Celebratio Mathematica

Joel Hass: Let’s start at the be­gin­ning. What led you to be­come a math­em­atician? When did you start think­ing about be­com­ing a math­em­atician and what do you re­mem­ber about your days in col­lege and gradu­ate school?

Rob Kirby: The short story is I star­ted out at school, when I was five and a half and it went well, so I nev­er left. But the pre­amble to gradu­ate school is kind of in­ter­est­ing; it leads me to go back even a little fur­ther, be­cause it will tell you something about how I learned and how I do math.

My moth­er was a de­voted moth­er and she got me read­ing, writ­ing, and do­ing arith­met­ic well be­fore I was five. The arith­met­ic by play­ing board games with dice where we ad­ded for a while and then to spice it up we mul­ti­plied. Things like that, mak­ing puzzles, geo­graphy, mem­or­iz­ing states and cap­it­ols. She got me in first grade when I was five and a half, that took a little bit of ar­gu­ment in those days, that would be 1943. After about three weeks, the teach­er called my moth­er and said this isn’t work­ing, he just sits in his seat and looks out the win­dow all the time and doesn’t par­ti­cip­ate. So my moth­er then talked to her and found out that I was just bored be­cause I knew what they were do­ing already. So she sug­ges­ted that the teach­er give me something ex­tra and that worked. I still spent a lot of time look­ing out the win­dow, and this has be­come a habit, to this day dur­ing sem­inars, and it star­ted way back then.

Here’s a little an­ec­dote. We lived in Yakima, Wash­ing­ton for two months in Janu­ary, Feb­ru­ary 1945. I took a bus to school and got a free school lunch. In those days they ex­pec­ted you to fin­ish your plate be­fore you got desert, so you didn’t waste food. I hated ma­car­oni and cheese and that’s what they served every Thursday, so one Thursday I de­cided to play hooky and I went up to the school bus and didn’t get on., and then went out to the back woods to a creek and played in the creek all day long, and came back as though I just got off the school bus. Nobody was wiser, in­clud­ing my moth­er.

This shows you something about my char­ac­ter, play­ing hooky and spend­ing all day at the creek, prob­ably in Feb­ru­ary, when it’s cold in Yakima, and all be­cause I don’t like ma­car­oni and cheese.

In Novem­ber 1946 the fam­ily moved to Far­ragut, Idaho, where my Dad de­cided to try his hand at teach­ing at a new col­lege on the premises of the de­com­mis­sioned nav­al base on Lake Pend Or­eille. I went to a three room school­house where the third, fourth and fifth were in one room. I was in fourth grade so I also listened to the fifth graders and it was nat­ur­al that I would skip fifth grade, so I was young­er than most of my class­mates from then on.

I then went to Chica­go as an un­der­gradu­ate. I had al­ways liked play­ing games. So I get to Chica­go and Bur­ton Jud­son Courts (dorm­it­ory) and here were a few hun­dred boys all ready to play games. That was won­der­ful and that’s ba­sic­ally what I did, for the first two years in the dorms, and I didn’t go to class much. Very little in fact, ex­cept for math class. I think I pretty much at­ten­ded those.

My grand­fath­er had taught me to play chess when I was eight; ap­par­ently I pestered him con­stantly to play games and was re­stric­ted to one game per day. At the dorm I met a fel­low, Mike Robin­son, and we star­ted play­ing chess. For two years, it typ­ic­ally went like this; lunch would be over and we’d say let’s play a quick game be­fore class. So we’d sit down and about four or five hours later we’d fin­ish the game, which nev­er really fin­ished be­cause whenev­er we made a mis­take we would take it back and ana­lyze the po­s­i­tion. So there’s lots of ana­lys­is and not so much is­sue with win­ning. And that really was how I be­came a good chess play­er.

In Decem­ber 1956 the bi­an­nu­al in­ter­col­legi­ate chess cham­pi­on­ship was held in Man­hat­tan (Chica­go to New York was my first plane trip). The cen­ter of chess was Man­hat­tan and East coast schools and they did not ex­pect to lose to some kids from the boon­docks, which was what Chica­go was ac­cord­ing to them. So we won, and there’s an in­ter­est­ing pho­to­graph of the four of us (in­clud­ing Robin­son) in Cel­eb­ra­tio Math­em­at­ica. We went back two years later, and we won again, so that was fun.

So dur­ing those years as an un­der­gradu­ate I got more F’s than A’s. I think I got three A’s if I re­mem­ber right and def­in­itely more F’s. Even with grade in­fla­tion that was still not a good re­cord.

So one of the key things that happened was that you had to pass a for­eign lan­guage ex­am in those days. Chica­go had a year-long course and there was a big long ex­am at the end of the year that de­term­ined your grade. In my third year I took the Ger­man ex­am and failed. I took it again in my fourth year and I failed again, be­cause, for one thing, it’s very hard to pass the or­al part of the ex­am if you have nev­er gone to class and nev­er spoken any Ger­man. And that I just didn’t do. So my grades were such that I lost my fel­low­ship after three years. My dad gave me \$1,000, a fair amount in those days, and said “That’s it, you’re on your own,” which was com­pletely fair. Tu­ition, by the way, was \$690 a year and that was true for the en­tire time I was there. My fel­low­ship was twelve hun­dred dol­lars a year and that covered everything; its hard to be­lieve today, even with in­fla­tion.

Be­cause I had failed Ger­man twice I went back for a fifth year at the Uni­versity of Chica­go in or­der to pass Ger­man to get a bach­el­or’s de­gree. While there dur­ing that fifth year I took sev­en gradu­ate courses in math. The cru­cial one was a course from Kelly’s book, Gen­er­al To­po­logy, with lots of great prob­lems. I re­mem­ber the prob­lems and the pleas­ure of chew­ing on a prob­lem for a while be­fore I got a proof of something.

