Celebratio Mathematica

Colin P. Rourke

My collaboration with Colin Rourke

by Brian Sanderson

Colin’s thes­is was about a highly tech­nic­al PL prob­lem that Hud­son and Zee­man ran in­to whilst at­tempt­ing to clas­si­fy tor­us knots (knots of a product of two spheres in Eu­c­lidean space):

If a sphere \( S^n \) is em­bed­ded in a ball \( B^{n+q} \) im­prop­erly (i.e., it is al­lowed to meet the bound­ary) then is it un­knot­ted in the sense that it bounds a ball \( B^{n+1} \) in \( B^{n+q} \)? As­sume for sim­pli­city that \( q > 2 \) (codi­men­sion \( \geq \) 3) so that by Zee­man’s the­or­em \( S^n \) in un­knot­ted in the con­tain­ing Eu­c­lidean space \( R^{n+q} \) of \( B^{n+q} \).

The un­ex­pec­ted an­swer that Colin found is that the an­swer is yes in the meta­stable range \( 2q > n+3 \) (and, as a res­ult of our col­lab­or­a­tion this is best pos­sible).

In or­der to ap­ply this un­knot­ting the­or­em to ob­tain a clas­si­fic­a­tion of tor­us knots, Colin needed a PL no­tion of the Grass­man­ni­an in the smooth cat­egory and after ask­ing Zee­man at the 1965 Ober­wolfach meet­ing for help with this, Zee­man poin­ted to me.

I had been at the In­sti­tute for Ad­vanced Study in Prin­ceton the pre­vi­ous year, after a year at Yale. At Yale I had learned PL the­ory by giv­ing a gradu­ate course on Zee­man’s PL notes. I came up with block bundles in or­der to re­duce an in­ter­lock­ing braid of ho­mo­topy groups of Lev­ine [e3], Sec­tion 2.2, to ho­mo­topy the­ory and Zee­man had come up with the name block bundles on the flight to Ober­wolfach. Meet­ing Colin was for­tu­it­ous as I was hav­ing prob­lems with the cor­res­pond­ing PL bundle the­ory. We star­ted a cor­res­pond­ence about this and next met in per­son at the Brit­ish Math­em­at­ic­al Col­loqui­um held in Im­per­i­al Col­lege in March 1966 and had sev­er­al long con­ver­sa­tions about it and ef­fect­ively sor­ted out nearly all the prob­lems. The PL tech­niques that Colin used in his thes­is — in par­tic­u­lar, the re­l­at­ive reg­u­lar neigh­bour­hood the­or­em or RRNT — were ideally suited to the de­tails needed for the the­ory. One key point was wheth­er to in­sist on charts for the new bundles and for a while we toyed with hav­ing block bundles without charts. We quickly real­ised that charts can be re­in­ser­ted block by block us­ing con­ic­al ex­ten­sion and the RRNT so the no­tions of block bundles with or without charts are the same.

One ser­i­ous prob­lem re­mained. We ex­pec­ted that the “group” would be a semisim­pli­cial group with all the usu­al face maps and de­gen­eracies. But the nat­ur­al way to define de­gen­eracies ran in­to a fam­ous PL prob­lem known as “the stand­ard mis­take”. The mis­take is to as­sume ra­di­al pro­jec­tion is piece­wise lin­ear. On the oth­er hand the ex­ten­sion con­di­tion (the Kan con­di­tion) was easy to prove by the same tech­niques used to re­in­sert charts, and we star­ted to think about set­ting up semisim­pli­cial the­ory us­ing just face maps and the Kan con­di­tion. We called this the­ory delta-sets [5], [6] and we used this for the fi­nal present­a­tion of the new the­ory. One ex­tra bo­nus of the new the­ory was that trans­vers­al­ity, which Zee­man had just proved us­ing the highly tech­nic­al PL no­tion of “transim­pli­ci­al­ity”, now had a nat­ur­al proof us­ing “block trans­vers­al­ity”.

The the­ory gave all the ex­pec­ted ana­logues of smooth char­ac­ter­ist­ic classes and huge braid dia­grams of as­so­ci­ated ho­mo­topy groups [2], [3], [4].

