act se­quence based on the as­sembly map in quad­rat­ic L-the­ory, which in par­tic­u­lar paved the way for at­tack­ing Borel’s ri­gid­ity con­jec­ture for as­pher­ic­al man­i­folds (pre­dict­ing that any ho­mo­topy equi­val­ence between such is ho­mo­top­ic to a homeo­morph­ism). As part of this pro­gram, An­drew showed that L-the­or­et­ic Poin­caré du­al­ity sets to­po­lo­gic­al man­i­folds apart from gen­er­al Poin­caré com­plexes. He ul­ti­mately used this rather subtle ob­ser­va­tion to de­vel­op the total sur­gery ob­struc­tion — a single in­vari­ant as­so­ci­ated to a Poin­caré com­plex which van­ishes if and only if it is ho­mo­topy equi­val­ent to a closed to­po­lo­gic­al man­i­fold (as long as the di­men­sion is at least 5, and along with a re­l­at­ive ver­sion de­tect­ing homeo­morph­isms among ho­mo­topy equi­val­ences). His found­a­tion­al in­sights in this dir­ec­tion are prob­ably most suc­cinctly sum­mar­ised by what we have come to call Ran­icki’s sig­na­ture square, that is, \[ \begin{CD} \Omega^\mathrm{STop} @>{{\sigma^\mathrm{s} }}>{{}}> \mathrm{L}^\mathrm{s}(\mathbb{Z})\\ @VV\mathrm{fgt}V @VV\mathrm{fgt}V\\ \Omega^{\mathrm{S}\mathrm{G}} @>{{\sigma^\mathrm{n} }}>{{}}> \mathrm{L}^\mathrm{n}(\mathbb{Z}) \end{CD} \]

where \( \Omega^\mathrm{STop} \) de­notes the ori­ented to­po­lo­gic­al cobor­d­ism spec­trum, \( \Omega^{\mathrm{S}\mathcal{G}} \) de­notes Quinn’s ori­ented nor­mal cobor­d­ism spec­trum, and the right-hand terms are the sym­met­ric and nor­mal L-spec­tra of \( \mathbb{Z} \) ana­log­ously defined by Ran­icki us­ing cobor­d­isms of chain com­plexes with du­al­ity in place of man­i­folds. Let us ex­pli­citly men­tion that the right-hand ob­jects evolved with An­drew’s work and con­sequently there are a mul­ti­tude of dif­fer­ent ver­sions of these spec­tra. This is a fant­ast­ic source of con­fu­sion and will be­come rel­ev­ant be­low; in fact, we man­aged to get ourselves rather con­fused yet again while writ­ing this very note. The spec­tra above are (hope­fully) de­noted \( \mathrm{L}^{\cdot}(\mathbb{Z}) \) and \( \widehat{\mathrm L}^\cdot(\mathbb{Z}) \) in An­drew’s Al­geb­ra­ic L-The­ory and To­po­lo­gic­al Man­i­folds. Re­gard­less, the top ho­ri­zont­al map in the above square is the fam­ous Sul­li­van–Ran­icki ori­ent­a­tion, which takes a man­i­fold to its sym­met­ric sig­na­ture, es­sen­tially its (co)chain com­plex to­geth­er with its Poin­caré du­al­ity equi­val­ence (Sul­li­van had done something sim­il­ar away from the prime 2 earli­er us­ing real to­po­lo­gic­al K-the­ory in­stead of sym­met­ric L-the­ory). It im­ple­ments the L-the­or­et­ic Poin­caré du­al­ity for ori­ented to­po­lo­gic­al man­i­folds men­tioned above, and there­fore provides the fun­da­ment­al class in \( \mathrm{L}^\mathrm{s}(\mathbb{Z}) \)-ho­mo­logy of ori­ented to­po­lo­gic­al man­i­folds. In par­tic­u­lar it lifts the sym­met­ric sig­na­ture of a to­po­lo­gic­al man­i­fold along the as­sembly map in sym­met­ric L-the­ory and the total sur­gery ob­struc­tion meas­ures ex­actly wheth­er such a lift ex­ists for a gen­er­al Poin­caré com­plex. The lower map as­signs to one of Quinn’s nor­mal spaces its nor­mal sig­na­ture and provides a fun­da­ment­al class in \( \mathrm{L}^\mathrm{n}(\mathbb{Z}) \)-ho­mo­logy for all ori­ented Poin­caré com­plexes via the map \( \Omega^{\mathrm{SP}} \rightarrow \Omega^{\mathrm{SG}} \) from ori­ented Poin­caré cobor­d­ism.

