#### by Joel Hass

**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.