The embedding without a normal microbundle

One ma­jor prob­lem that was left un­answered was wheth­er a new type of bundle the­ory was really ne­ces­sary. Mil­nor had in­ven­ted “mi­crobundles” [e1] in the hope of provid­ing nor­mal bundles in the PL and to­po­lo­gic­al cat­egor­ies and Kister [e2] had shown that mi­crobundles are really \( R^n \) bundles in dis­guise. Per­haps block bundles were also real bundles in dis­guise. This was per­haps the hard­est prob­lem that we tackled and many oth­er math­em­aticians were work­ing on it. We worked night and day on this for sev­er­al weeks. We struck gold us­ing a won­der­ful pa­per of An­dré Hae­fli­ger [e4]. This was pure PL ma­gic. Hae­fli­ger found his own no­tion of nor­mal bundle for the PL cat­egory via stable mi­crobundle pairs. He used this to clas­si­fy links of spheres in codi­men­sion \( \geq 3 \) which was com­plete in sev­er­al im­port­ant cases and con­tained some use­ful ex­act se­quences. Con­cen­trat­ing on links of two cop­ies of \( S^n \) in \( S^{n+q} \), Colin no­ticed that if you sus­pend a link (to get a link of two cop­ies of \( S^{n+1} \) in \( S^{n+q+2} \)) then the res­ult­ing link is the bound­ary of a rib­bon (a copy of \( S^{n+1}\times I) \). This proved by adding a few cones, see the dia­gram on page 296 of [1]. A link in codi­men­sion 3 has two “link­ing num­bers” (ele­ments of a cer­tain ho­mo­topy group of spheres) be­cause the com­ple­ment of one of the spheres has the ho­mo­topy type of a sphere by Zee­man’s the­or­em and the oth­er sphere then de­term­ines a link­ing num­ber. We also no­ticed that if a rib­bon has a nor­mal bundle with fibre a Eu­c­lidean space, then the rib­bon can be “turned over” in­side the fibres and this proves that if q is odd and one link­ing class of the link which bounds the rib­bon is zero then the oth­er one must have or­der 2. Now us­ing an ex­act se­quence found by Hae­fli­ger and a search through Toda’s tables of ho­mo­topy groups of spheres a link can be found of two 18-spheres in \( S^{27} \) whose sus­pen­sion has one link­ing num­ber zero and the oth­er not of or­der two. This proves that there is an em­bed­ding of \( S^{19} \) in \( S^{29} \) with no PL nor­mal mi­crobundle. Since the “turn­ing over” ar­gu­ment works equally well with to­po­lo­gic­al fibres, this em­bed­ding has no to­po­lo­gic­al nor­mal bundle either. By re­strict­ing the ex­ample to one end of the rib­bon, there is a PL em­bed­ding (ne­ces­sar­ily loc­ally flat by Zee­man’s the­or­em) of \( S^{19} \) in a 28-man­i­fold with no to­po­lo­gic­al nor­mal bundle.

About the Hauptvermutung

Our joint work on block bundles, delta-sets and the em­bed­ding without a nor­mal bundles was com­pleted whilst Colin was at Queen Mary Col­lege in Lon­don from 1964 to 1967. He moved to War­wick in 1968 and spent the in­ter­ven­ing year, half at the Prin­ceton In­sti­tute, and half on his fel­low­ship at Trin­ity Col­lege Cam­bridge. Whilst at Prin­ceton he gave a series of lec­tures in the Mont­gomery sem­in­ar on the ho­mo­topy Hauptver­mu­tung (a homeo­morph­ism between PL man­i­folds is ho­mo­top­ic to a PL homeo­morph­ism) a proof of which had re­cently been an­nounced by Den­nis Sul­li­van. In a tele­phone con­ver­sa­tion, Sul­li­van told Colin that he had a new, and much sim­pler proof which could be eas­ily un­der­stood and I sent Colin an out­line of this proof, which he used for his lec­tures. Later it was made clear by Terry Wall that Sul­li­van’s new proof fol­lowed a sim­pli­fic­a­tion of the ori­gin­al proof due to An­drew Cas­son and the title of the lec­ture notes was changed from “The Hauptver­mu­tung ac­cord­ing to Sul­li­van” to “The Hauptver­mu­tung ac­cord­ing to Cas­son and Sul­li­van”. Cas­son ex­ten­ded the no­tion of block bundle to al­low ar­bit­rary man­i­fold blocks (man­i­fold cross a base disc) an used this to for­mu­late Sul­li­van’s proof in a co­her­ent and el­eg­ant way.

In The Hauptver­mu­tung Book [10], An­drew Ran­icki as­sembled all the ori­gin­al pa­pers, in­clud­ing Cas­son’s fel­low­ship thes­is con­tain­ing the new proof, Sul­li­van’s rough notes on the sub­ject and much else.

