Celebratio Mathematica

We wish to hon­our the life and work of Pro­fess­or An­drew Ran­icki. An­drew ob­tained his BA in 1969 and PhD in 1972, both from Cam­bridge. He spent a fur­ther five years at Cam­bridge as a postdoc from 1972–1977. He was an as­sist­ant pro­fess­or in Prin­ceton from 1977–1982, be­fore mov­ing to Ed­in­burgh in 1982. An­drew worked at the Uni­versity of Ed­in­burgh from 1982 un­til his death in 2018. From 1995 he was the Pro­fess­or of Al­geb­ra­ic Sur­gery in the School of Math­em­at­ics, and from 1992 he was a Fel­low of the Roy­al So­ci­ety of Ed­in­burgh.

An­drew was awar­ded the Trin­ity Col­lege Yeats Prize in 1970, the Cam­bridge Uni­versity Smith Prize in 1972, and from the Lon­don Math­em­at­ic­al So­ci­ety, both the Ju­ni­or White­head Prize in 1983 and the Seni­or Ber­wick Prize in 1994.

From 2008 to 2017 he served as an ed­it­or of the journ­al Pro­ceed­ings of the Roy­al So­ci­ety of Ed­in­burgh, Sec­tion A: Math­em­at­ics. For four of those years An­drew was ed­it­or-in-chief, and was in­stru­ment­al in mod­ern­ising the journ­al’s ed­it­or­i­al pro­cesses and broad­en­ing its math­em­at­ic­al scope. A spe­cial is­sue in his hon­our re­cently ap­peared in the journ­al.

An­drew and his wife Ida’s le­gendary hos­pit­al­ity il­lu­min­ated Scot­tish to­po­logy, and at­trac­ted a range of pres­ti­gi­ous vis­it­ors to Ed­in­burgh, who en­joyed the food, the garden, the pic­tures, the in­fec­tious laugh, the whis­key, and above all the com­pany.

An­drew was an at­tent­ive and warm ad­visor for twelve PhD stu­dents, whom he sup­por­ted far bey­ond the norm, of­ten wel­com­ing them in­to his home and fa­cil­it­at­ing their in­tro­duc­tion in­to the math­em­at­ic­al world through his net­work of con­tacts.

Pro­fes­sion­ally, An­drew was a huge con­trib­ut­or to the com­munity. He was an ed­it­or at mul­tiple journ­als, wrote sev­en books, ed­ited twelve col­lec­tions of art­icles, and or­gan­ised nu­mer­ous con­fer­ences and work­shops. His web­site re­mains a re­source for the study of al­geb­ra­ic and geo­metry to­po­logy. He served for many years as Dir­ect­or of the Gradu­ate School of the Ed­in­burgh Uni­versity School of Math­em­at­ics.

An­drew loved sur­gery the­ory, which is the primary tool for clas­si­fy­ing man­i­folds, par­tic­u­larly those of di­men­sion five and high­er, the so-called high di­men­sions. He was also fas­cin­ated by knot the­ory, again par­tic­u­larly in high di­men­sions. Among to­po­lo­gists, An­drew was fam­ous for his al­geb­ra­ic skills. He pos­sessed an un­canny abil­ity to dis­cov­er the right al­geb­ra­ic for­mu­lae that both cap­tured the ne­ces­sary geo­met­ric phe­nom­ena and led to a clear­er un­der­stand­ing of a prob­lem.

An­drew’s PhD in the early 1970s was su­per­vised by Frank Adams, one of the lead­ers in Brit­ish to­po­logy at that time, and un­of­fi­cially by An­drew Cas­son, who did not him­self have a PhD (but whose con­tri­bu­tions have been worth many PhDs). An­drew stud­ied the sur­gery the­ory of Browder, Novikov, Sul­li­van, and Wall. In his early work he de­veloped the al­geb­ra­ic the­ory of sur­gery, after which his pro­fess­or­ship was later named.