Hav­ing got­ten my bach­el­or’s be­cause I passed Ger­man with a D in my fifth year, then what to do. Well I en­joyed be­ing in Chica­go, I was hav­ing a good time there, I liked it. And I don’t re­mem­ber ever really think­ing about this is­sue much. Any­way, I ap­plied to Chica­go to come back for a Mas­ter’s de­gree and the ar­gu­ment was that I had taken sev­en of the eight courses that were covered in the Mas­ter’s ex­am and wanted a chance to take that ex­am and get a Mas­ter’s de­gree. They said, “Okay, you are ad­mit­ted pro­vi­sion­ally if you get grades closer to B then C in the fall quarter.” So I took three courses in the fall (1959), and I got a B and a C and a Pass. That’s not strictly closer to a B than a C. Non­ethe­less they didn’t say any­thing, and so then I took the Mas­ter’s ex­am.

There were a few pos­sib­il­it­ies, you could pass and they’d give you money to con­tin­ue, or you could pass and they wouldn’t give you any money but they wel­comed you to con­tin­ue, or you could pass and they re­com­mend you go else­where, or you could fail. So I passed, but they re­com­men­ded I go else­where. I don’t re­mem­ber think­ing much about this, but I de­cided I wanted to stay in Chica­go, I liked it there and had friends there, and so on.

I needed some money, so I ap­plied for a job at Roosevelt Uni­versity, a down­town col­lege in Chica­go, where people took a course or two after work. Saun­ders Mac Lane, who I al­ways ap­pre­ci­ated (he was a good guy in my eyes), wrote a let­ter and I pic­ture the let­ter turn­ing up at the top of the pile just as the chair was about to make an of­fer, and he looks at that and makes me an of­fer, so I took it, and I taught there for four years. That’s how I man­aged to pay my way through gradu­ate school.

So it was ne­ces­sary that I fail Ger­man twice and take sev­en gradu­ate courses in my fifth year, oth­er­wise at the end of four years I nev­er would have got­ten in­to any de­cent gradu­ate school.

So now I’m in gradu­ate school. But first, in the sum­mer of 1960, I met In­grid whom I mar­ried a few years later; she was a stu­dent at Berke­ley in Ger­man Stud­ies, Ger­man lan­guage. And so I de­cided that, hav­ing met her, I would trans­fer to Berke­ley. I ap­plied and I don’t think I heard any­thing but I packed up all my stuff and drove out to Berke­ley and dis­covered I wasn’t ad­mit­ted. That sur­prised me a little, be­cause I thought get­ting a Mas­ter’s at Chica­go would get me in­to gradu­ate school at Berke­ley. But they had high­er as­pir­a­tions than people like me.

So I told Roosevelt Uni­versity that I would come back and teach again, I hadn’t done much dur­ing 1960–61, fig­ur­ing I was trans­fer­ring any­way and per­haps be­ing tired of tak­ing courses after 16 years. So now back in 1961–62 I had to take the qual­i­fy­ing ex­am. In Chica­go in those days, it was an or­al ex­am, 45 minutes each on two sub­jects that you could choose. I had chosen fi­nite groups from Mar­shall Hall’s book [e1] and most of the book on ho­mo­topy the­ory writ­ten by Hu [e2]. I took the ex­am in late fall, and of course I failed it. You could take it a second time. They give me a bit of ad­vice which was start talk­ing math­em­at­ics, be­come part of the com­munity; you don’t look like you can stand on your feet and speak math­em­at­ics, be­cause you’ve nev­er done it.

So I star­ted to go to Tea and something im­port­ant happened. Nor­man Steen­rod was vis­it­ing for the year and one day he was de­scrib­ing the Hopf map; you know the usu­al pic­ture with the \( (1,1) \) curves on the con­cent­ric tori around the unit circle and the \( z \)-ax­is, you know the pic­ture. I had nev­er seen that be­fore; to me it was just tak­ing a pair of com­plex num­bers and map­ping them to their pro­ject­iv­iz­a­tion, which just didn’t mean any­thing to me, not in­tu­it­ively, where­as this is won­der­ful. So I real­ized all the stuff you could pick up by word of mouth. I took the qual­i­fy­ing ex­am again, and this time they said “Okay, you pass; but you didn’t do very well in the ho­mo­topy the­ory part of it, so we re­com­mend you don’t write a thes­is in to­po­logy.”

So I thought about this a bit. One thing that had happened dur­ing that year was that there was a postdoc named George Mc­Carty, a stu­dent of Richard Arens at UCLA. He had writ­ten a thes­is [e9] on the ho­mo­topy groups of the space of homeo­morph­isms of a man­i­fold. He didn’t really have any­body to talk to there and so he asked me if I wanted to read his thes­is and talk about it. I looked at it, it wasn’t hard, in fact it was easy for me to un­der­stand. So this was in the back of my mind when, at the end of the year, I went in to see Saun­ders Mac Lane to ask him to be my ad­visor. He wisely asked what I was in­ter­ested in, and I said ho­mo­lo­gic­al al­gebra or group the­ory or this stuff on to­po­logy of man­i­folds. He said: “Go home for the sum­mer and think about those three and come back in the fall and we’ll talk about it.” So I went home for the sum­mer, had some sort of job, and all I thought about, not that much, was man­i­folds. Also I real­ized Mac Lane was the kind of ad­visor who would want you to come in once a week and say what you’d ac­com­plished, and I ab­so­lutely didn’t want that kind of set-up for me.