The total sur­gery ob­struc­tion now meas­ures ex­actly wheth­er this nor­mal fun­da­ment­al class can be lif­ted to a fun­da­ment­al class in sym­met­ric L-ho­mo­logy, and in the re­l­at­ive case of a de­gree 1 nor­mal map among man­i­folds, the in­vari­ance of the nor­mal sig­na­ture un­der nor­mal (and not just Poin­caré) cobor­d­isms al­lows one to ex­tract a class in \( \mathrm{L}^\mathrm{q}(\mathbb{Z}) \)-ho­mo­logy (the quad­rat­ic L-spec­trum \( \mathrm{L}^\mathrm{q}(\mathbb{Z}) \) is the fibre of the right-hand ver­tic­al map) whose im­age un­der the as­sembly map is the sur­gery ob­struc­tion. This is the in­road that con­nects the al­geb­ra­ic sur­gery se­quence to the to­po­lo­gic­al one.

Many years later and in some­what vague form, Hes­sel­holt, Mad­sen, Wil­li­ams and oth­ers con­jec­tured a re­la­tion between L-the­ory and al­geb­ra­ic K-the­ory — linked by what is re­ferred to vari­ously as real al­geb­ra­ic K-the­ory, her­mitian K-the­ory, or Grothen­dieck–Witt the­ory (the lat­ter is the ter­min­o­logy we ad­op­ted). It is in our work on these con­jec­tures that An­drew’s teach­ings really paid off: over the last two dec­ades the geo­met­ric cobor­d­ism spec­tra ap­pear­ing in the sig­na­ture square above have been re­fined to great ef­fect by the com­bined ef­forts of Gala­ti­us, Mad­sen, Ran­dal-Wil­li­ams, Till­mann, and Weiss to what is known as cobor­d­ism cat­egor­ies, and we (along with our col­lab­or­at­ors Calmès, Dotto, Harpaz, Moi, Nardin, Nikolaus and Steimle) real­ised that one could use An­drew’s ideas to pro­duce al­geb­ra­ic cobor­d­ism cat­egor­ies that sim­il­arly re­fine his L-spec­tra. Us­ing the meth­od of al­geb­ra­ic sur­gery he pi­on­eered for the ana­lys­is of L-groups, we showed that these al­geb­ra­ic cobor­d­ism cat­egor­ies pre­cisely give rise to Grothen­dieck–Witt spec­tra. Just as al­geb­ra­ic sur­gery is an im­it­a­tion of geo­met­ric sur­gery at the level of chain com­plexes, we could then trans­port well known geo­met­ric tech­niques con­nect­ing cobor­d­ism cat­egor­ies and cobor­d­ism spec­tra in­to the al­geb­ra­ic world, and con­nect Grothen­dieck–Witt and L-spec­tra. This cul­min­ated in a gen­er­al fibre se­quence of the form \[\mathrm{K}(R)_{\mathrm{hC}_2} \xrightarrow{\operatorname{hyp}} \mathrm{GW}^{\mathrm{s}}(R) \xrightarrow{\operatorname{bord}} \mathrm{L}^\mathrm{s}(R).\] A word of warn­ing is in or­der here. As in­dic­ated earli­er, An­drew gave two com­pet­ing defin­i­tions of sym­met­ric L-groups for gen­er­al rings, one 4-peri­od­ic and one not gen­er­ally so. After es­tab­lish­ing that these do not dif­fer for the in­tegers, at least in non­neg­at­ive de­grees, he even­tu­ally fa­voured the peri­od­ic ver­sion for its sup­posed sim­pli­city. In Lurie’s form­al­ism, on which our work is based, both of An­drew’s defin­i­tions have nat­ur­al and dis­tinct mean­ing. It turns out, though very much not by defin­i­tion, that it is the ori­gin­al non­peri­od­ic ver­sion which ap­pears in the fibre se­quence above; to dis­tin­guish them we usu­ally de­note this ver­sion by \( \mathrm{L}^{\mathrm{gs}}(R) \) out­side this note. This non­peri­od­ic ver­sion was far less known out­side Ran­icki’s in­ner circle, and per­haps this mis­match ex­plains why con­jec­tures about the re­la­tion between K- and L-the­ory had pre­vi­ously re­mained vague. Fur­ther­more, the non­peri­od­ic L-groups are only really defined in non­neg­at­ive de­grees, and in his early works An­drew noted that they can be spliced in an ad hoc man­ner with quad­rat­ic and sym­plect­ic L-groups to ex­tend cer­tain long ex­act se­quences also in­to neg­at­ive de­grees. To the best of our know­ledge, quite which long ex­act se­quences he meant was nev­er made ex­pli­cit, but our set-up ca­non­ic­ally provides a defin­i­tion of neg­at­ive L-groups as well. As a typ­ic­al in­stance of Ran­icki’s ma­gic abil­ity to guess for­mu­lae, these turn out to be ex­actly the quad­rat­ic and sym­plect­ic L-groups. Moreover, this iden­ti­fic­a­tion has very sur­pris­ing con­sequences: for ex­ample, it im­plies that the neg­at­ive sym­met­ric Grothen­dieck–Witt-groups identi­fy with quad­rat­ic and sym­plect­ic L-groups, and are not dir­ectly re­lated to the (peri­od­ic) sym­met­ric L-groups even in the case of the in­tegers.

