#### 1. Early work

The annulus conjecture (AC) as stated by Milnor
is: Is the region bounded by two locally flat __\( n \)__-spheres in
__\( (n+1) \)__-space necessarily homeomorphic to __\( S^n \times [0,1] \)__?

I was pleased to see problems that I could understand with my limited
knowledge of mathematics. But I did know
Morton Brown’s
proof of the
topological Schoenflies conjecture, and a canonical form of the
Alexander isotopy used by
Jim Kister
in his proof that micro bundles
contain bundles
[e4].
I also knew that a bounded
homeomorphism of __\( R^n \)__ was canonically isotopic to the identity, from
reading a paper of
Ed Connell,
and did not know until decades later
that this fact was first proved by Kister
[e1].

Occasionally I would have an idea for proving the annulus conjecture. One was embarrassing, for when I went to show it to my advisor, Eldon Dyer, we were joined by Saunders Mac Lane, and it was discovered that I had overlooked that the intersection of nested closed sets in a metric space might be empty if I did not know that the metric space was compact.

I did however prove that if one could nicely fit a
“pillar” between the two spheres in the annulus conjecture, then
there was indeed an annulus
[1].
Dyer seemed mildly
impressed, perhaps that this dubious student had managed to actually
prove something. He was an editor of the *Proceedings* of the
AMS and accepted the paper. Later I found out that this may have been
folklore in the Bing topology community. If so, it’s not surprising.

I finished my PhD in 1965 with a thesis on a different topic and went
to UCLA as an assistant professor. While there I read a paper of
Jim Cantrell
in which he showed that Morton Brown’s proof of the
topological Schoenflies conjecture still held when the embedded
codimension one sphere had a point not known to be locally flat, if
its dimension was not equal to 2 (the 2-sphere boundary of the
neighborhood of the Fox–Artin arc in __\( S^3 \)__ is not locally flat at the
wild point of the arc). The real point here was that a locally flat
embedding could not fail to be flat at just one isolated point (except
when __\( n=2 \)__). I managed to show that it could not fail to be flat at a
Cantor set if the Cantor set was tame in the embedded sphere and in
the ambient sphere. I submitted this to the *Annals*, and
Milnor obtained an excellent referee in
Dale Rolfsen
who improved the proof substantially
[3].

This paper led to a tenure track offer from the University of Wisconsin. I’d become a Westerner and did not wish to go back to the midwest, but I did spend the fall semester of 1967 at Wisconsin where I formed what became a lifelong friendship with Raymond Lickorish who was visiting for the year.

#### 2. The torus trick

In August 1968 I had gone to the yearly topology conference in Athens, Georgia, and returned with a preprint of Černavskiĭ [e6] showing that the space of homeomorphisms of a topological manifold is locally contractible.

I was home,
babysitting my 4 month old son, when something in the paper caused me
to consider the lift of a homeomorphism __\( h:T^n \to T^n \)__ of the torus
to its universal cover __\( H: R^n\to R^n \)__. Obviously __\( H \)__ is periodic, and
if __\( h \)__ was homotopic to the identity, then __\( H \)__ would be bounded by the
same constant that held for a fundamental domain.

But if __\( H \)__ is bounded, then it is isotopic to the identity (by a
version of the Alexander isotopy
[e1])
and
therefore *stable*. Stable means that __\( H \)__ is the composition of
finitely many homeomorphisms, each of which is the identity on some
open set. These had been carefully
studied in
[e3]
and stable homeomorphism and the annulus
conjecture were closely related. In particular, if __\( H \)__ is stable, then
the original __\( h \)__ is stable, and I knew that was a
very interesting fact (the requirement that __\( h \)__ be homotopic to the
identity was no problem, for one could arrange it to be so by
composing with a diffeomorphism).

That last paragraph was a quick observation if one had the right tools
and questions in hand. Now the obvious question was how to turn an
arbitrary homeomorphism of __\( R^n \)__ into a homeomorphism of __\( T^n \)__ so as
to show that those homeomorphisms were also stable.

I quickly thought of immersing a punctured __\( T^n \)__ into __\( R^n \)__
(nothing-else-to-do theory). Then the road forked, one fork leading to
Černavskiĭ’s
theorem on local contractibility and the other to PL
structures.

The easier fork was to assume that __\( h \)__ moved points less that some
__\( \epsilon \)__, for then __\( h \)__ lifted to an embedding of __\( T^n \)__ with a bigger
(roughly by __\( \epsilon \)__) puncture into __\( T^n \)__ minus a point. Then a
canonical version of the topological
Schoenflies theorem (see
[e8])
allowed the extension of the embedding to a
homeomorphism of __\( T^n \)__, which was stable, but also agreed with the
original __\( h \)__ on an open set, so __\( h \)__ was stable and isotopic
(canonically) to the identity, thus proving local contractibility for
__\( R^n \)__. I think I knew this still in August.