I had taken courses from El­don Dyer, who was a Moore stu­dent, but had turned very al­geb­ra­ic and his courses to me were form­al and not so in­tu­it­ive. I would have picked Dick Lashof who was more to­po­lo­gic­ally minded, but Lashof had been on my qual­i­fy­ing ex­am and pre­sum­ably had been the one who’d said “Don’t write a thes­is in to­po­logy.” So in­stead I did ask El­don Dyer if he’d be my ad­visor. We chat­ted a little bit, and then I left, think­ing the ball’s in his court, he prob­ably wants to find out more about me be­fore ac­cept­ing me as a stu­dent.

From then on, once a month or so, I’d drop in and ask a ques­tion on something, and we’d talk for a bit. At some point it was clear that I’d be­come his stu­dent. For a while my main in­ter­ac­tion with El­don was play­ing cut-throat hand­ball with him and John Polk­ing.

Now it was time to write a thes­is. I well knew that I could fail courses and ex­ams, but some­how I figured that I’d be able to do re­search. I’m not sure where that con­fid­ence came from. Two things were im­port­ant for the years 1962–64: my first pub­lic­a­tion and my study part­ner­ship with Wal­ter Daum, a fel­low stu­dent (about which more be­low). Here’s how the pub­lic­a­tion came about. I’d heard about the An­nu­lus Con­jec­ture and Mil­nor’s sev­en fam­ous prob­lems in geo­met­ric to­po­logy. I could un­der­stand some of these prob­lems, in par­tic­u­lar, the An­nu­lus Con­jec­ture which I liked and I thought about.

I had the idea that if you pic­tured the two loc­ally flat em­bed­ded 2-spheres, and if you could put a little tube between them, then the com­ple­ment was con­tract­ible and we now had Mort Brown’s Schoen­flies The­or­em so that the com­ple­ment is a ball and you put the ball to­geth­er with the tube and you’ve got an an­nu­lus. That’s a proof of the An­nu­lus Con­jec­ture, but with that big hy­po­thes­is. Dyer was ac­tu­ally a little bit im­pressed, and he was an ed­it­or of Pro­ceed­ings of the AMS, and so they ac­cep­ted it, and so I got a pub­lic­a­tion [2]. Dyer might have thought that, well you know there is a chance this guy can fin­ish a thes­is.

Now I’m pretty sure this res­ult was ac­tu­ally folk­lore in the RH Bing crowd, but I didn’t know, as nobody had writ­ten it down be­fore I did.

One of the things about that sum­mer of 1963 was that I was in­vited to go on the US team to the World Stu­dent Chess Cham­pi­on­ships in Dubrovnik but I couldn’t af­ford to take In­grid. Some­how it didn’t work out to leave her be­hind, so I didn’t go and I’ve al­ways re­gret­ted it. I would have liked to have seen Dubrovnik in 1963.

At some point around this time, in 1962–63, I met a fel­low grad stu­dent, Wal­ter Daum. He was a good stu­dent, and ac­com­plished in four years what took me sev­en. He was an in­ter­est­ing guy, and so we began a sem­in­ar: we met one hour every day of the week. And we would read through pa­pers to­geth­er, just the two of us. One of us would stand at the black­board and write a the­or­em on the board. We would sort of puzzle about it, how would we prove this, and if we needed to we would go get a hint, and think some more, and this I found to be a very con­du­cive way to learn. In fact this was the most im­port­ant learn­ing ex­per­i­ence I’ve had in my life. Just get­ting in­to this, learn­ing math­em­at­ics just the two of us every day for an hour. We didn’t pre­pare ahead of time, that was against the rules. Our rules.

So we covered pa­pers by Stallings [e6] [e10]. We read through Zee­man’s to­po­logy notes, which had all those no­tions, col­lapsing and so on, all those words that are kind of lost now. That was very, very use­ful, not just the PL case but all the to­po­logy that was in­volved, gen­er­al po­s­i­tion, for ex­ample. So this con­tin­ued with Wal­ter Daum for a couple of years and then in 1964 Dyer de­cided to go to Rice Uni­versity. But that year sev­en to­po­lo­gists turned up there, in­clud­ing Stallings (as a vis­it­or), John Hempel and Les Glaser. So it was a lively year. While I was there I sort of wound up writ­ing a Ph.D. thes­is on smooth­ing loc­ally flat em­bed­dings [1]. The idea was that if you had a smooth man­i­fold, loc­ally flat em­bed­ded in­to an­oth­er smooth man­i­fold, and the codi­men­sion was roughly one-third, then this em­bed­ding was iso­top­ic to a smooth em­bed­ding.

So then I have a thes­is, and, in­ter­est­ingly enough, Wal­ter Daum did not yet have one. He de­cided to take a job at CCNY, and fin­ish his thes­is there, which he nev­er did. I haven’t seen him since, but over 30 years later, I tried to find out what happened and I kept com­ing across this Wal­ter Daum who’s pub­lish­ing pa­pers on Marx­ism, Sta­lin­ism and so on and it turned out that this was the same guy. Work­ing with him had been cru­cial to my pro­gress.

Those were my gradu­ate stu­dent days.

JH: After gradu­ate school you went to UCLA and made some break­throughs there. Tell us the back­ground and how that came about.

RK: First of all, there’s a funny story about get­ting my job at UCLA, a per­fect ex­ample of the old boy’s net­work. Be­cause my ad­visor El­don Dyer was at the an­nu­al AMS meet­ing in Den­ver and was hav­ing a beer with the chair at UCLA, Low­ell Page, a very nice guy as I found out. El­don came back and said “You’ve got an of­fer at UCLA.” So this was prob­ably done over a beer.

