Celebratio Mathematica

I ar­rived in Berke­ley as a new gradu­ate stu­dent in 1975 and gradu­ated in 1980. Dur­ing those years I had no idea of what a re­mark­able time and place it was and how much it was shap­ing my life. Rob Kirby was at the cen­ter of it, but what made it unique was the com­munity that Rob had built, a re­mark­able group of to­po­lo­gists, every one a mod­el of col­legi­al­ity as well as tal­ent.

My course in­struct­ors in­cluded Rob and most of the oth­er to­po­logy fac­ulty: Span­i­er, Stallings, Thomas, and Wag­on­er. An equal num­ber of my to­po­logy courses were taught by vis­it­ors, in­clud­ing Cas­son, Gor­don, Hatch­er, Ji­ang, and Ker­ck­hoff. I over­lapped with a start­ling num­ber of oth­er Kirby stu­dents. Paul Melvin and John Harer were my first ment­ors, wel­com­ing me in­to the group and spend­ing hours talk­ing math­em­at­ics with me. My friend from UCLA cal­cu­lus courses, Bill Menasco, entered the pro­gram with me, and we were soon joined by oth­ers, in­clud­ing (among a total of 15 who over­lapped with me) Iain Aitchis­on, Tim Co­chran, Bob Gom­pf, Joel Hass, John Hughes, and Danny Ruber­man. That list doesn’t in­clude stu­dents with com­mon in­terests who had oth­er ad­visors, in­clud­ing Marc Cull­er, Pat Gilmer, and Bill Gold­man. It also doesn’t in­clude Rob’s former stu­dents who were fre­quent vis­it­ors: Sel­man Ak­bu­lut, Mike Han­del, and Marty Schar­le­mann. Bey­ond them, Bob Ed­wards, Den­nis John­son, and Larry Taylor were a fre­quent pres­ence.

Be­ing a gradu­ate stu­dent in math­em­at­ics came with its wor­ries. At the wel­com­ing event for the 80 new gradu­ate stu­dents (with few ex­cep­tions, white men), the gradu­ate dir­ect­or, Cal Moore, warned us: “Look around you. One out of four of you are go­ing to make it through to a PhD, and for those who do, it’s not a pretty pic­ture.” Rob ad­ded to the weight of that mes­sage; he had pos­ted on his of­fice door graph­ics that dis­played the col­lapse of math­em­at­ics hir­ing and salar­ies. Yet all of us did well, some with aca­dem­ic ca­reers in to­po­logy, oth­ers, like John Hughes, branch­ing in­to al­lied fields, and some head­ing in­to dif­fer­ent realms, such as Cole Giller, who left his study of sur­faces in four-space to be­come a noted neurosur­geon.

Life out­side of the uni­versity also had plenty of chal­lenges. In­fla­tion at times ex­ceeded 10 per­cent a year, whit­tling away the value of our paychecks; that wasn’t helped by the “WIN” but­tons (Whip In­fla­tion Now) that Pres­id­ent Ford had dis­trib­uted a year earli­er. Gas­ol­ine was ra­tioned, only avail­able on odd or even days, de­pend­ing on your li­cense plate num­ber and re­quir­ing wait­ing in lines that stretched for blocks. Drought con­di­tions led us to save our bath wa­ter for our gar­dens. In­ter­na­tion­al polit­ics had calmed with the re­cent end of the war in Vi­et­nam, but events such as the So­viet in­va­sion of Afgh­anistan in 1979 kept the ten­sions high.

When I entered gradu­ate school, I planned on study­ing geo­metry. That changed as I was drawn to­ward to­po­logy; more ac­cur­ately, I was drawn to­ward join­ing the Kirby group by see­ing Rob work­ing daily with John Harer and Arnie Kas at the lounge black­boards, draw­ing 22 com­pon­ent handle­body dia­grams of the Kum­mer sur­face. The math­em­at­ics looked fas­cin­at­ing, but the ca­marader­ie shared by all the geo­met­ric to­po­lo­gists was equally ap­peal­ing. To­po­logy sem­inars were of­ten group dis­cus­sions and daily lunches were sel­dom missed.

The first pa­per that Rob sug­ges­ted I read was Fox’s “Quick trip” [e1]. The book that it was in was out-of-print, so it be­came the first art­icle I pho­to­copied; at five cents a page, it cost me an hour’s pay. That led me to writ­ing my first knot the­ory com­puter pro­gram, one that would com­pute the ho­mo­logy of branched cov­ers of closed braids; it was in For­tran, writ­ten on punch cards, with a turn-around time of sev­er­al hours for each (usu­ally failed) run.