The other fork was more complicated, for it involved using the
immersion of a
punctured __\( T^n \)__ into __\( R^n \)__ to pull back a PL structure from the
standard one on __\( R^n \)__ to one on the punctured torus. This could be
extended to a PL structure on __\( T^n \)__ when __\( n\geq 6 \)__, giving a
homeomorphism from __\( T^n \)__ with a possibly exotic PL structure __\( \Sigma \)__
to the standard __\( T^n \)__. Whether the identity homeomorphism was isotopic
to a PL homemorphism was a topic in nonsimply connected surgery
theory, which had just been worked out by
Terry Wall,
although the
calculations for fundamental group equal to a free Abelian group were
not entirely worked out or known by all.

I had already arranged to teach in the summer of 1968 so as to have a free fall quarter to spend at the Institute for Advanced Study, a very propitious decision.

Shortly after arriving at the IAS, I generalized the torus trick to a
handle version,
using __\( B^k \times R^n \)__, rather
than just the zero-handle case when __\( k=0 \)__. I circulated a one-page
document
[2],
that spawned a few
papers about my conjectures (see, for example,
[e7]).
In October I found myself sitting next to
Larry Siebenmann
at a
colloquium dinner. He started asking questions about the torus trick
and by the time dessert was well over, I thought he had drained every
bit of useful information out of my brain. That began our
collaboration. Larry worked late and I remember many mornings finding
a sheaf of notes inside my screen door. We progressed quickly, for
Larry knew all the PL to DIFF smoothing theory, and much else, that I
would have
needed months to absorb.

I listened in on discussions about nonsimply connected surgery;
Bill Browder
explained a lot to us and then Larry had the idea of first
taking a __\( 2^n \)__-fold cover of the homeomorphism __\( T^n_{\Sigma} \to T^n \)__,
which would then kill Wall’s obstruction, and then take the lift to
__\( T^n \)__. This was an added-in-proof paragraph to my *Annals*
paper, and settled the annulus conjecture except in dimensions __\( 4,5 \)__.

Looking back, the moment I considered the universal lift of a
homeomorphism of __\( T^n \)__ and observed it was bounded, an observation
known to many starting in the 19th century or earlier, I was on a
downhill slope in the sense that there was never a difficulty that was
a serious
long-term problem. I thought of immersing the punctured torus quickly
and it was easy to generalize to the handle version. So, a marvelous
torus trick just fell in my lap, an amazing piece of luck. When you
work on hard problems, it is often true that there is no nice solution
to be found. Think of the Poincaré conjecture which so many people
devoted years of their lives to, and a topological proof still hasn’t
been found;
Perelman was needed. It could have been the same with the annulus
conjecture.

And then I was lucky again to be at the IAS and meet the perfect collaborator. Our work would not have been done by me in isolation.

And I was lucky in another way, in that I didn’t think of the torus
trick four years earlier when I just as well could have. Then I’d have
written the *Annals* paper, but that would have only been the
next-to-last piece of the puzzle, and Wall would have done his surgery
theory during those four years and put in the last piece of the
puzzle, gaining most of the glory. So,
I was lucky to be smart but not too smart.

Larry and I had our key results and wrote them up in late December,
1968, as a *Bulletin* of the AMS article. I went back to UCLA
to teach, but met Larry again (with a few others) to ski at Heavenly
Valley at the south end of Lake Tahoe in February, 1969. We skiers
didn’t see much of Larry, for it turned out he was finishing up his
argument that
__\( \pi_3 (\mathrm{TOP}/\mathrm{PL}) = Z/2 \)__. I, in a desultory sort of
way, had been trying to prove the opposite, but for Larry his result
was crucial, for otherwise there would have been no pathology, no
obstruction to triangulation, and the subject would have boiled down
to just the torus trick.

Back at UCLA, I received a letter from Larry explaining his proof. I remember well sitting at the kitchen table of my apartment beginning to read his proof when lightening struck and I saw a simple proof, the one in the literature, and stopped reading. Recently, over 40 years later, I looked for the letter and was surprised I couldn’t find it. Bob Edwards was in Paris so I asked him to ask Larry if he had a copy. Larry’s response was: “There’s no letter. I came to UCLA in spring 1969 and gave a lecture on the proof.”

Perhaps Larry is wrong, but I suspect that something that was very clear in my memory was wrong, an example of the fallibility of memory. That then raised the question of whether my memory is wrong in claiming that the simple proof is mine, but here I think I will rely on my memory. For one thing, the proof which is sketched in [4] is not what I would have written (it was submitted March 3, 1969, soon after our ski trip).