I was ac­tu­ally hop­ing to go to the In­sti­tute for a year, or to Michigan, where Mort Brown was; I don’t know if I said much at all about this to Dyer, but I cer­tainly got the im­pres­sion from him that I should ac­cept UCLA and be thank­ful. So I was thank­ful, be­cause when I went to UCLA, there really wer­en’t many to­po­lo­gists of note in South­ern Cali­for­nia at that time. There was Hu, who wrote the ho­mo­topy the­ory book. He was at UCLA, but his wife had died, and he re­treated to his study where he wrote some more books, and I’m not sure that I ever met him. He just wasn’t around. And in­stead, there were some young guys there: Bob Brown, Dav­id Gill­man, a couple of postdocs, in­clud­ing Ned Staples and my two Ph.D. stu­dents, Dav­id Gauld and Ted Turn­er. And so we were just a young group who had a good time to­geth­er; we ran sem­inars. It was a nice place, and it re­minded me of Chern’s ad­vice. Chern once said that it’s not a bad idea after you get your Ph.D. to go to some pro­vin­cial uni­versity where it’s quiet and you can do your work. And that’s what he did after he got his Ph.D. in Ger­many. He did this and it worked for him and that’s prob­ably why he said it. But any­way, this worked for me, I think, be­cause if I had gone to the In­sti­tute for Ad­vanced Study, I would have felt that there was this op­por­tun­ity, I should go to all of these lec­tures by these fam­ous math­em­aticians like Borel and so on. As I said earli­er I was not good at learn­ing via lec­tures. I would have been listen­ing rather than try­ing to do something. So ac­tu­ally I think it turned out very that well that I went to UCLA.

The next thing I did math­em­at­ic­ally was — well, Mort Brown had proved the Schoen­flies The­or­em [e4], and then Jim Cantrell at Geor­gia had shown that if you as­sume that the sphere is loc­ally flat ex­cept at one point, then it still bounded a ball on both sides, ex­cept in di­men­sion 3, where a neigh­bor­hood of the Fox–Artin arc will give you a loc­ally flat em­bed­ding ex­cept at that bad point [e11].

Aside: I had gone up to teach at Berke­ley for the sum­mer quarter. I didn’t have an NSF and this was some ex­tra teach­ing and I needed the money. I gradu­ated with six thou­sand dol­lars in debt, mostly for tu­ition for In­grid at Chica­go. In today’s money, that’s about 40 thou­sand. I fi­nally real­ized it was pretty easy to pay off if I just con­tin­ued liv­ing like a gradu­ate stu­dent for the next year or two. Be­cause my salary tripled. My fel­low­ship had been \$3,000 and I star­ted at UCLA at \$8,900 per year and then went up a little bit slowly. So I was teach­ing up at Berke­ley in Tony Phil­lips’ of­fice, which was dec­or­ated with all kinds of pic­tures from a Sci­entif­ic Amer­ic­an art­icle on turn­ing the 2-sphere in­side out. We were in the T4 build­ing, with worn wooden steps and so on. It was a great place be­cause all sorts people had of­fices there. All the young people.

Any­way, I ended up fig­ur­ing out how to show that if you had a loc­ally flat codi­men­sion one sphere in an­oth­er sphere, loc­ally flat ex­cept maybe at a Can­tor set of points, which had to be tame, then the sphere bounded a ball any­way, so ac­tu­ally it was loc­ally flat. This was a con­sid­er­able gen­er­al­iz­a­tion of Cantrell’s The­or­em. I sent it to the An­nals and Mil­nor handled it, and I got a very good ref­er­ee­ing job, with one par­tic­u­lar im­prove­ment. I think the ref­er­ee was Dale Rolf­sen. And so they ac­cep­ted it [3], and that was enough that I did get an of­fer from the Uni­versity of Wis­con­sin in Madis­on, the main school that no­ticed this.

By then I had real­ized that I liked liv­ing in the West and didn’t really want to live in Madis­on. But I did ac­cept the job for a semester in the fall of 1967. One nice thing was that Ray­mond Lick­or­ish was there and we got to know each oth­er and be­came good friends. This helped ac­count for all the great times I vis­ited Cam­bridge in the fol­low­ing years. Also one oth­er thing, show­ing how na­ive I was about things back in those days, the Uni­versity of Chica­go knew I was vis­it­ing Wis­con­sin and asked me to come down to give a col­loqui­um. Well, I figured that this pre­print, this stuff on the loc­ally flat Schoen­flies Con­jec­ture, I figured that was old hat be­cause it had been out, not at something like the arX­iv, but it had been out for a little over a year. So I couldn’t talk about that. Well, of course, in real­ity, that’s why they in­vited me — it was to talk about something good like that. In­stead, I gave some un­mem­or­able talk about an­oth­er top­ic that nev­er went any­where. So it shows you that I was still na­ive about how the math world worked.

And then in 1967 I went to the Geor­gia To­po­logy Con­fer­ence, and you have to re­mem­ber that in those days, this was about the only to­po­logy con­fer­ence in the whole year. There were big ones like the Bechtel con­fer­ence in 1963 and the AMS meet­ings. But this was really the only to­po­logy con­fer­ence. It star­ted in 1961 and it was a one-week con­fer­ence ex­cept two weeks every eighth year.

I went back the next year, and I got a copy of Cernavskii’s pa­per [e13] prov­ing the loc­al con­tract­ib­il­ity of the space of homeo­morph­isms of \( \mathbb{R}^n \). This was very in­ter­est­ing to me be­cause for years I had been try­ing to say something about an ar­bit­rary homeo­morph­ism and couldn’t un­less it had some ad­di­tion­al prop­erty like be­ing smooth at a point or PL on an open set.