In sur­gery the­ory, one starts with a fi­nite CW-com­plex \( X \) that looks to the eyes of glob­al al­geb­ra­ic to­po­logy like an \( n \)-di­men­sion­al man­i­fold, in oth­er words, \( X \) sat­is­fies Poin­caré du­al­ity. Such a com­plex is called a Poin­caré com­plex. The driv­ing ques­tion in man­i­fold clas­si­fic­a­tion is: does there ex­ist an \( n \)-man­i­fold ho­mo­topy equi­val­ent to \( X \), and if so how many are there up to homeo­morph­ism/dif­feo­morph­ism? Sur­gery the­ory of­fers a three-step ap­proach to an­swer this, when the di­men­sion \( n \geq 5 \). A Poin­caré com­plex \( X \) has a ca­non­ic­al spher­ic­al fibra­tion called the Spivak bundle, and step one is to see if it re­duces to a \( \mathbb{R}^n \)-bundle. If so, one can ap­ply trans­vers­al­ity to ap­prox­im­ate \( X \) by a de­gree one nor­mal map \( f \colon M \to X \). Bor­d­ism classes of de­gree one nor­mal maps over \( X \) are de­noted \( \mathcal{N}(X) \). The second step is to per­form sur­gery on \( M \) be­low the middle di­men­sion to make \( f \) in­to an \( [n/2] \)-con­nec­ted map \( f^{\prime} \colon M^{\prime} \to X \). One ob­tains an ele­ment in the \( L \)-group \( L_n(\mathbb{Z}[\pi_1(X)]) \), which is the re­main­ing ob­struc­tion to do­ing sur­gery in the middle di­men­sion. This gives rise to the sur­gery ob­struc­tion map \[\sigma \colon \mathcal{N}(X) \to L_n(\mathbb{Z}[\pi_1(X)]).\] If the sur­gery ob­struc­tion \( \sigma(f^{\prime}) \) van­ishes, then one can pro­ceed with the third step and sur­ger \( M^{\prime} \) along \( [n/2] \)-spheres to ob­tain a ho­mo­topy equi­val­ence \( f^{\prime\prime} \colon M^{\prime\prime} \to X \), giv­ing a pos­it­ive an­swer to the ex­ist­ence part of the man­i­fold clas­si­fic­a­tion ques­tion above.

Via the \( s \)-cobor­d­ism the­or­em, unique­ness is a re­l­at­ive form of ex­ist­ence, and so the three-stage ob­struc­tion the­ory just out­lined can in prin­ciple clas­si­fy the homeo­morph­ism classes in a fixed ho­mo­topy type too.

We are now in a po­s­i­tion to in­dic­ate An­drew’s early con­tri­bu­tions to sur­gery the­ory: (1) the \( L \)-the­ory of Laurent poly­no­mi­al rings; (2) the no­tion of an al­geb­ra­ic Poin­caré com­plex; and (3) the al­geb­ra­ic bor­d­ism ex­act se­quences. All three have been fun­da­ment­al tools for sur­geons ever since.

Shaneson and Novikov com­puted \( L_n(\mathbb{Z}[\pi_1(X \times S^1)]) \) us­ing geo­met­ric tools and vocab­u­lary, while An­drew re­did these fun­da­ment­al com­pu­ta­tions purely al­geb­ra­ic­ally, com­put­ing \( L_n(R[x,x^{-1}]) \) for a ring with in­vol­u­tion \( R \). This was done in his four pa­pers on al­geb­ra­ic \( L \)-the­ory [2], [3], [1], [4].

Mo­tiv­ated by Steen­rod’s earli­er work on co­homo­logy op­er­a­tions, An­drew al­geb­rat­ised the no­tion of a Poin­caré com­plex, de­fin­ing a sym­met­ric al­geb­ra­ic Poin­caré com­plex. Not only did this elu­cid­ate the key al­geb­ra­ic prop­er­ties of a man­i­fold, it led to the no­tion of a quad­rat­ic Poin­caré com­plex, which al­lowed his for­mu­la­tion of the in­stant sur­gery ob­struc­tion, com­bin­ing steps two and three of the sur­gery pro­gram above. The defin­i­tion and the­ory of al­geb­ra­ic Poin­caré com­plexes were de­veloped in his pa­pers “The al­geb­ra­ic the­ory of sur­gery: I, II” [5], [6].