Here is what I would have written: Start with __\( Q \)__, an exotic, rel
boundary, __\( [-1,+1] \times T^n \)__, meaning that the standard PL
structures on the boundary components do not extend to a PL
homeomorphism from __\( Q \)__ to the product structure. But the
__\( s \)__-cobordism
theorem tells us (__\( n \geq 6 \)__) that __\( Q \)__ is PL homeomorphic to the
product if we do not require that it be the identity on *both*
ends, but just at __\( -1 \)__. Measure the difference at the __\( +1 \)__ end by a PL
homeomorphism __\( h \)__. Now a large odd cover leaves __\( Q \)__ still exotic, but
__\( h \)__ is now arbitrarily close to the identity on __\( +1 \times T^n \)__, and
is thus topologically isotopic to the identity. But this cover of __\( h \)__
cannot be PL isotopic to the identity. This is the difference between
TOP and PL.

In general we do not pick apart contributions to joint papers, but in this case it seems to me in retrospect that after I met Larry, everything was really due to him, with the exception of two of his proofs which I substantially improved upon. One I have just mentioned, and the other is the proof of the product structure theorem using the “windowblind” lemma [5], p. 35.

Larry and I intended to get a full paper out fairly quickly, for I visited IAS again in fall 1969, and then we met in Cambridge, England, in summer 1971, but by then our goals had diverged. I would have settled for a shorter paper along the lines of Chapters 1 and 3 of our eventual book, but Larry had greater goals including parameterized versions of our theorems and more. So he eventually wrote the vast majority of our book.

#### 3. The calculus

In early 1973, I went to a conference in Tokyo and while there
overheard
Takao Matumoto
talking to someone and mentioning the Kummer
surface. I was curious and luckily had been jogging with
Arnold Kas
who was a fairly recent student of
Kodaira.
It was natural to ask him
about this interesting 4-manifold. He taught me many things about
complex surfaces, and did so in a way that I could turn much of it
into handlebody theory. Our AMS memoir
[14]
with
John Harer
contains much of this material. The algebraic geometers had a way of
showing that various descriptions of the complex surface __\( E(1) \)__ as a
Lefschetz fibration with various singular fibers, were all
diffeomorphic to __\( CP^2 \)__ blown up nine times. The methods for doing so
translated into connect summing with __\( \pm CP^2 \)__ and sliding
2-handles over 2-handles.

At some point I began to wonder if different ways of adding
2-handles to __\( B^4 \)__, which gave the same boundary, could be
equivalent under the same moves that the complex geometers used. This
worked nicely in some examples, in particular when the boundary was
the Poincaré homology 3-sphere. By summer 1974, when I was
visiting Warwick, the conjecture was firmly in my mind, and I
explained it to
Colin Rourke.
That fall, the proof was boiled down to
some Cerf theory. I queried
Jack Wagoner
at length for he was our
local expert, having just sorted out the pseudo-isotopy problem with
Allen Hatcher
[e9]
using advanced Cerf theory. He
told me enough, and I lectured on the theorem in
the fall of 1974. It took me a while to write it up, as usual, and
then I sent it to the
*Annals*. Browder
then suggested it wouldn’t look proper if they accepted my paper after
rejecting a related paper, for I was an
associate editor of the *Annals* at that time. So it went to
*Inventiones*, where the referee (Allen Hatcher I believe)
found an error which took me a few weeks to sort out. The paper
[6]
appeared in 1978.

Meanwhile Rourke, with Roger Fenn, was working on a PL version of a proof, and got stuck with 3-handles. Cerf theory had been an easier path for me. But they did see a neat way to combine my two moves into one [e11].

This theorem was really a uniqueness theorem to go with the existence
theorem (every oriented 3-manifold can be obtained by surgery on a
framed link in __\( S^3 \)__) proved by Lickorish
[e2]
in
the early 60s. Nowadays an existence statement automatically generates
a uniqueness question, but back then I was not aware of anyone asking
about uniqueness.

This theorem and the framed link pictures became known as the Kirby
calculus (it rolls easily off the tongue) despite the fact that
others, especially
Akbulut,
developed it far beyond what I had done.
Oddly, it had no applications (other than giving moral support for
trying to prove manifolds were the same with these methods) until
Reshetikhin
and
Turaev
[e17]
used the moves to show that
certain combinations of their invariants for framed links were
unchanged under the calculus moves,
with the result that they had 3-manifold *quantum invariants*, as they came
to be called.