I knew that if the homeo­morph­ism is bounded at in­fin­ity — that is, no point is moved more than a fixed con­stant, over all of \( \mathbb{R}^n \) — then it is iso­top­ic to the iden­tity by the Al­ex­an­der iso­topy [e3]. So we knew that but not too much else. And then here was this pa­per by Cernavskii [e13] show­ing that the space of homeo­morph­isms was loc­ally con­tract­ible. I think that pa­per had a flaw but he re­paired it. But I nev­er read the pa­per that care­fully al­though I think I know what he was do­ing. Any­way, I brought this back to UCLA and was sit­ting there in Au­gust 1968, my six-month-old son was asleep and I was babysit­ting. Something in that pa­per made me pic­ture a lift of a homeo­morph­ism of the \( n \)-tor­us to \( \mathbb{R}^n \). Don’t know what it was that made me think of that. And so of course, I pic­tured the two-di­men­sion­al case. And if you do that, you eas­ily see that it’s peri­od­ic. Ob­vi­ously lots of people knew that; it’s ba­sic. If you com­pose with a dif­feo­morph­ism, you can make it ho­mo­top­ic to the iden­tity. And there­fore, when you lift that homeo­morph­ism to \( \mathbb{R}^n \) it will be bounded and hence iso­top­ic to the iden­tity.

It was known from the work of Brown and Her­mann Gluck [e8] that there is a close re­la­tion­ship between the an­nu­lus con­jec­ture and the stable homeo­morph­ism con­jec­ture. And so the fact that this uni­ver­sal cov­er of the homeo­morph­ism of the tor­us was bounded meant that it was iso­top­ic to the iden­tity and there­fore stable. And stable is a loc­al prop­erty, so that meant the ori­gin­al homeo­morph­ism was stable. This all happened that even­ing, and I real­ized that I had a good the­or­em. Just the fact that the homeo­morph­isms of the \( n \)-tor­us are stable was clearly strik­ing. Just be­cause this was close to the an­nu­lus con­jec­ture. Noth­ing was known. So not that even­ing, but shortly, with­in a day or two or three, the nat­ur­al ques­tion was: well, if you start with a homeo­morph­ism of \( \mathbb{R}^n \), how do you get a homeo­morph­ism of the \( n \)-tor­us, whose sta­bil­ity im­plies that the homeo­morph­ism of \( \mathbb{R}^n \) is also stable?

I knew about im­mer­sion the­ory from read­ing pa­pers about that [e12]. And so I knew that the punc­tured tor­us can im­merse in \( \mathbb{R}^n \). If you im­mersed the punc­tured tor­us in \( \mathbb{R}^n \) and you have a homeo­morph­ism which is very close to the iden­tity, no more than ep­si­lon, then a slightly smal­ler im­mersed tor­us, which is moved only by ep­si­lon, a little bit, lies in­side the slightly lar­ger ver­sion of the im­mersed tor­us. So that gives you an em­bed­ding of the punc­tured tor­us in­to a slightly big­ger punc­tured tor­us. But what’s miss­ing is a ball, an \( n \)-handle. Us­ing the Schoen­flies The­or­em, you can fill in the ball and you have a homeo­morph­ism of the tor­us. And then that’s stable and there­fore the ori­gin­al homeo­morph­ism of \( \mathbb{R}^n \) is stable. And fur­ther­more, you can fill in that \( n \)-ball ca­non­ic­ally. As you vary the \( n \)-ball you put in by the Schoen­flies The­or­em will vary con­tinu­ously with the para­met­riz­a­tion of the sphere. So it’s all a ca­non­ic­al ver­sion of this. So then I had a proof of the loc­al con­tract­ib­il­ity of \( \mathbb{R}^n \). And I was very pleased, as it was quite easy to do.

What else can you do with the tor­us trick, what can you say about a homeo­morph­ism which is not close to the iden­tity? Well, you can then im­merse this punc­tured tor­us in \( \mathbb{R}^n \) and use the homeo­morph­ism to pull back the stand­ard PL struc­ture on \( \mathbb{R}^n \) to a pos­sibly non­stand­ard PL struc­ture on the punc­tured \( n \)-tor­us. And then from high­er-di­men­sion­al to­po­logy, I knew that you could fill in the punc­ture with a PL \( n \)-ball in di­men­sion great­er than five. Which is where di­men­sion­al re­stric­tion be­gins to come in. So now we had a PL struc­ture on the \( n \)-tor­us that wasn’t the same as the stand­ard one. Or are they the same?

Wall was still in the midst of writ­ing his book on nonsimply con­nec­ted sur­gery [e14], but he had not yet ap­plied it to the \( n \)-tor­us. And so that wasn’t in print any­where. So it’s just left as a con­jec­ture in the pa­per, that if this pos­sibly non­stand­ard PL struc­ture on the \( n \)-tor­us was ac­tu­ally PL, homeo­morph­ic to the stand­ard \( n \)-tor­us then I would have proved that a homeo­morph­ism of \( \mathbb{R}^n \) was al­ways iso­top­ic to the iden­tity without know­ing that it was close to the iden­tity. So that was the pa­per that was sent to the An­nals [4].

By good luck I had taught that sum­mer of 1968 at UCLA and then I had the fall free so I had ar­ranged to go to the In­sti­tute. So I’m there and in the first week or two, I gen­er­al­ize the tor­us trick to the \( n \)-tor­us cross a \( k \)-ball, a ball of di­men­sion \( k \), be­cause now there’s a whole the­ory for handles, \( k \)-di­men­sion­al handles, as the pre­vi­ous case was just the zero-handle case. So I had done that.

