Celebratio Mathematica

Colin P. Rourke

Colin Patrick Rourke: a short biography

by Rob Kirby

Colin Patrick Rourke was born on 1 Janu­ary 1943, in Al­trin­cham, a sub­urb of Manchester. It was there that his fath­er, Fran­cis Louis Al­bert Rourke, in­jured in the Dunkirk evac­u­ation, was con­vales­cing. His moth­er, Gladys Pau­line Rourke (née Levy), had moved up from Lon­don to be with her hus­band along with Colin’s older sis­ter, Sue, then a baby.

Fran­cis was a “quant­ity sur­vey­or”; he worked on build­ing sites mak­ing sure that the build­ers were us­ing the cor­rect amounts of ma­ter­i­als. One might think that Colin’s le­gendary build­ing skills came from his fath­er, but Fran­cis left the fam­ily when Colin was an in­fant.

Gladys’ fam­ily were all tail­ors, second gen­er­a­tion Jew­ish im­mig­rants from Rus­sia, who had shortened the fam­ily name to Ley to sound less Jew­ish. Gladys and her chil­dren even­tu­ally moved back to Lon­don, near Hamp­stead Heath, and she went to work in High Hol­born (near Lin­coln’s Inn Fields) at a shop spe­cial­iz­ing in clothes for the leg­al pro­fes­sion. That shop was owned by a broth­er, Stan­ley Ley, and still ex­ists, though it has since re­lo­cated to Fleet Street, closer to the Roy­al Courts of Justice. Colin grew up close to the Par­lia­ment Hill Lido, a large open air swim­ming pool, and he swam there fre­quently. He at­ten­ded Highg­ate, a pub­lic school, with a schol­ar­ship. At Highg­ate it was known that Colin was good at maths, so he was en­cour­aged to try for a Trin­ity schol­ar­ship a year early. His math teach­er gave him a long list of past ex­ams and Colin would sit in a corner dur­ing math classes do­ing prob­lems from these ex­ams. When it came time to sit the Trin­ity geo­metry ex­am, he had already seen eight of the ten prob­lems (from his hours in the corner), so he had plenty of time to work on the oth­er two!

Colin once ob­served that he was not “pushed” as a child, was not a prodigy, and thus was not “scarred for life”. Yet in 1960 at the age of 17 he ma­tric­u­lated at Trin­ity Col­lege in Cam­bridge. Moreover, be­cause he had done so well on the ex­ams, he skipped Part I of the un­der­gradu­ate cur­riculum, did Part II over the course of two years, and then Part III (gradu­ate courses) in his third year. Life im­proved dur­ing Colin’s second year, for Daphne Cale came to Newn­ham Col­lege as a maths un­der­gradu­ate. They were mar­ried in 1963.

Among his teach­ers were Chris­toph­er Zee­man, Terry Wall and Dav­id Ep­stein. (The “great ex­odus” had not yet happened; at that time Cam­bridge lim­ited it­self to four pro­fess­or­ships in math­em­at­ics, so Zee­man left to found the Maths In­sti­tute at War­wick in 1964, Wall went to Liv­er­pool, Ep­stein to War­wick, Mi­chael Atiyah to Ox­ford, and Frank Adams to Manchester.

While a gradu­ate stu­dent Colin took a job as as­sist­ant lec­turer at Queen Mary Col­lege (QMC) in Lon­don in 1964. He had not yet re­ceived his PhD, but needed the money as he and Daphne had their first child, Jane, in 1964. Jane was fol­lowed by Jon in 1965, Si­mon in 1966, and then Beth in 1973.

Colin be­came a full lec­turer after win­ning the Raleigh Prize. While at QMC he began col­lab­or­at­ing with Bri­an Sander­son, got a PhD in 1965, and re­ceived a prize fel­low­ship at Trin­ity (taken up for only six months).

Colin met Sander­son when Zee­man asked Colin to come along to an Ober­wolfach meet­ing. Colin’s thes­is was (in his own words) “something ob­scure”, but he needed a no­tion of PL Grass­mani­ans and Zee­man told him to con­tact Bri­an. It turned out that Bri­an did not have an an­swer, but their col­lab­or­a­tion led to block bundles [2] [3], a PL ana­logue of vec­tor bundles in the smooth case and to­po­lo­gic­al bundles in that cat­egory. This work, plus a loc­ally flat em­bed­ding without a nor­mal mi­crobundle [1], was com­pleted be­fore Colin left Queen Mary and Zee­man’s of­fer to come to War­wick.