In the mid 70s John Harer joined Kas and me in working on handlebody
descriptions of complex surfaces. This became an AMS memoir
[14]
with an expository first chapter written by Kas
describing complex surfaces from a topological point of view, followed
by handlebody/calculus pictures of __\( E(1) \)__ and __\( E(2) \)__
(we called them the half-Kummer and Kummer (K3) surfaces), and then a
chapter drawing the 3-handle attaching maps for __\( E(1)_{2,3} \)__, which
is __\( E(1) \)__ with two
logarithmic transforms of degrees 2 and 3. We
hoped to see that the 3-handles could not be canceled and thus that
these logarithmic transforms were not diffeomorphic to __\( E(1) \)__, but
this had to wait for Donaldson and gauge theory. Our work was done by
1979, but didn’t appear in a timely fashion.

In the latter 70s, Selman and I wrote a paper [9] on Mazur manifolds, as we called them, which is noteworthy because these examples turned into Akbulut’s “corks” which he used to give the smallest exotic 4-manifold [e15], [e16]. Selman also introduced the notation for a 1-handle, namely an unknotted circle with a dot on it. If you want to trade in a 1-handle for a 2-handle, just replace the dot with a zero.

Cappell
and
Shaneson
constructed smooth 4-manifolds homotopy
equivalent to __\( RP^4 \)__ but not diffeomorphic
[e10].
This raised the
question of whether the double cover was diffeomorphic to __\( S^4 \)__ and
that these
were exotic involutions or, less likely, that the double cover was a
counterexample to the smooth 4-dimensional Poincaré conjecture.
Selman and I drew handlebody pictures of the double cover and showed
that it was indeed __\( S^4 \)__
[8].
Meanwhile,
Fintushel
and
Stern
had elegantly constructed exotic involutions on __\( S^4 \)__ by different
methods
[e12].
However,
Iain Aitchison,
then a Master’s student in Melbourne, read our paper carefully and
pointed out that a key framing was 1 rather than 0.

So, the double covers of the Cappell–Shaneson exotic __\( RP^4 \)__s were
still only homotopy spheres. We, meaning Selman with me cheering from
the sideline, learned a lot more about the first interesting case,
showing that it was also the Gluck construction on the knotted
2-sphere made from the two distinct slices of the __\( 8_9 \)__ knot, that
there was a natural presentation of the trivial fundamental group
given by __\( \{x,y\mid xyx=yxy, x^4 = y^5\} \)__, and there was a
comparatively simple description with two 1-handles and two
2-handles. We published this
[13].
A few years later,
Bob Gompf
found an elegant way to add a
__\( 2-3 \)__-handle pair and show that
this homotopy 4-sphere was indeed __\( S^4 \)__
[e13].

In the summer of 1976 a group of us met in Cambridge, imposing on the
hospitality of Lickorish. While there,
Paul Melvin
and I concocted a
very simple argument (a three-page paper) showing that if 0-surgery
on a knot __\( K \)__ gave __\( S^1 \times S^2 \)__, then __\( K \)__ had to be a slice knot.
In fact, __\( K \)__ had to be the intersection of the equatorial 3-sphere
in __\( S^4 \)__ with an unknotted __\( S^2 \)__ in __\( S^4 \)__. This proves Property R for
almost all knots, but the result was soon forgotten, not because
Dave Gabai
proved Property R for the remaining cases, but because his
methods in
[e14]
were so far reaching.

From Berkeley, I sent the
three-page paper to
Frank Adams,
the topology editor at
*Inventiones*. Three days later I received an acceptance
letter. How’s that for turn-around time?! I surmise that Frank
received the submission, opened it around tea-time in Cambridge, saw
Lickorish, who vouched for the result, and replied by return mail.

In the mid-70s Siebenmann was visiting Berkeley and he gave a few lectures on descriptions of the Poincaré homology 3-sphere. Marty Scharlemann and I added other descriptions to produce [10]; the contents are nicely described in a math review by Anatoly Libgober.

In 1978 during another summer visit to Cambridge, I had a few conversations with Raymond Lickorish and suddenly there was a preprint of a nice theorem: every knot is concordant to a prime knot — with my name on it. I’m very pleased to have a joint paper [11] with Raymond, but I could have done more to earn it.

Akbulut and I saw a nice way to see the double branched cover of a
link __\( L \)__ in __\( S^3 \)__ bounding a Seifert surface __\( F \)__ pushed into __\( B^4 \)__,
using the calculus
[12].
This worked well for complex
curves in __\( CP^2 \)__, and led to my drawing the quintic with its 53
2-handles, using plastic forms. The senior editor at *Math
Annalen* wrote back, “Are all these cartoons really necessary?”

