#### by Rob Kirby

In a recorded interview between Bob Edwards and myself on May 9, 2019, the subject turned to Mike Freedman’s work on 4-manifolds, in particular the four-dimensional topological Poincaré conjecture — work for which Freedman won a Fields Medal in 1986. Freedman announced this work at a CBMS (Conference Board of Mathematical Sciences) conference at UC San Diego in August 1981, where Dennis Sullivan gave a ten-lecture series titled “Hyperbolic geometry, three-dimensional topology, and Kleinian groups”. Edwards recounts here the story of how Freedman’s ideas were initially presented, puzzled over, scrutinized, and finally accepted as the momentous breakthrough they really were. What follows is a lightly edited transcript of our exchange.

**Kirby:** *At the end of the
two-month-long gathering of low-dimensional topologists in Cambridge,
UK, in the summer of 1981, you received in the mail, just as you were
leaving, a twenty page handwritten manuscript from Mike Freedman
outlining a proof of the 4-dimensional topological Poincaré
conjecture. You hardly had time to look at it before going to the CBMS
lecture series at UCSD. There Mike, as conference organizer, scheduled
himself to give two evening talks. You took a look at what he had sent
you, and it all just seemed crazy.*

**Edwards:** I looked at Mike’s manuscript
on the flight from London back to LA. It was full of tantalizing
claims and comments, but overall I couldn’t make any sense of it. But
what the heck, by then I was used to Mike’s expository style, and knew
that you had to hear it from him in person. As soon as we got back
home (in Pacific Palisades, CA, probably on Thursday or Friday 20–21
August 1981) I got a phone call from Mike, asking if I could come down
to San Diego a day or two early, to take part in a preconference
introduction to his work. He said that
Ric Ancel
would be there, among
others. That was good news, for Ric was well versed in decomposition
space theory (à la
Bing
and others), which Mike seemed to be invoking
to finish his argument. So I agreed to come down early on Sunday. The
session that day, as I dimly recall, basically just gave us all a
chance to get warmed up for the task ahead.

**Kirby:** *And so the conference
began on Monday morning, with*
*Dennis Sullivan*
*giving the first of his
ten lectures.*

**Edwards:** Yes. Dennis was wonderful, as always. Then late
Monday or early Tuesday, to everyone’s surprise, Mike (who served as
the conference MC) announced that he was scheduling himself for two
evening talks. I was flabbergasted, since we hadn’t begun to digest
his claims. I remember thinking that quite possibly he was setting
himself up for a career-busting disaster. Oh well, that’s Mike.

Mike talked on Tuesday and Wednesday evenings. I think it’s fair to say that everyone in the audience found his presentations to be both mind-boggling and incomprehensible, thinking that his ideas were harebrained and crazy. It was sort of a kitchen sink kind of proof. He was hauling in Cerf theory at one point, and decomposition space theory, which he didn’t really know all that well. (He had heard me talk about various results in this theory at Southern California get-togethers throughout the 70s.) It all just sounded kind of wacko. If you look at the pages that he sent me, much of it reads like it is delusional. It was hard to extract a concrete, well-explained sequence of steps. At any rate, I remember after the second lecture Raoul Bott coming up to me and asking, “Is there any hope for this stuff?” All I could do was raise my arms and shrug. I didn’t know at that point. Wednesday we started having lunchtime working seminars, attended by people who were interested. Ric Ancel, Jim Cannon and Larry Siebenmann are the three attendees I remember, but there were others. I seem to recall maybe five to ten people sitting around the outdoor picnic table each day.

**Kirby:** *You were doing this, after all it’s the Poincaré
conjecture, and were willing to spend some time on it, so it had some
plausibility.*

**Edwards:** Yes, his analysis of Casson handles, which was the
start of his new work, definitely showed promise.

**Kirby:** *The reimbedding
arguments?*

**Edwards:** Yes, the thirteen-into-six stage reimbedding
theorem. Even that by itself was a significant advance. I think we had
those details of Mike’s program figured out by Thursday morning. That
gave me some confidence. The next step for us was to understand what
Mike called the “design” (a not-very-meaningful term IMHO, but my
proposed alternatives are maybe not much better. The “common core” was
one of them.). The design is a locally compact space which is common
to both a Casson handle and the standard handle (i.e., it can be
imbedded in each). This was tricky. But, with Mike’s help, we came to
understand its details. Wow! Very pretty. Then we began trying to
understanding the common quotient space. Mike’s idea was that, using
the design, one could construct two surjective maps, one whose source
was a Casson handle, the other whose source was the standard handle,
with both maps having a common target (quotient) space, and with all
of the point-inverses of each map being cell-like. So brilliant! Who
would have dreamed?