Fol­low­ing that in­tro­duc­tion to knot the­ory, Rob sug­ges­ted that I read Kauff­man and Taylor’s pa­per on sig­na­tures of links [e2], in which the link sig­na­ture is in­ter­preted in terms of branched cov­ers of four-man­i­folds. I told Rob I didn’t have time for it, I was study­ing for my to­po­logy or­al ex­am. Rob re­spon­ded that wasn’t a prob­lem: “We’ll ask you ques­tions about the pa­per.” That soun­ded good to me un­til a week be­fore the ex­am when it was pos­ted that my ex­am­iners were Chern and Wolf, who knew noth­ing of Rob’s plan. I made it through re­l­at­ively un­scathed and in ret­ro­spect had the be­ne­fit of learn­ing math­em­at­ics as it is best learned, not by spend­ing months cram­ming from text­books. (Al­though “un­scathed,” I still shud­der at the memory of telling Chern that one of his ques­tions about bor­d­ism didn’t make sense.)

The first two pa­pers I wrote came about by study­ing Rob’s work. Rob had writ­ten a pa­per with Ray Lick­or­ish [1] an­swer­ing to the af­firm­at­ive a ques­tion of Camer­on Gor­don: Is every knot con­cord­ant to a prime knot? Their proof was the first that used tangles in the con­text of con­cord­ance. I man­aged to find an al­tern­at­ive proof, us­ing com­pan­ion­ship, that had the ad­vant­age of gen­er­al­iz­ing to three-man­i­folds: every three-man­i­fold is ho­mo­logy cobord­ant to an ir­re­du­cible three-man­i­fold [e3]. (I in­cluded this res­ult in my Rice re­cruit­ment talk while Bobby My­ers was vis­it­ing. With­in a few months, Bobby out­did me [e5], re­pla­cing “ir­re­du­cible” with “hy­per­bol­ic.”) That pa­per of Rob and Ray also high­lighted the prob­lem, first raised by John Con­way, of find­ing con­cord­ance re­la­tions between low cross­ing num­ber knots. This is a top­ic I have re­peatedly re­turned to, cul­min­at­ing in joint work [e9] with Paul Kirk and Ju­lia Collins (a stu­dent of An­drew Ran­icki) of­fer­ing a con­cord­ance clas­si­fic­a­tion of the sub­group of the knot con­cord­ance group gen­er­ated by low cross­ing num­ber knots.

The second of these early pa­pers of mine con­cerned sur­faces in the four-ball. Rob taught a course about four-man­i­folds in which he out­lined a proof of a res­ult he and Sel­man were work­ing on: a four-man­i­fold that res­ults as the cyc­lic cov­er of the four-ball branched over a pushed-in Seifert sur­face for an un­link is de­term­ined en­tirely by the to­po­lo­gic­al type of the sur­face and the de­gree of the cov­er. At the time, some of the de­tails of the proof had not been filled in. As I tried to work out those de­tails, I dis­covered that the iso­topy class of the pushed-in sur­face is, in fact, de­term­ined by the to­po­lo­gic­al type of the sur­face. After writ­ing that up (ac­tu­ally typ­ing it for my thes­is on Bill Gold­man’s Se­lec­tric type­writer) and send­ing a short­er ver­sion to a journ­al [e4], I didn’t re­turn to the top­ic. Yet the main res­ult of the Ak­bu­lut–Kirby pa­per [2], giv­ing an al­gorithm for branched cov­ers of the four-ball branched over Seifert sur­faces for knots, has re­appeared for me fre­quently; for in­stance, the con­struc­tion is es­sen­tial in a joint pa­per I wrote with Matt Hed­den and Danny Ruber­man [e8] show­ing that to­po­lo­gic­ally slice knots need not bound smooth disks with isol­ated sin­gu­lar­it­ies that are cones on knots with Al­ex­an­der poly­no­mi­al one.

I have vis­ited Rob in Berke­ley most years since my gradu­ate school days. The most math­em­at­ic­ally in­tens­ive stretch was dur­ing my sab­bat­ic­al year, 2001. By luck, Rob was teach­ing a course on Hee­gaard Flo­er the­ory, work­ing through the first two pre­prints from Oz­sváth and Szabó [e6], [e7]. Not only was it a chance to dig in­to that amaz­ing work, but, just like when I was a gradu­ate stu­dent, I was sur­roun­ded by a won­der­ful group of math­em­aticians. Thomas Mark was there as a postdoc, ready to help out with the chal­lenges. Among Rob’s stu­dents, I got to spend time with Eli Grigsby and Lawrence Roberts. (At the time, Eli was already mas­ter­ing Hee­gaard dia­grams for sur­gery on knots, filling me in on the de­tails of those con­struc­tions; Lawrence taught me all about con­tact struc­tures and Thur­ston–Ben­nequin the­ory.)