#### 4. A book and low-dimensional bordisms

By 1987 I had taught courses on 4-manifolds several times and in particular had talked with Iain Aitchison about proofs of the classical 4-manifold theorems using geometric techniques in dimension four. When Chern invited me to visit Nankai, his Institute in Tianjin, China, for the month of May, 1987, I agreed and offered to teach a course on 4-manifolds, with the idea that the students would help with notes.

I was pleased with the proof I’d worked out for Rohlin’s
theorem. In it’s generalized form it states that the extra __\( Z/2 \)__
factor comes from
the spin bordism class (__\( \Omega^{\mathrm{Spin}}_2 = Z/2 \)__) of a
characteristic surface __\( \Sigma \)__ in __\( X^4 \)__; __\( \Sigma \)__ is dual to __\( w_2 \)__
and the spin structure on __\( X-\Sigma \)__ descends to the
(codimension one) normal circle bundle to __\( \Sigma \)__, and this descends
to a spin structure on a section of __\( \Sigma \)__ into the circle bundle;
this is independent of the choice of section because __\( \Sigma \)__ is
characteristic
[15],
pp. 64–71.

I also included
chapters sketching an outline of Freedman’s great work on topological
4-manifolds and included some history of how the existence of exotic
smooth structures on __\( R^4 \)__ came about. More history is given in
Freedman’s volume at Celebratio Mathematica,1
but I can add yet one more bit.

In fall 1982 it became clear that there would be no paper announcing
one of the more remarkable results in topology, the existence of
exotic structures on __\( R^4 \)__ (in all other dimensions exotic smooth
structures did not exist). To remedy this, I wrote to Atiyah
suggesting that a paper be written announcing this result authored by
Casson,
Donaldson,
Freedman,
Taubes
and
Uhlenbeck.
The authors could
have been just Donaldson and Freedman, but I felt that Casson deserved
credit for his Casson handles, and that Taubes and Uhlenbeck also
deserved credit for the foundations that Donaldson built upon. But
Atiyah
was the opposite of enthusiastic, probably not wanting to
dilute the credit due his PhD student, and the idea died. But two
Fields Medals (Donaldson and Freedman), and later an Abel Prize to
Uhlenbeck gave each a good dose of glory.

In the late 1980s,
Larry Taylor
spent time in Berkeley during his vacations and we frequently met for
lunch at the Musical Offering café.
We began discussing
low-dimensional bordism groups in the Spin and Pin__\( _\pm \)__ cases. The
Arf invariant and the Brown invariant with values in __\( Z/8 \)__, used in
the nonorientable cases, were key ingredients. We would work out a
case, in a very geometric fashion, and the next day Larry would turn
up with a well-written
version typeset in __\( \mathrm{\TeX} \)__. This happened lunch after lunch and magically
(in my eyes) our paper
[16]
was done. One cannot ask for a
better collaborator than Larry.

This is probably a good point at which to admit than in all of my collaborative publications, it is my coauthor who has done the lion’s share of the writing, and perhaps the lion’s share of the thinking as well. Yet another way I’ve been lucky.

#### 5. Quantum invariants

In March 1989 Paul Melvin was visiting Stanford, and we had begun to
think about projects when Kolya Reshetikhin came to talk there about
his new work with Vladimir Turaev on the subject now known as
*quantum invariants of 3-manifolds*. Paul and I were curious
and wondered whether these invariants were a repackaging of known
invariants or if not, what their topological nature might be.

We did show
[17],
[18]
that they were known at the
roots of unity, __\( 2\pi i/q \)__ for __\( q = 2,3,4,6 \)__ and we developed some
properties of the invariants. The fact that the paper is so often
cited is due to Paul’s great care with accuracy, for the subject was
bedeviled by easy-to-get-wrong signs. Paul is more algebraically
minded than I am, and he taught me more (elementary) representation
theory than I ever expected to know.

This work lead to our being invited to the opening semester of the Newton Institute in Cambridge in the fall of 1992 for a program involving outgrowths of the Reshetikhin–Turaev methods. Besides the math, the visit included some British tradition.

Prince Philip (the Queen’s husband) came by for an hour to “open” the Institute. He made the rounds of some of the offices, and told Paul and me of an old monastery in Greece (his grandfather was George I of Greece) which had preserved bits of mathematics during a dark time. He made a good impression.

I was the Rothschild Visiting Professor and at an appropriate function at Trinity College (where Atiyah, Director of the Institute, was Master), I was briefed on how to thank Lady Rothschild and to be aware that she would let me know when our conversation was to end. She did, gracefully, after a few sentences, and awkwardness on both our parts was avoided.

And I was a Fellow at Emmanuel College which has a fantastic Oriental plane tree (google it!) in a back garden.