And then one day I found my­self at a col­loqui­um din­ner sit­ting next to Larry Sieben­mann, and we got to talk­ing, and we stayed well after the din­ner was over. And by the time we’d fin­ished, I think Larry had drained and ab­sorbed everything I knew, in par­tic­u­lar, all the stuff about the An­nu­lus Con­jec­ture. And he was very in­ter­ested in this, and that’s how our col­lab­or­a­tion began. We would meet in the af­ter­noon and the next morn­ing I would wake to find notes be­hind my screen door at the In­sti­tute hous­ing, and I’d see Larry in the af­ter­noon and talk awhile and then more notes the next morn­ing. So this one went on for a bit.

But we were learn­ing about sur­gery and es­pe­cially the nonsimply con­nec­ted case. Bill Browder gave some very help­ful lec­tures. Larry knew lots of stuff that I didn’t know about high-di­men­sion­al sur­gery and such things. And at one point we real­ized from Wall’s work that there was just a \( \mathbb{Z}/2 \) ob­struc­tion for the 0-handle case. And I re­mem­ber I said something about “what would hap­pen if you double that?” — think­ing what would hap­pen if put­ting two one-handles end-to-end, doub­ling that way. But Larry thought about doub­ling in a double cov­er. And Larry’s idea was a great idea be­cause if this ob­struc­tion was just a \( \mathbb{Z}/2 \) ob­struc­tion in the third co­homo­logy of an \( n \)-tor­us, then you could double cov­er every gen­er­at­ing circle. So take a \( 2^n \) fold cov­er. So now the pos­sibly non­stand­ard PL struc­ture on the \( n \)-tor­us was ac­tu­ally stand­ard. Then con­tin­ue on to the uni­ver­sal cov­er and you get a PL homeo­morph­ism. This im­plies that the ori­gin­al homeo­morph­ism of \( \mathbb{R}^n \) is stable. So that was an ad­ded note in my An­nals pa­per [4]. Larry had come up with this idea and now we had a proof of the An­nu­lus Con­jec­ture in all di­men­sions oth­er than four.

So that’s how that happened. And then we set out (Larry mostly) to elab­or­ate a ver­sion of the smooth­ing PL man­i­folds the­ory ad­ap­ted to put­ting PL struc­tures on TOP man­i­folds, e.g., the product struc­ture the­or­em, con­cord­ance im­plies iso­topy, etc., in try­ing to tri­an­gu­late man­i­folds. When the sur­gery the­ory be­came clear, we ended up with the the­or­em, in Decem­ber or so, that man­i­folds have PL struc­tures in di­men­sion big­ger than five if and only if a cer­tain ob­struc­tion van­ishes. That ob­struc­tion is in \( \pi_3(\operatorname{Top}/\operatorname{PL}) \), which at that time was either 0 or \( \mathbb{Z}/2 \).

And then we par­ted ways by Janu­ary 1st, and I went back to teach­ing at UCLA. Over the next couple months, I was the one who wanted to knock off this ob­struc­tion, to try to show that \( \pi_3(\operatorname{Top}/\operatorname{PL}) = 0 \), be­cause everything else was zero so why not that; that was about the level of my think­ing. And so I thought about that a little bit, where­as for Larry, if that was zero, then all this elab­or­ate ob­struc­tion the­ory and so on would be un­ne­ces­sary, be­cause you could just straight­en any handle it­self. So he was mo­tiv­ated to show that \( \pi_3(\operatorname{Top}/\operatorname{PL}) \) was not equal to zero.

Then a bunch of us met at Heav­enly Val­ley ski re­sort near Lake Tahoe, and Larry was there. This was in March, in spring break. Larry was sleep­ing in the base­ment, be­cause of his al­ler­gies and we no­ticed that he wasn’t get­ting out on the slopes. And it turned out af­ter­wards that he was work­ing hard on show­ing that \( \pi_3(\operatorname{Top}/\operatorname{PL}) \) was ac­tu­ally \( \mathbb{Z}/2 \). He did so, and came to UCLA and gave a talk. There is an an­nounce­ment in the No­tices about his ar­gu­ment.1 But then I did one of the few things I re­mem­ber that ac­tu­ally con­trib­uted to our joint work, which was to come up with a simple proof of that,2 which is in the lit­er­at­ure. So that was sort of the end of our joint work.

So then it needed to be writ­ten up. I wanted to write something to­po­lo­gic­al, in­tu­it­ive, and get it out pretty quick (something like my UCLA notes but bet­ter). Larry really wanted to write the manuscript that had all the i’s dot­ted and all the t’s crossed, whereby you didn’t have to go back and gen­er­al­ize it any­more. We were to­geth­er at the In­sti­tute in the fall of 1969, the sum­mer of 1970 in Eng­land and then again in the sum­mer of 1971 in Cam­bridge. And we were go­ing to fin­ish the book but it just didn’t hap­pen. Mainly be­cause I wasn’t do­ing my part, or I wasn’t do­ing it the way he hoped to do it. So he ended up really writ­ing the An­nals of Math book [5] which came out in 1977, and it really does have a lot more in it. So that’s really the story of put­ting PL struc­tures on man­i­folds. Of course, it left out the case of non­com­bin­at­or­i­al tri­an­gu­la­tions, which was re­duced to a fas­cin­at­ing con­jec­ture in 4-man­i­folds by Galewski and Stern, and in­de­pend­ently Matumoto. And that was even­tu­ally taken care of by Cipri­an Man­oles­cu eight or ten years ago [e18]. So that’s kind of the end of the story.

