#### by Charles Livingston

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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