#### 6. Gökova

In 1992 Selman Akbulut organized the first Gökova geometry/topology conference during the week of Memorial Day. It was held at the Hotel Yucelen in Akyaka at the east end of the Bay of Gökova in southwest Turkey. It was a beautiful site and an excellent group of mathematicians came, so it became a yearly conference which continues to this day. Besides Selman, the local organizer is Turgut Önder, an Emery Thomas PhD from Berkeley, now at Middle East Technical University, METU. Funding came from TUBITAK, the Turkish equivalent of the US National Science Foundation, and (eventually) from the NSF itself. Initially, the rooms were heavily subsidized by the hotel owner, but as outside support for the conference has declined, the hotel now is the biggest supporter of the conference.

I’ve been to the conference about a dozen times, partly for the math and camaraderie, and also for the Wednesday afternoon hikes to beautiful spots, the Saturday boat trips in the Bay, and for an exceptional swim or kayak down a one mile river formed by underground springs. There are some photos in the gallery.

In 1998, Selman convinced the METU math department to invite an outside committee (Avner Friedman, Robert Langlands, Ron Stern and myself) to review the department, perhaps the one and only time this has been done. It turned out that Langlands had visited Turkey for a year in the late 1960s, learned the language, and was curious to meet old friends and see what had changed.

#### 7. Problem lists

The AMS held its 24th Summer Research Institute at Stanford, August 2–21, 1976, and the topic was algebraic and geometric topology. Besides myself, the speakers included Raoul Bott, Greg Brumfiel, Sylvain Cappell, Julius Shaneson, Bob Edwards, Cameron Gordon, Allen Hatcher, Jack Wagoner, Wu-Chung Hsiang, William Jaco, Max Karoubi, Dick Lashof and Mel Rothenberg, Ronnie Lee, James Lin, Ib Madsen, Jim Milgram, John Morgan, Bob Oliver, Ted Petrie, Dan Quillen, Larry Siebenmann, Emery Thomas, Friedhelm Waldhausen, Terry Wall, and James West.

This conference may have been the last of the all-inclusive topology conferences with an all-star line-up of lectures, for within six to eight years low-dimensional topology had exploded in size with hyperbolic 3-manifolds (Thurston) in one constellation, topological 4-manifolds (Freedman) in another, exotic smooth 4-manifolds (Donaldson) in a third, and new knot invariants connected to operator algebras and physics (Jones) in a fourth.

It was still possible to have a problem session followed by a problem list of manageable size. This I did by 1978, with much help from many people [7].

This is the review by Louis Kauffman:

This is an excellent survey of problems in knot theory, surfaces, three-manifolds and four-manifolds. For specific topics and questions we can do no better than to refer the reader directly to the article. Here are a few up-dates and comments: Problem 1.1 is true. It follows from a remark and results of Thurston. The top part of the diagram for Problem 1.37 should contain left-half-twists. Problem 2.3 is false, by methods of Johnson and Johanson. In Problem 2.6, the conjectured mapping exists for a finite subgroup via results of Steve Kerkhoff. The answer to Dale Rolfsen’s question in Problem 3.13 is: exactly the commutators (by John Harer). The answer to Problem 3.33 (B) is “yes” (Thurston). The answer to Problem 3.38 (the Smith conjecture) is “yes” (solution by Thurston, Meeks–Yau, Gordon, Litherland, Bass, Shalen and Otto Schmink). Problem 3.43 has been proved for various cases where the manifold has a “geometric structure”. Problem 4.8 has been answered affirmatively by M. Freedman. In general the would-be problem solver should beware (be aware) of the presence of Thurston, Jaco, Shalen and Johannson.

The reader may never have heard of Otto Schmink, who exists solely as an inside joke. In 1954 a Canadian group satirized Senator Joe McCarthy by postulating that he died and went to heaven and then began to investigate the process by which dubious (in his mind) figures such as John Stuart Mill, John Milton, etc., had been admitted. Soon they were removed from “up here” and sent to “down there”. The distinguished names began to dwindle, and eventually even one Otto Schmink was also demoted, just as McCarthy went mad while investigating God herself. The satire is well done and can be found at Nick Voss’ blog. Recommended.

In 1993 I agreed to an update of the “old” problem list, not realizing that such a project would end up at 438 pages [19]. Of course I had even more help this time, especially from Geoff Mess who essentially wrote the updates for most of the old 3-manifold and knot theory problems.

The National Academy of Sciences has a yearly Prize for Scientific Reviewing. This refers to the practice in most other disciplines of writing reviews on a particular topic which brings researchers up to date in that topic. Mathematicians don’t really write such reviews, but the NAS did not wish to leave mathematicians out of competition, so they decided that my problem list(s) were the next best thing to a “review”. The Prize was awarded in April 1995, before the second list was published although there were drafts circulating. Perhaps the Prize was for past and future problem lists?