**Kirby:** *So at this point, Mike’s claim maybe became reasonable?*

**Edwards:** Maybe so, but there still was a long way to go. We were impressed
by the new structure that he had found in a Casson handle. We were
hooked, and wanted to understand more. Friday (as I recall) we spent
coming to grips with the details of the two cell-like decompositions
(as they are called classically; they give rise to the two quotient
maps mentioned above). The first decomposition, the one on the
standard handle, was tricky, but had some resemblance and connections
to/with several classical decompositions. You had to take each “gap”
(which visibly was homotopy-equivalent to a circle, indeed
homeomorphic to __\( S^1 \times B^3 \)__) in the complement of the design, and cap it
off with a disc to make it cell-like. This was a bit tricky, for you
had to carefully thread each disc back and forth through the wild line
to make it embedded, meanwhile making all of these discs disjoint. But
eventually it made sense to us. Then our attention in our post-lunch
seminar turned to understanding the details of Mike’s second
decomposition, the one on the Casson handle. Here the “gaps”
represented completely unknown territory, and all one could say about
them was that they were circle-like. Still, by using the design, one
could find discs to cap off (attach to) these mysterious gaps in the
Casson handle, by mimicking how it was done in the standard handle.

**Kirby:** *What next?*

**Edwards:** At this point (on Friday), having come to an
understanding of the descriptions of the two decompositions, we had
some time left to begin thinking about how Mike was proposing to
“shrink” them, i.e., how to show that each quotient map was ABH
(Approximable, arbitrarily closely, By Homeomorphisms). We decided to
look at the more daunting one, the one on the Casson handle. Here Mike
was proposing an argument that seemed completely out of nowhere. It
was just mind-blowing, and initially it made no sense whatsoever. It
didn’t help that Mike had presented it with minimal clarity; he sort
of thought and explained things in terms of ideas and perceptions
rather than hard details. After Mike’s first attempt at explaining
this part of his argument, Jim Cannon exploded (that was completely
out of character for gentleman Jim), “Mike you can’t shrink a
decomposition this way — it’s total nonsense, it will not work.
This is balderdash!”, in so many words. I heard that later, when he realized it
all worked, Jim wrote a letter of apology to Mike.

And so finally by the end of the conference on Friday we understood Mike’s setup, seeing that now “all” he had to do was to show that his two decomposition maps (above) were each ABH. And so I went back home, anxious to continue thinking about this work (and happy to see my family again).

**Kirby:** *And then?*

**Edwards:** On Saturday I decided to confront what was clearly the more
questionable, indeed outrageous, ABH claim, the one whose source-space
was the Casson handle. Here Mike was proposing a completely new method
of shrinking, where in order to make progress you made your
decomposition worse and worse (i.e., made more and more nontrivial
point inverses), meanwhile making the biggest elements become smaller
and smaller. In the end the process converged to produce what you
wanted, a homeomorphism. It was a nonisotopy argument which,
incidentally, we realized in retrospect in 1982, that it had to be.
For a classical decomposition argument
would violate Donaldson’s thesis. (I return to this topic below, near
the end.)
I’ve often thought to myself that if
Donaldson
had come before Freedman, we wouldn’t have paid any
attention to Freedman, because those of us in decomposition space
theory knew that we wouldn’t be able to shrink Mike’s decomposition by
classical methods. Mike proved his result just in time! So I thought
about it and thought about it, and by Saturday night I finally
realized that this crazy-ass argument of Mike’s in fact worked. It was
revolutionary. When Mike phoned on Sunday morning I told him that now
I understood this part of his argument. Simply amazing. So on Sunday I
turned my attention to the final hurdle, namely how to shrink Mike’s
decomposition on the standard handle. This issue was kind of sloughed
over in Mike’s notes and talk, but he had seen sufficiently many
classical arguments on similar decompositions that it seemed plausible
to him. And likewise to me. By Sunday night I was able to shrink
Mike’s decomposition on the standard handle, using a mild adaptation
of a wonderful argument of Bing. At this point I was blown away. Mike
had a theorem! I couldn’t believe it! I think back with some
satisfaction that, for twelve hours that night, I was the only one in
the world who understood all of the details of Mike’s proof (Mike
included). When Mike called the next morning I told him that he had a
proof. He understood my explanation of the standard-handle shrinking
argument immediately.