So, the oth­er part of the story was my jobs and so on. It seems Larry and I had fin­ished off half of Mil­nor’s fam­ous sev­en prob­lems. Of Mil­nor’s prob­lems, there was the 3- and 4-di­men­sion­al Poin­caré con­jec­tures and you know about that. The first to be solved was the to­po­lo­gic­al in­vari­ance of ra­tion­al Pontry­agin classes by Novikov, done in about 1965. Den­nis Sul­li­van got part of the Hauptver­mu­tung work­ing up to ho­mo­topy. An­oth­er prob­lem was: “Does a to­po­lo­gic­al man­i­fold have a simple ho­mo­topy type?” and that was a nat­ur­al, not-too-hard co­rol­lary of our work. An­oth­er prob­lem was the to­po­lo­gic­al in­vari­ance of White­head tor­sion and that was an­oth­er co­rol­lary. And then of course, the ex­ist­ence and unique­ness, the Hauptver­mu­tung, and that was an­oth­er one of Mil­nor’s prob­lems. And the An­nu­lus Con­jec­ture. So some­how we got three and a half out of sev­en.

I’d gone back to UCLA in the spring of 1969 and my salary was \$11,000, and I promptly got an of­fer from Rut­gers for \$22,000, which doubled my salary. I didn’t really want to go live in New Brun­swick, but when they double your salary, that was something. So I told UCLA and figured that, well I don’t know just what I said to them but something like “I can’t just turn that down”. And then to my sur­prise they matched it. It was surely be­cause of Mil­nor’s prob­lems: they could point to that, and make an ar­gu­ment to the ad­min­is­tra­tion for the salary to double, which was oth­er­wise un­heard of. And with the step sys­tem, I went to step five, the highest step of full pro­fess­or, and I had been a lowly as­sist­ant pro­fess­or, at step two, maybe, or three. So this was un­heard of. It be­came a well-known case about cam­pus be­cause in most fields, what is it you could do that would get you such a raise? If you were in Eng­lish you could write the equi­val­ent of a Shakespeare play, or a couple of them, and that wouldn’t get you ad­vanced that much that fast. So I stayed at UCLA, quite hap­pily. I liked UCLA, the fac­ulty was a con­geni­al bunch.

I went to Europe in Janu­ary 1970 and I spent a nice month in Geneva with An­dré Hae­fli­ger, who was a very kind, nice man. Even­tu­ally in June I went to the Arbeit­sta­gung that Hirzebruch ran. And they al­ways have a boat ride, and I found my­self stand­ing at the rail next to Steve Smale, and some­how the con­ver­sa­tion ended up with my say­ing that I would be in­ter­ested in an of­fer at Berke­ley. Who wouldn’t have been in those days? Well, that star­ted the wheels turn­ing and I did get an of­fer there in early spring of 1971. The curi­ous thing about that of­fer was this: John Ad­dis­on was the chair, a very good chair and ex­tremely good re­cruit­er, and he called me up to say that the de­part­ment had voted for the of­fer, but there’s some seni­or people here who are not yet at step five and this is a prob­lem. Would you be will­ing to drop to step four and take a salary cut? I said sure, for in those days (not now) step four at Berke­ley was com­par­able to step five at UCLA. So he was pleased but then the ad­min­is­tra­tion didn’t want to raid UCLA by of­fer­ing less salary, that would be an in­sult, so it nev­er happened.

So I came to Berke­ley but took leave in or­der to spend fall 1971 at Har­vard, where I taught a course on tri­an­gu­la­tions. Peter Shalen, then a grad stu­dent, took the course. One of the main things I re­mem­ber about that fall was play­ing pickup bas­ket­ball with Iz Sing­er and John Tate. This was fun. Sing­er was very com­pet­it­ive, and Tate had played bas­ket­ball in high school and was one of the first to de­vel­op a jump shot. Be­fore that every­body had set shots. So that was a nice fall.

Then I came to Berke­ley, and im­me­di­ately star­ted to pick up Ph.D. stu­dents. Marty Schar­le­mann was the first. Soon af­ter­wards Sel­man Ak­bu­lut, Mi­chael Han­del, Paul Melvin and John Harer was a little later. Soon I had 10 stu­dents. So that was fine.

A dec­ade or more earli­er, grad stu­dents might get their Ph.D. in three years, that was some­what typ­ic­al, be­cause they came, took the qual­i­fy­ing ex­am in the first year, pre­sum­ably passed and then got an ad­visor and the ad­visor says, “Oh, why don’t you read these pa­pers and work on this prob­lem?” And gen­er­ally the ad­visor knew how to do that prob­lem and didn’t have time for it, or wasn’t sure or whatever. And so then the stu­dent would read pa­pers and work on that prob­lem and have a res­ult by the fall of their third year. And then they would ap­ply for jobs, and they’d write up their thes­is, which used to have a lot of schol­ar­ship in it, so long pieces. And then they were fin­ished. And then I’d heard that three-quar­ters of Ph.D’s nev­er went on to pub­lish an­oth­er pa­per. I don’t know if that’s true, but that’s what I heard. And it seemed to me that I didn’t do that. I found my own prob­lems. And so I thought that was a much bet­ter way to do things. Find­ing your own prob­lem rather than hav­ing it giv­en to you by your ad­visor was much bet­ter for the stu­dent, even if it took a longer time. So that was my philo­sophy. And so when I had 10 stu­dents, I didn’t feel like I had to give them ten prob­lems, quite the op­pos­ite. I felt that there should be an at­mo­sphere where we had reg­u­lar sem­inars, stu­dents talk to each oth­er just like I had talked to Wal­ter Daum, that we’re just a bunch of people — I used to say this to them, that we’re “just a bunch of people try­ing to prove our next the­or­em”. It’s true that some already had and new stu­dents hadn’t yet, but we’re still try­ing to do the same thing. We’re part­ners in this. And at that time I think that worked quite well. Those stu­dents thrived.