Colin’s ori­gin­al con­tri­bu­tion to the area was an ac­cur­ate proof of the product for­mula as used in the proof. Sul­li­van knew it was right but did not have a proof and Cas­son didn’t either (he re­lied on Wall’s more gen­er­al product for­mula which was not yet writ­ten up). The proof of the product for­mula was pub­lished in On the Ker­vaire ob­struc­tion by Rourke and Sul­li­van which also con­tained de­tails of the peri­od­icity map for G/PL [7].

After Colin came to War­wick we pro­duced what is now a clas­sic text on piece­wise lin­ear to­po­logy [9]. It in­cluded the then re­cent handle the­ory and was pub­lished in 1972 and re­prin­ted in pa­per­back in 1982.

Mock bundles and the geometrisation of generalised homology theories

More gen­er­al than Cas­son’s man­i­fold block bundles, mock bundles (mock for man­i­fold-block) provide a nat­ur­al geo­met­ric set­ting for ho­mo­logy the­or­ies. A cycle for bor­d­ism was giv­en by a map of a man­i­fold in­to a space but there was no geo­met­ric view of a cobor­d­ism rep­res­ent­at­ive. This turned out to be simply a block bundle with blocks re­placed by man­i­folds. At this point Cas­son had used the more re­strict­ive no­tion of man­i­fold block bundles We called the gen­er­al­ised bundles mock bundles, the name sug­ges­ted by Mar­shall Co­hen who was vis­it­ing War­wick at the time. Colin and I worked on these col­lab­or­at­ing with Colin’s PhD stu­dent Sandro Buon­cris­ti­ano Buon­cris­ti­ano [8]. The the­ory gives geo­met­ric rep­res­ent­at­ives for all gen­er­al­ised ho­mo­logy the­or­ies (an­swer­ing a ques­tion posed by Sul­li­van) and al­lows a no­tion of sheaves of coef­fi­cients, again for all gen­er­al­ised ho­mo­logy the­or­ies.

Fenn Rourke Sanderson papers 1993–2007

In the late sev­en­ties I pub­lished two pa­pers with Ul­rich Kos­chorke [e6], [e5], and one on Ma­howald ori­ent­a­tions [e7]. The res­ults came from con­sid­er­ing situ­ations in­volving con­fig­ur­a­tion spaces and em­bed­dings of a man­i­fold \( M \) in a \( Q\times R^n \) which pro­jects to a trans­verse im­mer­sion in \( Q \). The mul­tiple point man­i­folds can rep­res­ent char­ac­ter­ist­ic classes or gen­er­al­ised Hopf in­vari­ants and we are in­to al­geb­ra­ic to­po­logy. Ul­rich and I could prove that the em­bed­ding would pro­ject after a bor­d­ism. This got Colin and I think­ing about pro­ject­ing after iso­topy. We proved that it could be done in a way that gave a new proof of the im­mer­sion the­or­em with the ad­vant­age of an ex­pli­cit de­scrip­tion of the res­ult­ing im­mer­sion. The de­tails are in our com­pres­sion the­ory pa­pers [12], [13], [14].

The col­lab­or­a­tion with Colin and Ro­ger Fenn began when Ro­ger came to give a lec­ture at War­wick. Ro­ger and Colin had been work­ing for some time on knots and racks and they were puzz­ling over the sig­ni­fic­ance of the rack space — a cu­bic­al com­plex without de­gen­eracies de­term­ined by a rack. We call these com­plexes square sets. I could see that for the trivi­al one point rack the space has the ho­mo­topy type of the double loop space on the 2-sphere and our work to­geth­er began.

In [15] we in­vest­ig­ate square sets. Any square set C de­term­ines a fam­ily of square sets \( J^n(C) \) by re­pla­cing each \( k \)-cube in \( C \) by the dis­joint uni­on of the codi­men­sion \( n \) sub­cubes through its centre. These sub­cubes have a nat­ur­al or­der and so we get an em­bed­ding of the real­isa­tion of \( J^n(C) \) in \( C\times R \). This leads to James Hopf in­vari­ants. By tak­ing real­isa­tions we get a map \( |J^nC| \rightarrow |C| \) which is the pro­jec­tion of a mock bundle with framed man­i­fold blocks. Now we have an ele­ment in stable co­homo­topy which turns out to be a Steifel Whit­ney class of a vec­tor bundle over \( |C| \) after ap­ply­ing the Hurewicz ho­mo­morph­ism. Fur­ther for any \( C \) there is an as­so­ci­ated cobor­d­ism the­ory. A \( C \)-man­i­fold \( M \) has a codi­men­sion-1 im­mersed sub­man­i­fold \( Q \) which has trans­verse self in­ter­sec­tions. Each mul­tiple point man­i­fold is mapped to the centre of a \( k \)-cube in \( |C| \) and has framed nor­mal bundle in \( M \) pulled back from the \( k \)-cube. Fur­ther the stable nor­mal bundle of \( M \) is re­quired to be trivi­al away from a neigh­bour­hood of \( Q \) and has twist­ing in the neigh­bour­hoods of the self in­ter­sec­tions pulled back from the as­so­ci­ated \( k \)-cube. There are cor­res­pond­ing \( C \)-mock bundles and a \( C \)-ho­mo­logy/co­homo­logy the­ory.

