#### by Daniel Ruberman

When I arrived in Berkeley in 1977, I had a general notion of studying
topology, but only the vaguest of ideas of what that might actually
entail. At some point, I began to wonder about an advisor, and noticed
the group of students standing at the blackboard in the common room,
often drawing complex diagrams and talking animatedly. These were not
exclusively Kirby students, but whoever they were, they seemed to be
having a lot of fun doing math. In the spring, I started going to the
topology seminar that Rob ran; of course as a beginner I was usually
lost after a few minutes. The last talk of the spring was by Rob
himself, and he presented his proof with
Selman Akbulut
that the
double cover of a homotopy __\( \mathbb{R}P^4 \)__ constructed by
Cappell
and
Shaneson
was in fact __\( S^4 \)__.

It was a beautiful and eye-opening lecture; Rob started with a description of handle calculus (still only a few years old) and showed how to build a handlebody for the double cover. (Nobody noticed that the framing of a last 2-handle was incorrect — this was pointed out later by Aitchison and Rubinstein [e1], and repaired only after strenuous work of Akbulut–Kirby and Gompf.) It all went by very fast, but by the end he had arrived at Figure 13 on the last page of [2]. The instructions in the paper read, “Figure 13 is the unlink! Get 3 colors of chalk and a large blackboard; have fun.” And that is just what we did, calling out simplifications of the picture until it fell into three components. Walking out of the room, I knew that I would be a Kirby student. I think the same was true for my classmates Tim Cochran, Dave Schorow, and John Hughes, although perhaps they felt this earlier.

Rob’s seminar was a focal point for our learning, and most of us gave talks on a regular basis on our readings and later on our research. Much more took place in informal reading groups and seminars. Tim, Dave, and I were already reading the basic texts (Milnor’s h-cobordism theorem lectures; Rourke–Sanderson on PL topology; Rolfsen; Casson–Gordon) that were considered the prerequisites for further study. Rob’s handle calculus was still fairly new, and the generation of students just before mine (Akbulut, Melvin, Kaplan, Harer) had played an important role in its development and refinement. Those more senior graduate students took it as their responsibility to train us in the largely unwritten folklore of the subject. I remember especially that John Harer, who was in his last year, would buttonhole one of us in a hallway, and say something like “Wait, you don’t know about spin structures…,” and give an impromptu lecture at a blackboard with exercises to be performed on the spot. We could go to Chuck Livingston or Pat Gilmer for further help and instruction and indeed my thesis topic grew out of conversations I had with them. Occasionally one of us would go ask Rob a question; he was generous with his time and expertise when asked, but sometimes you could get the message that it was best for you to just dope out some of these things on your own. Once I sat down next to Rob and pulled out a reprint while we waited for a lecture to start. Rob looked over my shoulder at what I was reading and told me that I shouldn’t read papers like that, but should just figure out the proof myself. In time I could recognize close parallels between the way that Rob dealt with his students and his style of raising his kids to be independent and self-sufficient.

I don’t know that Rob deliberately organized his graduate students to pass along knowledge in this fashion, or whether it developed organically. It seems to have been widely understood that one was supposed to do the same thing for the next generation. I think I was a little disappointed that the students who arrived after me, Ian Aitchison and Bob Gompf, seemed to already know a lot and didn’t need much tutoring. Bob Penner turned up for a few months as a sort of refugee from the rather nongeometric MIT grad program, and I remember us collectively teaching him about handle calculus. This entailed explaining to him that the 3-manifold invariant he’d discovered by using Rob’s theorem was just the order of the first homology, a sad lesson that we all had to learn at some point.

As far as I know, few of Rob’s many students in that era had a
scheduled weekly meeting with him; I think this may have changed in
subsequent years. The understanding was that when you had something to
discuss, you dropped by or made an appointment. Similarly, in that
era, we were not given (or even suggested) problems to work on; part
of your education was learning how to find your own problems.
Fortunately, there were lots of sources: Rob’s desk — a distant
forerunner of the ArXiv — always had a pile of recently arrived
preprints, and we were encouraged to grab something to read. If you
got interested in a paper, then you could give a talk on it in the
seminar.
My favorite memory of this was finding Cameron Gordon’s paper
n the __\( G \)__-signature theorem in low dimensions in that pile. A few weeks
later, I ran up to Rob in the hall and exclaimed about how great it
was. He smiled at my enthusiasm, and asked, ”would you like to meet
the author”, who was standing right there with him. There were often
visitors; Cameron was around quite a bit, as was Larry Taylor. Larry
was very helpful to me at a late stage in writing my thesis, and also
provided a key suggestion to Tim Cochran that made his thesis much
stronger.

Once I found my way to a thesis topic, I met with Rob more frequently. I
usually wanted to launch straight into my newest results, but after a
few minutes, Rob would usually ask, “Remind me what is a doubly slice
knot again…?” or some other question about the basics of what I was
doing. I might have found this a bit exasperating but by the end of my
graduate years I developed more skill at giving a quick gloss over the
background material, a skill that translated well into seminar talks
out in the world. In general, Rob didn’t like “machinery”, although
(or perhaps because) turning his geometric ideas into his famous work
on triangulations relied on sophisticated results of surgery theory. I
remember lecturing in some seminar (about a fake
__\( \mathbb{R}P^4 \)__) where I
needed to explain about normal maps, a part of surgery theory. Rob
kept pressing me for a more geometric explanation, and in the end gave
up, saying with a laugh that I was so earnest about it all that he
would just accept my version.