The same problem arose the next time it was math’s turn, and it was solved by awarding the Prize to Bruce Kleiner and John Lott for their opus on Perelman’s solution to the 3-dimensional Poincaré conjecture and Thurston’s geometrization conjecture [e18].

#### 8. Journal publishing and MSP

In 1996 Colin Rourke wrote to me proposing a new, very low cost
journal provisionally titled *Geometry & Topology*,
to be edited by
Brian Sanderson
and himself.
They had designed an excellent editorial
system in which an editor
would handle a paper and obtain referee reports and then make a
recommendation; if to accept, two seconders were needed, and the paper
would be under consideration for four weeks. A journal
tradition thus began: handlers gave serious arguments for their
recommendations, and were rarely overturned.

My only task was to help Colin recruit a
distinguished board of editors (which we did) in order to send a
message that *G&T* was at the level of the best specialized
journals. Topologists were unhappy with the subscription price of
Elsevier’s *Topology*, and we wanted to give authors an
excellent alternative. We succeeded a decade later when
*Topology* came to an end after the Oxford stalwarts resigned
and Elsevier failed to recruit other editors.

*G&T* thrived and in 2000 we established a new journal,
*Algebraic & Geometric Topology* with Bob Oliver and Marty
Scharlemann as chief editors. By 2003 it became clear that the
journals needed a legal home and
subscriptions. Up until then, prices were hard to set because we
didn’t know how many pages we would publish in a given year. Moreover,
we printed and bound the journals only at the end of the year, and
this meant that libraries tended to purchase them out of their book
budget rather than renew them yearly from their journal budget.
At a propitious moment,
Paulo Ney de Souza
(my PhD student and the
systems administrator for the math department) mentioned that he was
interested in publishing, and agreed to run the servers for the
electronic version of the journals. He persuaded
Silvio Levy,
already
well known as an editor of mathematics, to take on a role as
production editor with us. Ron Stern, president of the *Pacific Journal of Mathematics*,
let us know that he was ready to move production of *PJM* and was
willing to take a chance on our team. With servers, high-level copy
editing, and a source of money, we were ready to establish
Mathematical Sciences Publishers as a nonprofit California company.

MSP began a period of growth in which its “core” mathematics journals were founded:

*Communication in Applied Mathematics and Computational Science*(CAMCoS) in 2006 by Alexander Chorin and John Bell;*Algebra & Number Theory*in 2007 by David Eisenbud and Bjorn Poonen;*Analysis & PDE*in 2008 by Maciej Zworski.

In 2006, the editors of an Elsevier journal in mechanical engineering,
led by chief editor Charles Steele (of Stanford University), were
ready to resign en masse; they heard of our work as an independent
nonprofit and decided to bring their journal to us. They founded the
*Journal of Mechanics of Materials and Structures*, which MSP
is still publishing today. And in 2008 Ken Berenhaut founded our
journal of undergraduate research papers, *Involve*.

In the next decade MSP
started or adopted nearly ten more
journals.
It continues to operate on a principal of lean efficiency, its aim
being to set an example of sustainable publishing practices in a
climate dominated by corporate giants. MSP is now very ably
managed by Alex Scorpan (of
*The Wild World of
4-Manifolds* fame).

Another endeavor integral to MSP’s success has been EditFlow®, our editorial software for journal management. It is fair to say that it is easily the best software for math journals, having been designed (originally by de Souza) for and by mathematicians. Early on it was licensed to the AMS and then to the London Math Society, and it is now used by well over 100 other journals.

Last but not least, MSP is the home of *Celebratio Mathematica*
in which this article appears.

#### 9. Morse 2-functions and trisections

My collaboration with
Dave Gay
began with a visit to Tucson (Arizona
State) in 2003. It had recently been shown by Cliff Taubes and
Ko Honda
that a smooth 4-manifold __\( X \)__ with __\( \beta_2(X) \geq 1 \)__
has a
symplectic structure on the complement of a 1-manifold. We thought
it would be interesting to construct such almost symplectic forms,
perhaps on the complements of the cores of round 1-handles
[20].

This effort led to a second paper [21] in which we constructed broken Lefschetz fibrations on all smooth, closed, oriented 4-manifolds; at the time we thought we needed both Lefschetz and anti-Lefschetz singularities, but Yanki Lekili soon noticed that the anti-Lefschetz singularities were not in fact needed. The circles where the fibration “broke”, that is, where the genus of the fibers changed by one, were roughly the same as the circles on which an almost symplectic form vanished.

