Celebratio Mathematica

The fol­low­ing frag­ments are my per­son­al take on my good friend and math­em­at­ic­al col­lab­or­at­or. I first saw Colin strid­ing along King’s Parade, Cam­bridge, in his duffel coat. It was about 1961, “Between the end of the Chat­ter­ley ban // And the Beatle’s first LP.”1 I was a stu­dent at Clare Col­lege and Colin was a year or two ahead of me at Trin­ity Col­lege. We were both math­em­at­ics stu­dents but we didn’t really get to know each oth­er un­til I was a PhD stu­dent at King’s Col­lege, Uni­versity of Lon­don, un­der John Reeve and he was a ju­ni­or lec­turer at Queen Mary, Uni­versity of Lon­don. Reeve ran a to­po­logy sem­in­ar for his and oth­er stu­dents and be­cause geo­met­ric to­po­logy was not com­mon in Lon­don at the time, Colin was tasked with giv­ing a course of lec­tures on geo­met­ric to­po­logy and we sat at his feet, lap­ping it up. We then formed a non­mathem­at­ic­al bond since we were both young par­ents and keen on DIY. Later Colin went to the In­sti­tute for Ad­vanced Study at Prin­ceton and a new po­s­i­tion at the Uni­versity of War­wick’s math­em­at­ics de­part­ment. This was an ex­cit­ing time for Colin and the new de­part­ment, and al­though we of­ten met be­cause of our shared love of DIY, it wasn’t un­til the late 70s that we fi­nally got round to writ­ing math­em­at­ics in­stead of just talk­ing about it. We wrote a pa­per on spines of 3-man­i­folds [2] pub­lished in the pro­ceed­ings of a con­fer­ence I or­gan­ised in 1997. This didn’t make many waves at the time but some­time later an ex­ten­sion of this res­ult was used in the doc­tor­al thes­is of Ruben Vigara, a stu­dent of Montesinos. The res­ult was a num­ber of re­lated pa­pers; see [12], [e8], [e9], [e10], [e11], [e12], [e13], [e14].

There fol­lowed a pa­per on the Kirby cal­cu­lus and the so called K moves which was pub­lished in the journ­al, To­po­logy [1]. Colin clearly re­calls one im­port­ant con­ver­sa­tion when he ex­plained the K-move (twist­ing around an un­knot­ted curve) and the oth­er Kirby moves in­clud­ing handle slides. Ap­par­ently I poin­ted out at one cru­cial point that if you twis­ted around a curve par­al­lel to the one be­ing used for the K-move, then the ef­fect was a handle slide across the K-move. With a fair bit more work, we used this to prove that all handle slides can be achieved by K-moves and the cal­cu­lus re­duced to a single move. This is a key res­ult be­cause it makes the cal­cu­lus loc­al — handle slides are glob­al moves. The first real ap­plic­a­tion of the cal­cu­lus (to the con­struc­tion of the Resh­et­ikh­in–Tur­aev in­vari­ant) used this loc­al ver­sion of the cal­cu­lus.

Rather than just write a short note on the K-move and handle slides, we de­cided to write a com­plete proof of the cal­cu­lus, set­ting it in the PL cat­egory, rather than the smooth set­ting that Kirby used. Be­cause there is both a tri­an­gu­la­tion the­or­em and a Hauptver­mu­tung the­or­em in di­men­sion 3, there is no dif­fer­ence between smooth and PL to­po­logy for 3-man­i­folds, so there should be a sim­pler proof us­ing PL meth­ods rather than the “Cerf the­ory” used in Kirby’s proof. We al­most achieved this in our pa­per on the cal­cu­lus [1]. However, there is no proof in the lit­er­at­ure that shows that two PL handle de­com­pos­i­tions of a giv­en man­i­fold are equi­val­ent by handle moves (in­tro­duc­tion and can­cel­la­tion of com­ple­ment­ary pairs and handle slides) though one could be read­ily giv­en us­ing PL handle the­ory. So to avoid a lengthy di­gres­sion at this point, we moved in­to the smooth cat­egory just for this res­ult. The pa­per con­tains the “loc­al” ver­sion of the cal­cu­lus. However the im­port­ance of our con­tri­bu­tion of­ten es­capes the read­er. See, for ex­ample, the re­view of this pa­per by Wil­bur Whit­ten in Math­em­at­ic­al Re­views. The fun­da­ment­al the­or­em that Whit­ten states in the first para­graph is this loc­al ver­sion of the cal­cu­lus and he needs the Fenn–Rourke pa­per for its proof. Rourke rue­fully re­marks that in­stead of call­ing the loc­al move a K-move (for Kirby) we should per­haps have called it an FR-move! However I am not that bothered.