Once he had al­geb­ra­ic ver­sions of Poin­caré du­al­ity, he could define al­geb­ra­ic bor­d­ism groups. This was in­cred­ibly im­port­ant. Wall’s defin­i­tion of his \( L \)-groups de­pended on the con­gru­ence class of \( n \) mod­ulo 4, but An­drew’s bor­d­ism defin­i­tion of the \( L \)-groups was in­de­pend­ent of the di­men­sion. Also, giv­en the par­al­lel work of Bass, Quil­len, and Wald­hausen in al­geb­ra­ic \( K \)-the­ory, and geo­met­ric work of Shaneson, Novikov, and Cap­pell in al­geb­ra­ic \( L \)-the­ory, it was clear that ex­act se­quences were needed as a found­a­tion­al com­pu­ta­tion­al and geo­met­ric tool. An­drew’s bor­d­ism the­ory provided the re­quis­ite ex­act se­quences of pairs and maps. The mod­estly, but aptly, named book Ex­act Se­quences in the Al­geb­ra­ic The­ory of Sur­gery [7] has been a vi­tal ref­er­ence ever since its pub­lic­a­tion.

Not con­tent with com­bin­ing two out of three steps in the sur­gery pro­gramme, An­drew stud­ied Quinn’s sur­gery spec­tra \( \mathbb{L} \), and, by trans­lat­ing them in­to al­gebra, for­mu­lated the al­geb­ra­ic sur­gery ex­act se­quence and the total sur­gery ob­struc­tion. The total sur­gery ob­struc­tion gives a single ob­struc­tion that en­cap­su­lates the en­tire sur­gery pro­gramme, and was the top­ic of his book Al­geb­ra­ic \( L \)-The­ory and To­po­lo­gic­al Man­i­folds [8].

One of the key dif­fi­culties in ap­ply­ing the sur­gery pro­gramme to clas­si­fy man­i­folds is to eval­u­ate the sur­gery ob­struc­tion map \( \sigma \). As part of his work on the total sur­gery ob­struc­tion, An­drew re­fined Quinn’s geo­met­ric as­sembly maps in­to al­geb­ra­ic ver­sions. As a res­ult the map \( \sigma \) factors in­to a com­pos­i­tion of two maps: the first com­pares the \( \mathbb{L} \)-ho­mo­logy of \( X \) and its clas­si­fy­ing space \( B\pi_1(X) \), and is in prin­ciple com­put­able us­ing spec­tral se­quences. The second map is An­drew’s al­geb­ra­ic as­sembly map. This led to a re­volu­tion in our un­der­stand­ing of \( \sigma \), through sub­sequent work by many oth­ers on iso­morph­ism con­jec­tures in \( K \)- and \( L \)-the­ory, in par­tic­u­lar the Far­rell–Jones con­jec­tures. These con­jec­tures state that a suit­able re­fine­ment of the as­sembly map is an iso­morph­ism, and have been proven for many fun­da­ment­al groups. Armed with this know­ledge, in for­tu­it­ous cases one has a chance of cal­cu­lat­ing the map \( \sigma \) and ex­pli­citly clas­si­fy­ing man­i­fold struc­tures on a giv­en Poin­caré com­plex \( X \).

An­drew’s goal was usu­ally to con­vert to­po­logy in­to al­gebra, however his form­al­ism of sym­met­ric and quad­rat­ic com­plexes has re­cently been re-pur­posed in the op­pos­ite dir­ec­tion, in the drive to “spa­ci­fy” al­geb­ra­ic \( K \) and \( L \)-the­ory in­to the lan­guage of the \( \infty \)-cat­egory of spec­tra. He showed great pres­ci­ence in mak­ing defin­i­tions that math­em­aticians would con­tin­ue to use dec­ades later.

While the al­geb­ra­ic the­ory of sur­gery and the total sur­gery ob­struc­tion are his most fam­ous legacies, An­drew had broad in­terests in al­geb­ra­ic as­pects of man­i­fold to­po­logy. Oth­er not­able out­put in­cludes his books High Di­men­sion­al Knot The­ory [9] and Al­geb­ra­ic and Geo­met­ric Sur­gery [10] and his res­ults on loc­al­isa­tion in \( K \)- and \( L \)-the­ory, the al­geb­ra­ic the­ory of tor­sion, con­trolled \( K \)- and \( L \)-the­ory, mul­ti­plic­ativ­ity of the sig­na­ture, the ad­vent of al­geb­ra­ic trans­vers­al­ity, cal­cu­lat­ing \( \operatorname{Nil} \) and \( \operatorname{UNil} \) groups, and Morse the­ory for man­i­folds with bound­ary.

An­drew was an in­cred­ibly so­cial in­di­vidu­al, which in­flu­enced his ap­proach to math­em­at­ics. He loved to col­lab­or­ate; in fact, he col­lab­or­ated with 37 math­em­aticians al­to­geth­er.