In addition to the talks and seminars, Rob had just finished compiling the first version of his problem list, started in 1976 at a conference in Stanford, and expanded over the years by him calling or writing to his many friends and asking for input. This was a fantastic resource for us as students, especially as there were no texts or even survey articles to help get a grasp on a rapidly developing field. Most of the problems were too hard and fundamental for graduate students, but I thumbed through the problem list frequently to get ideas for questions that I might be able to approach. Even seeing how to phrase a question, as in the short write-ups for most of the problems, was a big help. Although most of the problems were suggested by others, Rob put it together in a way that reflected his point of view that there was a unified field of low-dimensional topology.

Rob gave some memorable topics courses, most notably a class on topological manifolds that he gave towards the end of my years in Berkeley and a course on topological 4-manifolds in the immediate aftermath of Freedman’s announcement of his classification results in the summer of 1981. The latter was Rob’s way of learning the very complicated proof that Casson handles are topologically handles. (This was a great trick that I learned from Rob and put into practice in my career: if you want to learn something new, give a course on it. You’d learn more than the students!) Of course we got bogged down in the details by the end, and it was very much Rob’s style to get us to work through those details as a group.

The course on high-dimensional topological manifolds felt like a trip down memory lane; Rob declared that it was the last time he would ever give a course on the subject, since he couldn’t see what good it would do anyone. The major problems had been solved, but the ideas were still fresh and for me it was a great experience. We were all eager to learn Rob’s famous torus trick, but Rob’s presentation of some of the more foundational ideas, such as the proof of the Schoenflies theorem, made them feel fresh and new. I had the sense that he had great affection for some of that material (“meshing” stands out in this way) and maybe was a bit sad that it was viewed as being somewhat dated. I think that Rob’s presentation followed a book that he was writing with Jim Kister that was intended as an alternative to the more hi-tech treatment in the famous Kirby–Siebenmann book [1].

We did learn many pieces of current research from courses given by postdocs and visitors. Steve Kerckhoff gave a couple of courses related to Thurston’s work on hyperbolic 3-manifolds and on surface diffeomorphisms in connection with his resolution of the Nielsen realization problem. Allen Hatcher gave a semester course on his proof of the Smale conjecture, teaching us a lot of more traditional 3-manifold theory along the way. During one quarter, Rob traded courses with Jim Milgram, going down to Stanford to teach 4-manifolds while Milgram taught us surgery theory. Rob told us rather pointedly that we were to attend all of the lectures no matter how confused we were. I thought at the time that the Stanford students got the better of the deal, but Milgram’s course came in handy later in my career.

Low-dimensional topology felt like a fairly coherent subject in this period, and Rob felt strongly that we should know about 3-manifolds and 4-manifolds in equal measure. The ground-breaking ideas of Freedman and Donaldson in 4-manifold theory were just over the horizon, but Thurston had already started a revolution in 3-dimensional topology by introducing vast new horizons in hyperbolic geometry and other geometric methods. We did our best to learn this, with Rob organizing regular study groups to try to read Thurston’s notes that were arriving in installments (mimeographed, no less!) from Princeton. We put a lot of effort into this, but somehow didn’t learn as much as we might have. In retrospect, I believe that we didn’t recognize how many details in those notes needed to be worked out by the reader, and that Thurston was trying to convey his visionary geometric viewpoint rather than laying out a guided route to the spectacular results he’d announced. Still, some students, like Joel Hass and Bill Menasco, wrote theses on geometric topics, and many of us eventually learned some portions of Thurston’s work.

Our group was very cohesive; we went to tea most days and either talked math or played speed chess, at which Rob excelled. Rob would sometimes challenge us to a contest to see who could hold their breath the longest; he could go more than 2 minutes. We thought this was aided by his kayaking practice but he said it was a matter of willpower. We often went out for a cheap lunch as a group; the Mexican restaurants that gave you free salsa and chips were a favorite for obvious reasons. We argued about the political issues of the day (rent control, nuclear arms, feminism…) more than we talked math, and I recall rarely being on the same side as Rob. We also did intramural sports together, soccer (coed in the fall and in the spring men only) and basketball. Rob was a full participant in all of these, and brought endurance and doggedness to the soccer field. I told him once (truthfully) that I’d heard one of the opposing team ask at half-time to be passed the ball because he could “beat the old guy”. Rob was mad enough that not much got past him in the second half. We fielded some strong teams and won our league championship one term when José Carlos Gomez-Larrañaga, who was an instinctive goal-scorer, was visiting.