**Kirby:** *And so word quickly spread that Mike had a theorem.*

**Edwards:** Yes, and it was quickly recognized as a monster advance. I
particularly enjoyed participating in the October 1981 Austin, TX
gathering, where Bing and others savored this triumph of decomposition
space theory (of which Bing was the Progenitor-in-Chief).

**Kirby:** *Mike quickly wrote up his proof, with applications to many of the
outstanding problems in dimension four, and submitted it to the
Journal of Differential Geometry*
[1]
*(perhaps a surprising choice as
there is no differential geometry in the paper). For*
*Yau**,
after
hearing that Mike had solved the topological 4-dimensional Poincaré
conjecture, had solicited the paper. Is this when Yau called you up
and asked you to referee the paper for the JDG?*

**Edwards:** Not quite. I never talked to Yau, but I received the paper from
the JDG and a request to referee it, in late October or early November
of 1981. I was amazed that Mike had gotten the paper written up so
quickly. That was a busy term for me. Minimizing my other obligations,
I started to look at the paper, and write notes. My notes began to be
as long as the paper. For each page in the paper I generated at least
a page of notes, questions, corrections, and typos. Weekly I got a
phone call from the JDG secretary asking how I was doing. After about
four weeks of this, I threw up my hands and said to myself, “Look,
there’s no way I can properly referee this paper that quickly; there
would be tons of back and forth.” The next time the secretary called I
said “Yes, the paper is correct, I assure you. But I can’t generate a
proper referee’s report any time soon.” So they decided to accept and
published it as it was.

**Kirby:** *Ah, so that explains why I couldn’t read it* (*ha ha!*). *Mike’s paper
was formally accepted in December 1981, but then there was the normal
publication delay. And then?*

**Edwards:** In July 1982 along came the major Durham, NH
conference “Four-Manifold Theory”.
For me, the highlight there was
Frank Quinn
making his
dramatic announcement that he had cracked the 4-dimensional annulus
conjecture. Mike and I had worked on this throughout the spring, but I
too often allowed myself to be diverted to other things, as I came to
regret. Frank gave a talk providing some details of his result. They
were sufficiently intriguing that Mike and I became committed to
understanding it. That led to an intense 48 hours, as we picked
Frank’s brain and fleshed out his details and finally realized that he
really did have it. This was major. The topological annulus conjecture
was now established in all dimensions!

**Kirby:** *There was a push to have
Frank’s paper appear back-to-back with Mike’s paper in JDG, and this
led to a further delay in publishing Mike’s paper.*

**Edwards:** Yes, I heard of this, but didn’t pay attention. I spent my time
writing up Frank’s 4-dimensional annulus conjecture in the way that I
viewed it. This appeared in the proceedings of the Durham conference
[e4].
I remember
Cameron
(Gordon and Kirby were editors of the
proceedings) getting on my back for being so slow, delaying the
publication of the proceedings. Those were amazing days, two one-week
long conferences that produced earth-shaking results.

**Kirby:** *Anything else you’d like to add?*

**Edwards:** For recreation I (like other mathematicians, I suspect) sometimes
like to ponder and compare various aspects of great advances
(theorems). One favorite question I ask myself is: If Person(s) X
hadn’t proved breakthrough Theorem Y when they did, how much time
would elapse before some Person(s) Z came along and proved it? I can’t
help but think: If Mike Freedman hadn’t proved, when he did, that
Casson handles are topologically standard, then likely it *never*
would have been proved (and so: No uncountably many smooth __\( R^4 \)__s,
etc.). My main point here is: An immediate consequence of Donaldson’s
breakthrough 1982 theorem was that (not all) Casson handles can be
smoothly standard.

My thinking goes as follows: Sure, absent Mike, before long someone
would have proved Mike’s 13-into-6 stage reimbedding theorem. Then
later someone would have come up with (something like) the “design”,
and then someone would have come up with the two cell-like
decompositions, one on the standard handle and the other on a Casson
handle, whose quotients are the same, and shown that the first
quotient map was ABH. But then, after all of this masterful work,
never would anyone come up with Mike’s Section 9 argument for showing
that the Casson handle quotient map was ABH. For experts would have
said: All shrinking so far (since Bing invented the concept in 1952)
has been accomplished by near-diffeotopies. But that can’t be done in
this case, consequence of Donaldson. And there is no other conceivable
method for shrinking a cell-like decomposition. But there was:
Freedman’s brilliant back-and-forth *inversion* method.1
What genius!