This work gradually morphed over time
into our paper on Morse
2-functions
[22],
an analog of Morse functions in which the
reals __\( R \)__ are replaced by the plane __\( R^2 \)__. Then Dave noticed a clever
way to trisect a Morse 2-function, and thus trisections of smooth
4-manifolds were born
[23].

A striking, but easy to prove, application is this
[24]:
If we
take the fundamental group of the various pieces of a
trisection — the
central fiber, the three slices, the three pieces of pie and the whole
4-manifold __\( X \)__ — then we get a cube of groups with epimorphisms
called a *trisection of* __\( \pi_1(X) \)__. Then there is a
bijection between
smooth, orientable closed 4-manifolds and trisections of finitely
presented groups, up to stabilization. Most striking is that the
countably infinite exotic smooth structures on simply connected
4-manifolds correspond to different trisections of the trivial
group!

#### 10. Mathematical descendants

Over 53 years I’ve greatly enjoyed my 54 PhD students and their descendants. I knew early on that I would enjoy having students, from the first two at UCLA, Ted Turner and David Gauld, to the many more at Berkeley. I left UCLA for Berkeley primarily to escape the southern California smog and for the opportunity to have more grad students, and so I left behind many friends on the UCLA math faculty.

I’ve written elsewhere2 about my philosophy of advising students, but I’ll restate it here. Briefly, grad students should find their own problems, if possible; some problems won’t work out (this is also valuable knowledge), and others will. I’ve said to many students in their post-qual, thesis-writing phase that we are now embarking on the same adventure, proving our next theorem (never mind that I have already done so a few times).

When I was a student in the 1950s, I’d heard that 75% of new PhDs never write another paper after their thesis. This seemed to be a consequence of the grad student “three-year track”: that is, taking grad courses the first year, passing a qual in the second year, and then working on a problem assigned by an adviser — with hints as to how to solve it. The result was thus a thesis by spring of the third year. Naturally, many who have no experience of doing research independently are able to publish again.

In the 1970s problems were easy to find in low-dimensional topology, and it was not so hard to be familiar with most aspects of the subject. My first successful group of students at Berkeley — Scharlemann, Akbulut, Handel, Kaplan, Melvin and Nordstrom — could all work on different topics and yet still talk to each other. They were aided by many visitors, for in those days there was enough money for people like Edwards, Lickorish, Siebenmann, Larry Taylor, Hatcher, John Morgan and others to visit for up to a quarter.

There followed another excellent cohort of students, sometimes taught by their older “brothers”, including Harer, Giller, Livingston, Hass, Menasco, Cochran, Hughes, Ruberman, Gompf and Aitchison.

But then low-dimensional topology was dramatically changed, first by Thurston, then Freedman and Donaldson in 1981–2, and Jones in 1984. More advances followed, e.g. contact and symplectic topology, Floer theory, pseudoholomorphic curves, Heegaard Floer homology by Ozsváth and Szabó, geometric group theory and Perelman’s contributions. There was much more to be learned and better advice than I could give was needed. By now there must be at least a half dozen subfields of low-dimensional topology and geometry that are as lively as all of low-dimensional topology in the 1970s.

I tried teaching some of these subjects in graduate courses, rather poorly in some cases. That lead students into these newer subjects, but often I couldn’t help them much. Tom Mrowka bailed me out with several of my students in gauge theory as did other mathematicians in other areas.

I haven’t mentioned all of my graduate students, but I could include Kevin Walker whom I introduced to the Canyonlands and now he is the County Commissioner in Moab, Utah, looking after the environment, and also once hiked across Utah for 35 days without seeing another person (it was summer but he knew the geology well enough to find seeps, and most days there were thunderstorms near enough to get water in the water pockets); Joanna Kania-Bartoszynska who keeps NSF on their toes; William Chen who has won millions at poker and wrote a book which got Amazon ratings of zero (“hated the book, too much math”) or five (“you must buy this book if you wish to win at poker”); Stephen Bigelow who won the Blumenthal Prize for the best PhD thesis in five years, worldwide, for, “The Braid Group is Linear”; Rob Schneiderman who remains a world-class jazz pianist while writing fine papers in 4-manifolds, often with Peter Teichner; Eli Grigsby who deigned to jog with me and ran the Boston Marathon; and Andy Wand who was heading up a popular rock band in Istanbul before rediscovering topology in Gökova, and several more whose weddings I was privileged to attend.

So, I’ve been lucky in math (and in many other things); lucky that the torus trick existed and that I stumbled on it and then met Larry Siebenmann; lucky that I jogged with Arnold Kas which led to many discussions about complex surfaces and to the Kirby calculus; lucky to have had so many excellent collaborators who did more than their share of the work; and lucky to have had such an excellent bunch of mathematical sons and daughters, many of whom have remained good friends, as are the editors of this volume.