At any rate, us­ing this new re­la­tion between Grothen­dieck–Witt and L-the­ory, we were able to al­most com­pletely de­term­ine the quad­rat­ic and sym­met­ric Grothen­dieck–Witt groups of num­ber rings and re­solve peri­od­icity con­jec­tures of Ka­roubi — in fact, these con­jec­tures trans­late ex­actly to state­ments in L-the­ory An­drew had provided dec­ades ago, when com­par­ing his two com­pet­ing defin­i­tions of sym­met­ric L-groups. Un­sur­pris­ingly, a pleth­ora of open ques­tions and on­go­ing pro­jects re­main, not least of which is a re­fine­ment of his sig­na­ture square to the level of cobor­d­ism cat­egor­ies. We would have loved to hear An­drew’s thoughts about all of them.

What we have de­scribed above is cer­tainly not the only point where our work in­ter­sec­ted An­drew’s, but we hope to have con­veyed an im­pres­sion of the feel­ing of work­ing un­der his wing and of the in­flu­ence he had on our math­em­at­ic­al think­ing.

Our last meeting

We told An­drew about early forms of our ideas (in fact as two sep­ar­ate pro­jects, whose over­lap and full im­pact we had not yet dis­covered), when we vis­ited him and Ida at their home in Feb­ru­ary 2018. Des­pite An­drew’s already ad­van­cing leuk­aemia their hos­pit­al­ity was simply stun­ning, as prob­ably every one of the many math­em­aticians (and prob­ably non­mathem­aticians) to stay at their house will have ex­per­i­enced as well.

Dur­ing our vis­it we joined An­drew to what we as­sumed would be a rather routine check-up in the hos­pit­al, and the three of us must have made a strange sight, ex­citedly chat­ting about math­em­at­ics in the can­cer ward, with his laughter filling more the just the room we oc­cu­pied. Un­for­tu­nately, the res­ults of the check-up ended up be­ing any­thing but routine and An­drew died just two days after we left.

That we can no longer share our ex­cite­ment for the de­vel­op­ments in the re­la­tion between high­er al­gebra and geo­met­ric to­po­logy, in­to our part of which An­drew was kind enough to pour so much of his en­ergy, leaves a hole that has not been filled to this day.

Du fehlst.