After three years at QMC, Colin spent a year at the In­sti­tute for Ad­vanced Study (IAS) in Prin­ceton. While there he lec­tured in Deane Mont­gomery’s sem­in­ar on an ex­pos­i­tion of Sul­li­van’s Hauptver­mu­tung, in a new ver­sion com­mu­nic­ated to him by Bri­an Sander­son, which re­lied con­sid­er­ably on An­drew Cas­son’s for­mu­la­tion of the proof from his fel­low­ship dis­ser­ta­tion. Colin’s notes on these lec­tures “The Hauptver­mu­tung ac­cord­ing to Cas­son and Sul­li­van” are pub­lished in [7], which also con­tains Cas­son’s fel­low­ship dis­ser­ta­tion. Dur­ing that year, Colin wrote a pa­per with Den­nis Sul­li­van [4] “On the Ker­vaire Ob­struc­tion” con­tain­ing a proof of the product for­mula needed for the Hauptver­mu­tung and the ho­mo­topy equi­val­ence between and \( \mathrm{G}/\mathrm{PL} \) and \( \Omega^4 \mathrm{G}/\mathrm{PL} \).

In 1968 he joined Bri­an Sander­son at the Uni­versity of War­wick (Sander­son had ar­rived in 1965) and re­mained there, and this began a long peri­od of col­lab­or­a­tion.

Bri­an had ob­tained his PhD at Liv­er­pool in 1963, work­ing with Michaael Atiyah, while col­lab­or­at­ing with an­oth­er stu­dent of Atiyah, Rolph Schwar­zen­ber­ger. After a Mas­ters de­gree at Liv­er­pool in 1961, Rolph sug­ges­ted that Bri­an go to Ox­ford to work with Ioan James, but he was too busy, so he switched to Atiyah. After a year at Ox­ford, Bri­an re­turned to Liv­er­pool so that his wife Christine could fin­ish her un­der­gradu­ate Ger­man de­gree while Bri­an fin­ished his PhD in 1963. Then Atiyah sug­ges­ted that Bri­an learn something new when he went to Yale for a year, so he got Zee­man’s fam­ous notes on PL to­po­logy. Then came a year at IAS in Prin­ceton where Bri­an got the idea for block bundles. Fi­nally Bri­an went to War­wick in 1965.

Once ap­poin­ted to War­wick, Colin bought a house in Leam­ing­ton Spa which he prac­tic­ally re­built from scratch (he had pre­vi­ously had fixed up a house he’d bought in Wood­ford Bridge while at Queen Mary). After three years at War­wick, Colin went to the Uni­versity of Wis­con­sin Madis­on in 1971–72 and then to Stan­ford in the sum­mer of 1972 where he and his fam­ily lived with the Sander­sons in a sor­or­ity house. With a pleth­ora of avail­able rooms, oth­er math­em­aticians came, too — John Mor­gan, among them.

When Colin re­turned to Leam­ing­ton Spa, Daphne ex­pressed re­gret about not hav­ing a garden, so they began house-hunt­ing and ended up buy­ing a five-acre farm in Barby in 1973. While there he and Daphne raised chick­ens, two dairy cows (Jer­seys), and even­tu­ally, after ex­pand­ing to 20 acres, a few hun­dred Hebridean sheep. The sheep had thick wiry wool (which was not then mar­ket­able) but the lamb and mut­ton was ex­cel­lent. Colin and Daphne es­sen­tially fed a fam­ily of six from the farm, call­ing it “a way of life”, if not a com­mer­cial ven­ture.

After their work on block bundles, Colin and Bri­an wrote their book on PL to­po­logy [5]. It was ori­gin­ally meant to be a book with Zee­man, amp­li­fy­ing Zee­man’s fam­ous notes on PL to­po­logy but Zee­man was too busy to col­lab­or­ate. So their book was built from their own class notes.