Our pa­per “Racks and links in codi­men­sion two” [3], pub­lished in the early 90s, has con­sist­ently been the most quoted of all our pub­lic­a­tions. At the time it ap­peared, the Resh­et­ikh­in–Tur­aev in­vari­ant had had a gal­van­ising ef­fect on geo­met­ric to­po­lo­gists. There are many ex­amples of an in­vari­ant es­tab­lished us­ing moves and many to­po­lo­gists star­ted a new set of quests for in­vari­ants us­ing moves. In par­tic­u­lar we stumbled upon an in­ter­est­ing (and we thought new) in­vari­ant of knots and links in­vari­ant un­der just two of the Re­idemeister moves (R2 and R3) which, after some notes of Con­way and Wraith on the con­jugacy re­la­tion in groups, we called the fun­da­ment­al “rack” (a sim­pli­fic­a­tion of the name “wrack” used by Con­way and Wraith). The gen­es­is of this pa­per is in­ter­est­ing. Email had not been in­ven­ted and I was at the Autonom­ous Uni­versity of Mad­rid as a vis­it­ing aca­dem­ic. So our ideas were com­mu­nic­ated by the agency of a fax ma­chine. I no­ticed at the time that a rack had a nat­ur­al cu­bic­al struc­ture which defined the clas­si­fy­ing space for racks. It was three faxes sent from Mad­rid which defined this clas­si­fy­ing space: the first was the 1-sec­tion, the second the 2-sec­tion and the third was gen­er­al [e6].

Racks have a nat­ur­al al­geb­ra­ic struc­ture ana­log­ous to groups with which they were closely as­so­ci­ated. Every semi­framed link in a 3-man­i­fold has a fun­da­ment­al rack which, in the ir­re­du­cible case, clas­si­fies both the link and the 3-man­i­fold. There are sev­er­al stand­ard al­geb­ra­ic res­ults sim­il­ar to ones in group the­ory for ex­ample there is a Tiet­ze the­or­em and an ana­logue of Nielsen the­ory. Un­for­tu­nately soon after its birth, we quickly dis­covered that racks were not a new concept, hav­ing been an­ti­cip­ated by Joyce, [e3] Matveev [e4], and Brieskorn [e5], as well as Con­way and Wraith [e2] and the Ja­pan­ese kei pub­lished dur­ing the second world war [e1]. Thus the big pa­per that we wrote on the sub­ject is partly ex­pos­it­ory of this pre­vi­ous work. Des­pite this (or per­haps be­cause of this) this joint pa­per is the most widely cited of all of our pa­pers.

The ho­mo­logy of the cu­bic­al clas­si­fy­ing space is closely as­so­ci­ated to the in­vari­ants con­struc­ted by Scott Carter et al. [e7]. It turns out that the ho­mo­topy type of this space is not usu­ally in­ter­est­ing but the com­bin­at­or­i­al struc­ture car­ries the full in­form­a­tion of the un­der­ly­ing link. Cu­bic­al sets of this type have in­ter­est­ing ho­mo­topy-the­or­et­ic prop­er­ties. There is an as­so­ci­ated hier­archy of bundles called James bundles be­cause of their in­tim­ate con­nec­tion with the James–Hopf in­vari­ants of loops \( (S^2) \). The first James bundle can be in­ter­preted as the clas­si­fy­ing bundle for the link. The Fenn–Rourke–Sander­son pa­pers on the rack space are pub­lished in Ap­plied Cat­egor­ic­al Struc­tures [4], Pro­ceed­ings of the Lon­don Math­em­at­ic­al So­ci­ety [8], and Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety [9]. Be­cause of the con­nec­tion with the Carter et al. work area [e7], these pa­pers also are very widely cited. The pa­per on racks also let to many spin-offs in­clud­ing in­valu­able joint work with Bri­an Sander­son, which are lis­ted in Bri­an’s con­tri­bu­tion.

We also pub­lished a pa­per on the ideas of Kly­ach­ko which nearly proves the Ker­vaire con­jec­ture [5].

At this time, the late 70s, I was get­ting very in­ter­ested in gen­er­al­ised knot and braid the­or­ies and the pa­per on the braid-per­muta­tion group in­tro­duced wel­ded braids and knots which are a rarer form of vir­tu­al knots and braids in­tro­duced by Kauff­man [6], [10].

At a con­fer­ence at the Isle of Thorns which I or­gan­ised in 1998 and hos­ted by the Uni­versity of Sus­sex, Dale Rolf­sen an­nounced that Patrick De­hornoy had shown that the braid groups were left or­der­able us­ing eso­ter­ic meth­ods from lo­gic the­ory. This was a dir­ect chal­lenge to the to­po­lo­gists there and by the late sum­mer a gang of us had for­mu­lated a to­po­lo­gic­al proof [7]. This was the last (so far) of our col­lab­or­a­tions, as Colin de­cided it was time to con­cen­trate on cos­mo­logy in joint work with Robert MacK­ay [11]. He wrote a book [12] on a prop­er un­der­stand­ing of the “geo­metry of the uni­verse” without either big bang or dark mat­ter.