Re­cently, 40 years after my gradu­ation, I was again en­gaged with Rob in a series of math­em­at­ic­al con­ver­sa­tions. In 2020, four of his math­em­at­ic­al des­cend­ants (Blair, Camp­isi, Taylor, To­mova) pos­ted a pa­per re­lated to con­nec­ted sum de­com­pos­i­tions of sur­faces in four-space. I had worked on such top­ics in the 1980s, and this new pa­per led me to won­der about wheth­er the res­ults in this area car­ried over to the loc­ally flat cat­egory. I soon real­ized that simply show­ing that the con­nec­ted sum of loc­ally flat sur­faces in to­po­lo­gic­al four-man­i­folds is well-defined is highly non­trivi­al. In fact, the proof that the con­nec­ted sum of to­po­lo­gic­al man­i­folds in any di­men­sion is well-defined is a deep res­ult; it de­pends on An­nu­lus Con­jec­ture, proved in high­er di­men­sions by Rob in his 1969 pa­per and re­quir­ing the work of Freed­man and Quinn for di­men­sion four. Through much of the cov­id pan­dem­ic, I’d have Zoom con­ver­sa­tions with Rob as I tried to com­plete a re­l­at­ive form of the same res­ult. It did work out for codi­men­sion two, but we have yet to settle the gen­er­al ques­tion: Giv­en con­nec­ted ori­ented pairs of man­i­folds in the to­po­lo­gic­al loc­ally flat cat­egory, \( (X_1, F_1) \) and \( (X_2, F_2) \), is there a well-defined con­nec­ted sum op­er­a­tion yield­ing \( (X_1, F_1) \mathbin{\#} (X_2, F_2) \) in the case that the \( F_i \) have codi­men­sion great­er than 2?

As oth­ers will surely re­peat, Rob showed us all that math­em­at­ics was simply a part of our lives. His kids, Kate and Rolf, were part of our com­munity. One memory I have is of them com­ing to din­ner at my apart­ment. As con­ver­sa­tions lingered late in­to the night, Rolf fell asleep on a re­clin­ing chair, ly­ing back­wards. Even­tu­ally Rob at­temp­ted to wake him by re­turn­ing the re­cliner to its up­right po­s­i­tion, but Rolf slept on, up­side down on the chair. Mean­while, Kate had fallen asleep on a dingy, torn, old chair that I had wrapped in a bed­spread to make present­able. When Rob car­ried Kate away, she took the bed­spread with her.

There were also Rob’s out­door ad­ven­tures. My friend and of­fice mate, Kathy O’Hara, and her hus­band Fred Good­man would of­ten go on week­end kayak­ing trips with Rob. I’d be re­lieved on Monday morn­ing when she would be back in the of­fice, in one piece, but some­times barely so. More wor­ri­some was when Rob didn’t re­turn to teach on a Monday morn­ing. As it turned out, he and Den­nis John­son had been on an un­mapped river and had to take an un­ex­pec­ted de­tour that ad­ded a day. When I asked Rob if he had planned ahead to have suf­fi­cient food, he re­spon­ded that it wasn’t a prob­lem, he and Den­nis had a couple of ex­tra candy bars.

Re­gard­ing those week­end kayak­ing trips, I vividly re­call a Monday morn­ing when Kathy re­lated to me a con­ver­sa­tion she had with Rob while on a river. The dis­cus­sion was about the ad­vant­ages of gradu­at­ing as quickly as pos­sible as op­posed to linger­ing on. Rob told her the ad­vant­ages of each, and then told Kathy, “Look at Chuck, he can fin­ish whenev­er he likes.” She asked him if he had told me that, and he re­spon­ded that he as­sumed I knew. That is when I learned that I was go­ing to gradu­ate.

Charles Liv­ing­ston began his col­lege stud­ies at UCLA in 1971. Two years later he trans­ferred to MIT, where he re­ceived his math­em­at­ics de­gree in 1975. His gradu­ate work was done at the Uni­versity of Cali­for­nia, Berke­ley, dur­ing the years 1975–1980. From there he took a postdoc­tor­al po­s­i­tion at Rice Uni­versity and then moved to In­di­ana Uni­versity, Bloom­ing­ton. Be­gin­ning in 2019 he has held the title of Pro­fess­or Emer­it­us.