An­drew took par­tic­u­lar pride in his stu­dents and their achieve­ments, on which of course he had a great in­flu­ence. They worked on a range of top­ics in sur­gery the­ory and knot the­ory. The PhD thes­is of his first stu­dent Des­mond Shei­ham com­pletely com­puted the al­geb­ra­ic con­cord­ance group of bound­ary links. Jo­erg Sixt de­veloped the al­geb­ra­ic the­ory of mod­i­fied sur­gery, and wrote an in­flu­en­tial pa­per with Di­ar­muid Crow­ley on clas­si­fy­ing man­i­folds with­in their stable class. Jeremy Brook­man worked on split­ting ho­mo­topy equi­val­ences along codi­men­sion one sub­man­i­folds and the \( \operatorname{UNil} \) groups, and was part of a team with Jim Dav­is and Qay­um Khan that com­puted the struc­ture set of man­i­folds ho­mo­topy equi­val­ent to \( \mathbb{R}P^n \mathbin{\#} \mathbb{R}P^n \). An­drew Korzeniewski worked on ab­so­lute White­head tor­sion, and with Ian Hamb­leton and An­drew proved the mul­ti­plic­ativ­ity of the sig­na­ture in fibre bundles mod 4. Ju­lia Collins stud­ied knot con­cord­ance, and com­puted the sub­group of the knot con­cord­ance group gen­er­ated by the knots of cross­ing num­ber at most nine. Mark Pow­ell took a multi-stage ob­struc­tion the­ory for knot con­cord­ance, and cre­ated a single ob­struc­tion us­ing sym­met­ric chain com­plexes. Mark is cur­rently a pro­fess­or at the Uni­versity of Glas­gow, and an ed­it­or of both the spe­cial is­sue of the Roy­al Pro­ceed­ings of the Roy­al So­ci­ety of Ed­in­burgh hon­our­ing Ran­icki and of Cel­eb­ra­tio Math­em­at­ica’s An­drew Ran­icki volume. Spiros Adams-Flor­ou re­lated con­trolled and bounded to­po­logy, which are tech­nic­al in­gredi­ents un­der­ly­ing An­drew’s con­struc­tion of the \( \mathbb{L} \)-spec­tra, and ap­plied his res­ults to prove a re­cog­ni­tion the­or­em for ho­mo­logy man­i­folds in terms of bounded Poin­caré du­al­ity. Patrick Or­son de­veloped the the­ory of double \( L \)-groups, in­clud­ing a loc­al­isa­tion ex­act se­quence, and ap­plied it to study doubly slice knots. Patrick is now an as­sist­ant pro­fess­or at Cal­Poly in San Lu­is Obispo, Cali­for­nia. Chris Palmer ap­plied al­geb­ra­ic sur­gery to tri­sec­tions of 4-man­i­folds, the \( L \)-the­ory of mat­rix rings, and to Seifert matrices of braids. Car­men Rovi re­vis­ited the mul­ti­plic­ativ­ity of the sig­na­ture in fibre bundles, and char­ac­ter­ised ex­actly when one has mul­ti­plic­ativ­ity mod 8. Car­men is now an as­sist­ant pro­fess­or at Loy­ola Uni­versity in Chica­go. An­drew’s last stu­dent Supreedee Dang­skul worked on Seifert sur­faces through the lens of dif­fer­en­tial geo­metry. Supreedee is now an as­sist­ant pro­fess­or at Chi­ang Mai Uni­versity in Thai­l­and.

A test­a­ment to the kind of su­per­visor An­drew was could be seen in the num­ber of his former stu­dents who at­ten­ded both his 60th birth­day and re­tire­ment con­fer­ences, in­clud­ing sev­er­al former PhD stu­dents who had been work­ing in in­dustry for some time.

From the au­thors of this trib­ute, Mark Pow­ell is par­tic­u­larly grate­ful for An­drew’s con­fid­ence, sup­port, and hos­pit­al­ity, and Jim Dav­is is par­tic­u­larly grate­ful for the kind­ness and guid­ance An­drew showed him as a postdoc ment­or at Prin­ceton.

At the time of writ­ing, An­drew’s per­son­al web­site is still main­tained by the Ed­in­burgh School of Math­em­at­ics. It is well worth per­us­ing. Here is a link to the spe­cial is­sue of the Pro­ceed­ings of the Roy­al So­ci­ety of Ed­in­burgh fea­tur­ing art­icles writ­ten in his hon­our.