Rob took some of us on rafting trips on the American River in the summer, using an old Army surplus raft that was ill-suited to the rapids we encountered. (A friend who was taking a rafting class told me later that her instructor had made the class pause, saying, “Watch that raft; they’re going to do everything wrong” — which we proceeded to do.) We hit our first rapid without having really learned how to control the boat. Rob’s daughter Kate, who was quite young, was perched in the bow and was sent flying as the under-inflated raft essentially folded in half when we hit a big wave. My image is that Rob plucked her out of the air as she was heading overboard, and she spent the rest of the afternoon holding onto his leg. I didn’t appreciate at the time just how accomplished Rob was as a kayaker, so I was really impressed when he kayaked upstream to retrieve some lifejackets that had been set out to dry on the island where a few of the crew washed up after our initial capsize. Eventually Rob got us organized into some semblance of coherent paddling so we made it through the more fearsome S-shaped rapid unscathed.

In the summer of 1978, many of Rob’s students went to Cambridge (UK) for a small gathering. I didn’t know enough to go along, but when a similar event took place in 1981, I jumped at the chance. It was a large group, a mixture of graduate students from Berkeley and elsewhere plus numerous postdocs and more senior figures. (A group photograph from late in the summer can be found on the MSP website.) Ray Lickorish organized this, finding us rooms in Pembroke College; I think he and others were a bit overwhelmed by the somewhat indecorous and energetic American students. We played soccer as often as we could (even, shockingly, during Charles and Diana’s wedding) and did math the rest of the time. (There may have been some punting and trips to the pub mixed in.) The soccer games attracted quite a crowd; a favorite image was of Kate (or possibly Linda’s daughter Erica) who might have been 7 at the time, fearlessly tackling John Hempel, who towered over her. That summer was probably the origin of the (not entirely complimentary) reputation of Kirby students as ferocious soccer players. This might have been related to Tim, who brought his soccer ball to conferences for many years afterwards.

The time in Cambridge had an enormous influence on us graduate students. I made lifelong friends from all over, and was able to spend time with some of Rob’s former students, such as Selman Akbulut, Paul Melvin, and Marty Scharlemann, and to talk informally with any number of luminaries, including Andrew Casson, Cameron Gordon, and John Conway. Many others passed through for shorter stays. I met Jerry Levine there, which had a great impact on my subsequent career. Linda, who was to marry Rob the following spring, came for the summer. It was a nice chance to get to know her and her daughters as they learned to navigate the strange new world of mathematicians that they’d entered.

The months after Cambridge were as mathematically consequential as any
period I experienced in my professional life. Mike Freedman announced
his results on topological 4-manifolds in August (I learned of this
from Rob’s course description tacked up on a bulletin board in
Berkeley),
and in November or so Rob told us about a remarkable result
of Simon Donaldson using gauge theory to show that some 4-manifolds
(which Freedman had just constructed topologically) could not be
smoothed. The summary of Donaldson’s proof had been passed from
Atiyah
to Freedman to Kirby by telephone, and, just as in the
kids game, had
gotten completely garbled. Despite the incomplete summary, Rob and
Mike understood quickly that the combination of Donaldson’s and
Freedman’s theorems implied the existence of an exotic __\( \mathbb{R}^4 \)__.
(Rob’s recollections of this period can be found in Freedman’s
*Celebratio* volume, in the section on Exotic __\( \mathbb{R}^4 \)__s.)
A more coherent report emerged by the spring,
and some of us scrambled to learn the basics. During this entire
period, there was an active and influential seminar at Berkeley in
gauge theory run by
Is Singer
and featuring a bright young physics
postdoc named
Ed Witten.
Sadly, there was virtually no interaction
between Rob’s group and this seminar at a time when each might have
done well to learn from the other. Who knew!

Without knowing that these great achievements were about to occur, Rob and Cameron Gordon had already planned a conference on 4-manifolds for the following summer of 1982 in Durham, New Hampshire. They reorganized the program to feature lecture series by the principal figures in the new developments, including Freedman, Donaldson, and Cliff Taubes. At that conference, Frank Quinn proved the 4-dimensional version of the annulus conjecture that Rob had proved in higher dimensions in 1969. Because of the tight schedule, Quinn’s lecture had to be given in the evening; I recall a marathon session in a crowded and sweaty basement classroom that was mercifully brought to an end after many hours when Rob recognized that most of the audience was about to expire, and sent us home.

#### Coda: MSRI, 1984–85

The years that I’ve described were formative and special to me and my fellow students. The stresses of finding our way through graduate school and our early careers were occasionally overwhelming, but there was a great sense of generosity and openness in the field. There were many leaders (and future leaders) who contributed to that feeling, but Rob’s personality and leadership stand out and had a great influence even beyond his important mathematical contributions. Although his political and social views were all about individualism, he acted in a way to create and nurture a remarkable sense of community, where name and reputation were secondary to what you could bring to the study of a wonderful subject. Rob rarely told us directly that we were doing well, but by treating us as colleagues, he made us feel that we were already full members of that community.

*Daniel Ruberman received his Ph.D. from UC Berkeley in 1982 under the direction of Rob Kirby. After a postdoc at the Courant Institute and a year at MSRI, he joined the faculty at Brandeis University, where he is Professor of Mathematics.*