In [16] we find the rack space defined as the real­isa­tion of a square set where an \( n \)-cube giv­en by an \( n \)-tuple of rack ele­ments. We cal­cu­late ho­mo­logy and ho­mo­topy groups of the rack space for some ex­amples. The rack space clas­si­fies framed codi­men­sion-2 links with the giv­en rack up to cobor­d­ism. For clas­sic­al links the rack to­geth­er with the cor­res­pond­ing ele­ment in \( \pi_2 \) of the rack space clas­si­fies. The proof is in [11] along with cal­cu­la­tions us­ing a ho­mo­morph­ism in­to a simple rack space to dis­tin­guish the stand­ard tre­foils and the 2-twist spun tre­foils.

Personal memories

After col­lab­or­at­ing at a dis­tance I really got to know Colin after he came to War­wick in 1968. I had been there from the be­gin­ning in 1965. He seemed to just know how to do things. He be­came known as the go-to for gradu­ate stu­dents with broken down cars and he had to call an of­fi­cial halt on that activ­ity as it was get­ting to be too much. He bought a house in Leam­ing­ton and in­stalled the cent­ral heat­ing, be­ing a dab hand with a blow torch. Later we both moved to the coun­try in 1973, Colin to Barby and I to Wolf­ham­cote. Of course like White­head be­fore him, Colin had cows and sheep and still does. Again he in­stalled his cent­ral heat­ing. I paid a plumb­er to do mine. In ‘99 our boil­ers needed re­pla­cing. I got boil­ers from a plumb­er/hang-glid­ing friend. Colin in­stalled his and helped me in­stall mine.


[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 B. J. Sander­son: “Block bundles, III: Ho­mo­topy the­ory,” Ann. of Math. (2) 87 (1968), pp. 431–​483. MR 232404 Zbl 0215.​52302 article

[5] C. P. Rourke and B. J. Sander­son: “\( \Delta \)-sets, I: Ho­mo­topy the­ory,” Quart. J. Math. Ox­ford Ser. (2) 22 (1971), pp. 321–​338. MR 300281 Zbl 0226.​55019 article

[6] C. P. Rourke and B. J. Sander­son: “\( \Delta \)-sets, II: Block bundles and block fibra­tions,” Quart. J. Math. Ox­ford Ser. (2) 22 (1971), pp. 465–​485. MR 300282 Zbl 0226.​55020 article

[7] 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

[8] 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

[9] C. P. Rourke and B. J. Sander­son: In­tro­duc­tion to piece­wise-lin­ear to­po­logy, Re­vised re­print of the 1972 ori­gin­al edition. Ergeb. Math. Gren­zgeb. 69. Spring­er, 1982. Re­vised re­print of the 1972 ori­gin­al. MR 665919 Zbl 0477.​57003 book

[10] 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

[11] C. Rourke and B. Sander­son: A new clas­si­fic­a­tion of links and some cal­cu­la­tions us­ing it. Pre­print, 2000. ArXiv math/​0006062 techreport

[12] C. Rourke and B. Sander­son: “The com­pres­sion the­or­em, I,” Geom. To­pol. 5 (2001), pp. 399–​429. MR 1833749 Zbl 1002.​57057 article

[13] C. Rourke and B. Sander­son: “The com­pres­sion the­or­em, II: Dir­ec­ted em­bed­dings,” Geom. To­pol. 5 (2001), pp. 431–​440. MR 1833750 Zbl 1032.​57028 article

[14] C. Rourke and B. Sander­son: “The com­pres­sion the­or­em, III: Ap­plic­a­tions,” Al­gebr. Geom. To­pol. 3 (2003), pp. 857–​872. MR 2012956 Zbl 1032.​57029 article

[15] R. Fenn, C. Rourke, and B. Sander­son: “James bundles,” Proc. Lond. Math. Soc. (3) 89 : 1 (2004), pp. 217–​240. MR 2063665 Zbl 1055.​55005 article

[16] R. Fenn, C. Rourke, and B. Sander­son: “The rack space,” Trans. Amer. Math. Soc. 359 : 2 (2007), pp. 701–​740. MR 2255194 Zbl 1123.​55006 article