Celebratio Mathematica

Robion C. Kirby

Rob Kirby

by Raymond Lickorish

There are two big math­em­at­ic­al res­ults due to Rob Kirby. Firstly there is the 1968 proof of the An­nu­lus Con­jec­ture [1] in di­men­sions great­er than four. The oth­er is the 1976 proof us­ing four-di­men­sion­al Cerf The­ory that any two sur­gery present­a­tions of a giv­en 3-man­i­fold are re­lated by a se­quence of just two types of moves, the Kirby moves. Of course, in half a cen­tury of re­search, he has many oth­er fine res­ults; they are all lis­ted and read­ily avail­able, with re­views, on the world-wide web. The An­nu­lus Con­jec­ture as­ser­ted that between any two dis­joint loc­ally flat em­bed­dings of the \( (n-1) \)-sphere \( S^{n-1} \) in \( S^n \) there was a copy of the an­nu­lus \( S^{n-1} \times I \). Kirby’s work on this not only solved one of the then out­stand­ing prob­lems on to­po­lo­gic­al man­i­folds but it led, in joint work with Larry Sieben­mann, to a com­pre­hens­ive the­ory of the tri­an­gu­la­tion of to­po­lo­gic­al man­i­folds, of di­men­sion at least five, as piece­wise-lin­ear man­i­folds (there is one ob­struc­tion). To­geth­er with re­prints of their ori­gin­al pa­per this is all de­scribed in their book [2]. The work on the Kirby moves had a strange ef­fect. The ac­tu­al res­ult was not really used un­til the ar­rival of quantum in­vari­ants some dozen years later, but people’s at­ti­tudes were changed. Know­ledge that any­thing could be done with just two moves meant that those two were the ones to play with and many did so.

Rob and I first met as vis­it­ors to the Uni­versity of Wis­con­sin in the Fall Semester of 1967. He lec­tured on to­po­lo­gic­al man­i­folds, I on piece­wise-lin­ear ones. Born six days apart, so at the same stage in life, we began a life-time’s friend­ship with reg­u­lar cor­res­pond­ence. In math­em­at­ic­al re­search, per­son­al con­tacts were im­port­ant be­fore the days of e-mail, the arX­iv, Math­S­ciNet, TeX, Google and Zoom. Rob re­turned to UCLA at Christ­mas 1967, pre­fer­ring the moun­tains and rivers of Cali­for­nia to a Wis­con­sin winter. Over the years we vis­ited each oth­er many times in Berke­ley and in Cam­bridge. Some­times we were ac­com­pan­ied by stu­dents and Rob had many of them. At the last count he has had 53 stu­dents who, he claimed, taught each oth­er. The gath­er­ings at these vis­its even­tu­ally cli­maxed in two three-week con­fer­ences in Cam­bridge in 1981 and 1984. These meet­ings were en­tirely in­form­al. Any­one was wel­come for any length of time. There was no fund­ing (though per­haps Rob’s NSF grant helped some of his stu­dents) and no pro­ceed­ings. We had two talks every af­ter­noon fol­lowed for some, in­clud­ing Rob, by foot­ball on nearby Lam­mas Land. The Cam­bridge de­part­ment­al fa­cil­it­ies were made avail­able without charge and empty stu­dent ac­com­mod­a­tion was cheap. Rob found that he rather liked Great Bri­tain, in­clud­ing its weath­er and food. He has vis­ited of­ten, with Colin Rourke in War­wick and An­drew Ran­icki in Ed­in­burgh be­ing par­tic­u­lar friends. His Brit­ish con­nec­tion cli­maxed with his ap­point­ment in 1992 as a Roth­schild Vis­it­ing Pro­fess­or at the Isaac New­ton In­sti­tute dur­ing the In­sti­tute’s first ses­sion, en­titled Low Di­men­sion­al To­po­logy and Quantum Field The­ory. This was ac­com­pan­ied by a vis­it­ing fel­low­ship at Em­manuel Col­lege.

Rob’s first mar­riage broke up around 1978 leav­ing him in charge of two young chil­dren, a large house and not much money. There was pres­sure on time avail­able for math­em­at­ics. Our two fam­il­ies did however join in 1980 for a stay in the cab­in of Jim Van Buskirk (Uni­versity of Ore­gon) by the Met­oli­us River in the Ore­gon Cas­cades, with swim­ming in the loc­al Scout Lake.

In the Oregon Cascades, 1980 (from left to right): Raymond Lickorish, Rob Kirby (with Kate Kirby and Susie Lickorish), Josephine and Henry Lickorish, Rolf Kirby, and Ann Lickorish.