An­oth­er book fol­lowed (with Colin’s stu­dent Sandro Buon­cris­ti­ano) [6] which de­veloped the no­tion of mock bundles to give a more geo­met­ric ap­proach to gen­er­al­ized ho­mo­logy the­or­ies (and gen­er­ated a book re­view [e1] by Frank Adams, fam­ous for his sharp-tongued re­views). The book in­cludes trans­vers­al­ity for CW-com­plexes, work that Colin is still quite pleased with and feels is un­der­ap­pre­ci­ated even though it leads to a geo­met­riz­a­tion of ho­mo­topy the­ory.

Wolfgang Met­z­ler gave a nice de­scrip­tion of this book in Math Re­views:

The sin­gu­lar \( n \)-cycles of a to­po­lo­gic­al space \( X \) can be con­sidered as maps \( f : P \to X \), where \( P \) is a closed ori­ented pl \( n \)-man­i­fold with a codi­men­sion 2 sin­gu­lar­ity. Two such maps \( (P, f ) \), \( (Q, g) \) are ho­mo­log­ous if they are bord­ant. The cent­ral idea of this book is to treat any ho­mo­logy the­ory as a gen­er­al­ized bor­d­ism the­ory. Its cycles are man­i­folds with sin­gu­lar­it­ies de­pend­ing on the the­ory. As cocycles one uses mock bundles with the same “man­i­folds” as fibres. (A \( q \)-mock bundle \( \xi^q \) is a pl pro­jec­tion \( \xi: E \to |K| \), where \( K \) is a ball com­plex, and for each ball \( \sigma \in K \), \( p^{-1} (\sigma) \) is a com­pact pl “man­i­fold” of di­men­sion \( q + \dim \sigma \) with bound­ary \( p^{-1} (\sigma) \).) Two cocycles are co­homo­log­ous if they are cobord­ant, that is if there ex­ists a mock bundle over \( K \times I \) with the giv­en cocycles cor­res­pond­ing to the re­stric­tions on \( K \times \{0\} [K \times \{1\}] \).

A lot of (co-)ho­mo­logy the­ory eas­ily can be de­scribed with­in this frame­work: the cup product for in­stance cor­res­ponds to Whit­ney sum of bundles, the cap product between the cycle \( M \underset{f}{\rightarrow} |K| \) and the cocycle \( E \underset{\xi}{\to} |K| \) is a is a cycle \( g=f^*(\xi) \) giv­en by an in­duced mock bundle \[ \begin{CD} E^{\ast} @>{{}}>{{}}> E\\ @V{g}VV @VV{\xi}V\\ M @>{{{f} }}>{{}}> K. \end{CD} \] The Poin­caré du­al­ity map turns out to be simply the iden­tity on rep­res­ent­at­ives.

Coef­fi­cients are treated in Chapter III and IV by la­belling the com­pon­ents of man­i­folds with group ele­ments; the sin­gu­lar­it­ies that have to be per­mit­ted are de­term­ined by the re­la­tions of the coef­fi­cient group. Chapter V ex­tends the res­ults to equivari­ant the­or­ies in­clud­ing power op­er­a­tions and char­ac­ter­ist­ic classes. \( Z_a \)-op­er­a­tions on pl-cobor­d­ism are dis­cussed with re­la­tions to work of tom Dieck and Quil­len. Sheaves are treated in Chapter VI. In Chapter VII the main the­or­em is presen­ted, that an ar­bit­rary (co-)ho­mo­logy the­ory can be ob­tained by the geo­met­ric means in­tro­duced be­fore. The cent­ral step is that every CW spec­trum is the Thom spec­trum of a suit­able bor­d­ism the­ory. Some of the tech­niques of this last chapter (the no­tion of trans­vers­al­ity and du­al­ity for CW com­plexes to­geth­er with some ba­sic the­or­ems) prom­ise to be fruit­ful bey­ond the scope of this book.

The ideas are presen­ted in a stim­u­lat­ing way, al­though some of the proofs are only in­dic­ated. Be­cause the re­view­er didn’t check all the de­tails, he does not take re­spons­ib­il­ity for them. But he wants to em­phas­ize that this book fills a gap and does it in a beau­ti­ful man­ner be­cause (1) it con­nects ideas of vari­ous pa­pers and per­sons (for in­stance, D. Sul­li­van) in a uni­fied treat­ment without bur­den­ing them with too many tech­nic­al de­tails and (2) it proves not only the­or­ems but the fact that geo­metry may give vivid im­pulses (even) in mod­ern al­geb­ra­ic to­po­logy.