Paul Melvin took a little bit longer to fin­ish, six years (like me) be­cause he was busy play­ing the cello at a very high level. He’s a good teach­er and so he sort of taught young­er stu­dents, in par­tic­u­lar John Harer. Then the next group came along which in­cluded Cole Giller, Chuck Liv­ing­ston, Joel Hass, Bill Menasco, Tim Co­chran, John Hughes, Danny Ruber­man, Bob Gom­pf and Iain Aitchis­on. They were also very good at find­ing their own prob­lems; I don’t think I ever gave a prob­lem to any one of them.

Now let me go back and men­tion how I got go­ing with the cal­cu­lus of framed links. I went to a con­fer­ence in Tokyo in 1973 in hon­or of Kodaira’s re­tire­ment. Kodaira is a Fields medal­ist who did a huge amount on com­plex sur­faces and el­lipt­ic sur­faces. Dur­ing a tea between talks, I over­heard Takao Matumoto talk­ing to some­body and men­tion­ing the Kum­mer sur­face, which was a four-man­i­fold, now called the K3 sur­face.

I had already star­ted jog­ging with Arnold Kas, who was a postdoc and then ten­ure track at Berke­ley. He was a stu­dent of Kodaira, so I asked him about the Kum­mer sur­face and we began to talk. And he was de­scrib­ing all sorts of stuff about el­lipt­ic sur­faces and how they drew them, and what a van­ish­ing cycle was, and the mono­dromy go­ing around a sin­gu­lar fiber and all of this stuff. And I thought it was very to­po­lo­gic­al and so I was trans­lat­ing it in­to handle­bod­ies. Blow­ing up was just con­nect sum­ming with a \( -\mathbb{CP}^2 \). So we talked quite a lot. I learned a lot from him and then he de­par­ted to go to Ore­gon State. I was at IHES in spring of 1974 and Kas vis­ited and we talked more.

Then I was be­gin­ning to see that all of these moves they did on el­lipt­ic sur­faces, moves they were do­ing with a tu­bu­lar neigh­bor­hood of a cer­tain con­fig­ur­a­tion of spheres in the com­plex sur­face, were not chan­ging the bound­ary. And so I began to won­der if there was something more gen­er­al here, that if you had two framed links that gave the same bound­ary, could there be a set of moves tak­ing one framed link to the oth­er, ana­log­ous to the moves from one con­fig­ur­a­tion of com­plex curves to an­oth­er?

We knew that every three-man­i­fold was the bound­ary of a four-man­i­fold, defined by adding handles to a framed link in the bound­ary of the four-ball. That was Lick­or­ish’s The­or­em [e7] from the early 1960s. It was strange that nobody had thought of con­jec­tur­ing a unique­ness state­ment. So I began to work on this and was be­gin­ning to get there.

In sum­mer 1974 I was vis­it­ing War­wick and I talked to Colin Rourke about these ideas and he got in­ter­ested. I re­turned to Berke­ley and was think­ing smoothly with Morse the­ory and Cerf the­ory where­as Rourke was nat­ur­ally work­ing in the PL cat­egory. With help from Jack Wag­on­er about the hard parts of Cerf the­ory, I real­ized that I had a proof of what’s now called the Kirby Cal­cu­lus. I wrote it up and sent it to the An­nals. I was an as­so­ci­ate ed­it­or of the An­nals at that time. They said that they had turned down a pa­per that had some re­semb­lance to it, and they didn’t think it was ap­pro­pri­ate to take one of their as­so­ci­ate ed­it­or’s pa­pers and pub­lish it in­stead. So I sent it to In­ven­tiones. Then I got a ref­er­ee’s re­port back with an er­ror, and I sweated for two weeks and sor­ted it out [6]. And then I found out that Colin Rourke was still stuck on deal­ing with 3-handles in the PL case. It was more awk­ward in the PL case. Then he saw my pre­print and sim­pli­fied it and that’s where the Rourke–Fenn pa­per [e16] comes in where they also sim­pli­fied the two moves to one move. That’s how that got star­ted.

Well, there’s no im­me­di­ate ap­plic­a­tion of the unique­ness the­or­em. In fact there was no new ap­plic­a­tion un­til 1989, when Resh­et­ikin and Tur­aev used it to get their in­vari­ant [e17], some­times called the Wit­ten–Resh­et­ikin–Tur­aev in­vari­ant of 3-man­i­folds, which came from a totally dif­fer­ent dir­ec­tion. From R-matrices and rep­res­ent­a­tion the­ory and a bit of math­em­at­ic­al phys­ics be­hind it all. But they needed this res­ult. They had an in­vari­ant for framed links, which was quite nat­ur­al. And then they showed that it didn’t change un­der a K-move, as Rourke and Fenn called it, So that was the very first ap­plic­a­tion. On the oth­er hand, it turned out to be a very use­ful way to de­scribe a four-man­i­fold, now called a Kirby dia­gram. Well, they’re giv­ing me more cred­it than I de­serve. But any­way that was a con­veni­ent name.

Sel­man Ak­bu­lut began talk­ing to me about this cal­cu­lus stuff, and we wrote a pa­per, and then more pa­pers. He really took it to heart and be­came ex­tremely pro­fi­cient at it, and took the tricks of the cal­cu­lus way bey­ond what I did. That’s sort of the story on that. The pa­per in In­ven­tiones is ded­ic­ated to Arnold Kas who was the grand­fath­er of the cal­cu­lus. We wrote a Mem­oir, Kas and John Harer and I [7]. Kas wrote a first chapter in which he writes most of the stuff I learned from him, so it’s a great first chapter if you want to un­der­stand this stuff from that point of view.