A further comment: Taking a broader view, I guess I regard Mike’s key
revolutionary idea as: Try to relate a Casson handle to/with the
standard handle by examining and comparing them from their
*insides*, not their outsides. Explanation: Any Casson handle
is in a natural way a subset of the standard 2-handle. So the
“obvious,” indeed compelling, way to show that they are homeomorphic
(or rather diffeomorphic, which was the natural supposition) was to
somehow collapse (shrink) away the difference
StandardHandle __\( \setminus \)__ CassonHandle
(I tried very hard to do
this). Instead, Mike’s method of comparing them was to do so
*internally*, as opposed to externally. This was brilliant.

A final point: Rereading Mike’s paper [1] recently (for this interview), I am reminded how much I enjoy reading it from a narrative point-of-view. If you ignore certain somewhat confusing notation and various details of the proof, it’s a wonderful exposition.

I was, and will remain forever, awed by this advance in topology.

**Kirby:** *Thanks for your memories.*

#### Editor’s note

Ric Ancel was the first to write up a nice version of the surprising approximation theorem [e3] and John Walsh’s Math Review explains this well:

At a crucial juncture in Freedman’s analyses of various decompositions that he produced during his exploration of Casson handles (that ultimately leads to showing that they are topologically 2-handles), he was faced with a map

\( f:S^4 \to S^4 \)satisfying

\( S(f) ={y \in S^4: \text{ diameter } f^{-1}(y) > 0} \)is nowhere dense, that is, its closure has empty interior, and- the collection
\( {f-1(y):y \in S(f)} \)contains only finitely many elements of size greater than\( \epsilon \)for each\( \epsilon > 0 \).Compared to the other decompositions encountered, the one associated to

\( f \)is not “easily understood”, but has the feature that it is clear that the associated decomposition space is\( S^4 \). (Consequently, it appears difficult to attack using “standard” shrinking techniques.) Freedman’s approach was to exploit a technique used by M. Brown in his proof of the Schoenflies theorem in 1961 to arrive at the valuable conclusion that such a map\( f \)is the uniform limit of homeomorphisms. This paper contains a proof of a more general version of this result that, following a suggestion of R. D. Edwards, replaces the above conditions with:

\( \!\!\!^{\prime} \)\( S(f) \)is not dense in\( S^4 \), and\( \!\!\!^{\prime} \)\( S(f) \)is a tame 0-dimensional subset.

Larry Siebenmann gave an account of Mike’s work in a Bourbaki lecture [e2], which was subsequently translated into English by Min Hoon Kim and Mark Powell (and which will be added to this volume). Siebenmann’s exposition (in French) contains a complete proof of the most novel parts involving decomposition space theory.

In [2] Freedman and Frank Quinn covered some of Mike’s earlier work and addressed some later developments — Frank’s extension of topological transversality to dimension 4, for example — and made use of “gropes” to simplify the exposition.

Through the 1990s, Peter Teichner worked with Mike to extend the theory to fundamental groups of subexponential growth [3] and began to feel that young topologists were going to have a hard time learning Mike’s work if an up-to-date exposition was not created. He induced Mike to give a series of lectures in Santa Barbara in 2013 for a team at the Max Planck Institute in Bonn (where Teichner is a Professor). This team has now written a new exposition, “The disc embedding theorem” [e5], to be published in 2020 by Oxford University Press. The book will include 28 chapters, each one authored by subgroups of between one and four members of the team, whose names we list here: Stefan Behrens, Xiaoyi Cui, Christopher W. Davis, Peter Feller, Boldizsár Kalmár, Daniel Kasprowski, Min Hoon Kim, Duncan McCoy, Jeffrey Meier, Allison N. Miller, Matthias Nagel, Patrick Orson, JungHwan Park, Wojciech Politarczyk, Mark Powell, Arunima Ray, Henrik Rüping, Nathan Sunukjian, Peter Teichner, and Daniele Zuddas.

#### Addendum: a letter to Bob Edwards from Ric Ancel (in response to the interview)

Ric Ancelsent Bob Edwards the following response after reading the latter’s recollections of Freedman’s historic breakthrough in the above interview. Its vivid details are worth preserving and so we include it here verbatim.— Editors

Bob,

I just finished reading your recollections of Mike Freedman’s proof for the second time. They brought back to life my own filed and forgotten memories of that period. Unfortunately my memory is, in general, imprecise and spotted with gaps. Yours is much sharper. But I do still have vivid impressions of the period before, during and after the 1981 San Diego conference, since this was, mathematically speaking, one of the most important events I have directly participated in. If you or Rob are interested, here are my recollections — slightly blurred by passage of time — of my own connections with those events.