From 1976–1981 Colin was act­ing pro­fess­or of pure math­em­at­ics at the Open Uni­versity (on second­ment from War­wick) where he mas­ter­minded the re­writ­ing of the pure math­em­at­ics course.

In 1996, dis­sat­is­fied with the rap­idly rising fees charged by the ma­jor pub­lish­ers of math­em­at­ic­al re­search journ­als, Colin de­cided to start his own journ­al, and was ably as­sisted by Rob Kirby, John Jones and Bri­an Sander­son. That journ­al be­came Geo­metry & To­po­logy. Un­der Colin’s lead­er­ship, GT be­came a lead­ing journ­al in its field while re­main­ing one of the least ex­pens­ive per page. GT was joined in 1998 by a pro­ceed­ings and mono­graphs series, Geo­metry & To­po­logy Mono­graphs, and in 2000 by a sis­ter journ­al, Al­geb­ra­ic & Geo­met­ric To­po­logy. Colin wrote the soft­ware and fully man­aged these pub­lic­a­tions un­til around 2005 when he cofoun­ded Math­em­at­ic­al Sci­ences Pub­lish­ers (with Rob Kirby) to take over the run­ning. Math­em­at­ic­al Sci­ences Pub­lish­ers has now grown to be­come a for­mid­able force in aca­dem­ic pub­lish­ing.

Dur­ing this cen­tury, Colin has been work­ing in cos­mo­logy, of­ten with Robert MacK­ay, an ap­plied math­em­atician at War­wick. They have sev­er­al pa­pers to­geth­er and Colin has writ­ten a book [8] pro­pos­ing a mod­el for the uni­verse without either dark mat­ter or the Big Bang.1


[1] C. P. Rourke and B. J. Sander­son: “An em­bed­ding without a nor­mal mi­crobundle,” In­vent. Math. 3 (1967), pp. 293–​299. MR 222904 Zbl 0168.​44602 article

[2] C. P. Rourke and B. J. Sander­son: “Block bundles, I,” Ann. of Math. (2) 87 (1968), pp. 1–​28. MR 226645 Zbl 0215.​52204 article

[3] C. P. Rourke and B. J. Sander­son: “Block bundles, II: Trans­vers­al­ity,” Ann. of Math. (2) 87 (1968), pp. 256–​278. MR 226646 Zbl 0215.​52301 article

[4] C. P. Rourke and D. P. Sul­li­van: “On the Ker­vaire ob­struc­tion,” Ann. of Math. (2) 94 (1971), pp. 397–​413. MR 305416 Zbl 0227.​57012 article

[5] C. P. Rourke and B. J. Sander­son: In­tro­duc­tion to piece­wise-lin­ear to­po­logy. Ergeb. Math. Gren­zgeb. 69. Spring­er, 1972. MR 350744 Zbl 0254.​57010 book

[6] S. Buon­cris­ti­ano, C. P. Rourke, and B. J. Sander­son: A geo­met­ric ap­proach to ho­mo­logy the­ory. Lon­don Math­em­at­ic­al So­ci­ety Lec­ture Note Series 18. Cam­bridge Uni­versity Press; Lon­don Math­em­at­ic­al So­ci­ety (Cam­bridge; Lon­don), 1976. MR 413113 Zbl 0315.​55002 book

[7] A. A. Ran­icki, A. J. Cas­son, D. P. Sul­li­van, M. A. Arm­strong, C. P. Rourke, and G. E. Cooke: The Hauptver­mu­tung book: A col­lec­tion of pa­pers of the to­po­logy of man­i­folds. \( K \)-Mono­graphs in Math­em­at­ics 1. Kluwer Aca­dem­ic Pub­lish­ers (Dordrecht), 1996. MR 1434100 Zbl 0853.​00012 book

[8]C. Rourke: The geo­metry of the uni­verse. Series on Knots and Everything 71. World Sci­entif­ic Pub­lish­ing (Hack­en­sack, NJ), 2021. MR 4375354 book