The work that pro­duced the Kirby moves on 3-man­i­folds led to a Kirby nota­tion for present­ing handle de­com­pos­i­tions of 4-man­i­folds. Over the years Rob and sev­er­al of his stu­dents ex­plored the con­sid­er­able in­tric­a­cies of this. Berke­ley be­came a centre for this type of work. The cul­min­a­tion was Rob’s Spring­er Lec­ture Notes book The To­po­logy of 4-Man­i­folds [5] which in­cludes Mike Freed­man’s use of An­drew Cas­son’s handles and a short de­scrip­tion of some of Si­mon Don­ald­son’s res­ults.

A most im­port­ant part of Rob’s ca­reer has been not only do­ing re­search but en­cour­aging re­search by oth­ers. This was typ­i­fied by his 1976 com­pen­di­um en­titled Prob­lems in low di­men­sion­al man­i­fold the­ory [3]. This lis­ted 68 prob­lems, at­trib­uted to a wide range of au­thors, to­geth­er with re­marks on the ap­pro­pri­ate state of know­ledge at the time. Known as the Kirby prob­lem list, this set the agenda for many years of low di­men­sion­al to­po­logy re­search. Many of his gradu­ate stu­dents went on to their own dis­tin­guished uni­versity ca­reers. He kept in touch with them, of­ten by writ­ing pa­pers jointly with them. Rob and I wrote just one pa­per to­geth­er [4]; it gave us pleas­ure at the time but does not seem im­port­ant now! Back in those Wis­con­sin days of 1967 one con­sul­ted journ­als in a de­part­ment’s lib­rary, or bor­rowed them from the lib­rary, to learn of any re­cent pro­gress of rel­ev­ance to one’s re­search. The num­ber of journ­als has greatly in­creased and their pur­pose is much changed. In more re­cent years Rob has played a lead­ing role in pro­mot­ing elec­tron­ic journ­als and in cam­paign­ing for a fair fin­an­cial deal from journ­al pub­lish­ers for math­em­aticians and their uni­versit­ies.

Ray­mond Lick­or­ish is Pro­fess­or Emer­it­us of Geo­met­ric To­po­logy in the Uni­versity of Cam­bridge where, for five years, he was head of the De­part­ment of Pure Math­em­at­ics and Math­em­at­ic­al Stat­ist­ics. He is a Fel­low of Pem­broke Col­lege where, for many years he was Dir­ect­or of Stud­ies in Math­em­at­ics. He has held uni­versity vis­it­ing po­s­i­tions in Wis­con­sin, Berke­ley, Santa Bar­bara, Texas, Los Angeles and Mel­bourne. In re­search he in­vest­ig­ated twist­ing auto­morph­isms of sur­faces and their in­ter­ac­tion with sur­gery de­scrip­tions of 3-man­i­folds. He worked on piece­wise-lin­ear the­ory, par­tic­u­larly col­lapsing res­ults and un­knot­ting the­or­ems in high di­men­sions. He was also much in­volved with in­vari­ant poly­no­mi­als for links and knots, after V. F. R. Jones’ ini­tial dis­cov­ery, and he has au­thored a knot the­ory text.


[1]R. C. Kirby: “Stable homeo­morph­isms and the an­nu­lus con­jec­ture,” Ann. of Math. (2) 89 (1969), pp. 575–​582. MR 0242165 Zbl 0176.​22004

[2]R. C. Kirby and L. C. Sieben­mann: Found­a­tion­al es­says on to­po­lo­gic­al man­i­folds, smooth­ings, and tri­an­gu­la­tions. An­nals of Math­em­at­ics Stud­ies 88. Prin­ceton Uni­versity Press, 1977. With notes by John Mil­nor and Mi­chael Atiyah. MR 0645390 Zbl 0361.​57004 book

[3]R. Kirby: “Prob­lems in low di­men­sion­al man­i­fold the­ory,” pp. 273–​312 in Al­geb­ra­ic and geo­met­ric to­po­logy (Stan­ford Univ., CA, 1976), part 2. Edi­ted by R. J. Mil­gram. Proc. Sym­pos. Pure Math. XXXII. Amer. Math. Soc. (Provid­ence, R.I.), 1978. MR 520548 Zbl 0394.​57002

[4]R. C. Kirby and W. B. R. Lick­or­ish: “Prime knots and con­cord­ance,” Math. Proc. Cam­bridge Philos. Soc. 86 : 3 (1979), pp. 437–​441. MR 542689 Zbl 0426.​57001

[5]R. C. Kirby: The to­po­logy of 4-man­i­folds. Lec­ture Notes in Math­em­at­ics 1374. Spring­er (Ber­lin), 1989. MR 1001966 Zbl 0668.​57001