My relationship with Mike Freedman started when I invited him to give
some talks in Norman about his previous foray into Casson handles — the fake __\( S^{3} \times \mathbb{R} \)__ work. I had read this paper carefully and was struck
by its apparent potential to make progress in dimension 4, and
subsequently introduced myself to him at an AMS meeting (where he gave
a talk about some joint work with
Joel Hass
and
Peter Scott)
and I
invited him to Norman. He must have visited Norman sometime during
the 1980–81 academic year. (I remember that during his visit he was
treated to a typically Oklahoman phenomenon — a dust storm that turned
the sky red and coated every outdoor object with a thin layer of red
dust.) My impression is that preparing these talks forced him to
reimmerse himself in 4-manifolds — with an eventual spectacular
outcome.

In the months after his visit, my recollection is that we corresponded occasionally about 4-dimensional topology. He invited me to stay with him at his house in Del Mar for a few days before the San Diego conference because he wanted to show me some new mathematics, but I vaguely recall that he didn’t say exactly what it was. He kept it a mystery until I arrived. When I arrived he revealed that he thought he had a proof of the 4-dimensional Poincaré Conjecture and he started to outline the proof for me. I don’t remember details, but I do recall being very confused at first, and slowly over several days seeing the logical outline of the proof emerge. (I also recall that at that point his grasp of decomposition space theory, which played a major role in his proof, was rudimentary and harbored some basic misconceptions — which ultimately dissolved away because of the strength of his overall conception and, I suspect, some serious mentoring about decomposition spaces by you, Bob.) The outline was an impressively massive structure which you alluded to in your recollections. I recall that by the time I left San Diego at the end of the conference, I understood the outline and believed that the steps of the outline involving “traditional” decomposition space techniques were plausible, but there was one remaining step that was a big question mark. This was the result concerning approximating certain maps between spheres by homeomorphisms.

Our conversations before the conference and his subsequent lectures at the conference brought the overall outline of his proof of the Poincaré conjecture into focus for me, and revealed that the whole enterprise seemed to depend on the validity of the maps-between-spheres (MBS) result. At the time, like you, I found Mike’s argument for this result vague and baffling. I recall his invocation of “Cerf theory” in a topological context and wondered about the plausibility of this. My sharpest memory from that time is the outdoor afternoon conversation involving Mike, you, Jim Cannon, me and others about this result in which Jim gave a very forceful mini-lecture about why “traditional” decomposition space techniques could never prove the MBS result; Jim provided relevant theorems and examples — mostly from Bing’s work. Of course, Jim was right if one insists on “traditional” techniques. (Having known Jim for years — he was my advisor — I knew he was a passionate guy and I didn’t find his “rant” as uncharacteristic as you did.)

I believe I flew back to Norman after the conference with a
hand-written version of Mike’s proof of the MBS result which was as
mystifying to me as his verbal arguments had been. Then, a week or so
later, I received a hand-written exposition of a new proof from Mike.
This proof was a bolt from the blue, a stroke of true brilliance, a
totally innovative argument unlike any preceding “traditional”
decomposition proof and unlike his previous inconclusive argument as
far as I could tell. It was complex yet simple enough that I quickly
believed it. Since I had been pessimistic about the prospects for a
proof of the MBS result, I was floored. He must have sent this proof
to Jim Cannon as well because Jim and I independently wrote up
expositions of this proof and disseminated them to our topology
colleagues.
(Jim and I were in a unique position to write such
expositions because Jim had formulated his “theory of cell-like
relations” in the mid-70s and used it in several results, including
the double suspension theorem, and I had exploited it in my thesis and
some later papers, and the language of relations was the natural way
to communicate Mike’s proof. A revised version of my exposition of
Mike’s proof appeared in the issue of *Contemporary Mathematics* devoted
to the proceedings of the 1982 Durham, New Hampshire, 4-manifolds conference that
also contains your exposition of
Quinn’s
proof of the 4-dimensional
annulus conjecture.) Like you, I heard that Jim had written a letter
of apology to Mike sometime after the San Diego conference. Given
Mike’s force of conviction, I doubt that Jim’s expression of doubts
had much impact on him and the apology was unnecessary. I have never
heard Mike describe the thoughts that led him to the remarkable proof
of the MBS result; and though decades have passed, I would still be
interested in what he can tell us about this.

Following the San Diego conference, I immersed myself in the details of Mike’s proof, so that I felt I had a comfortable understanding of all of it by the time of the October 1981 conference in Austin devoted to it.

I had some contact with the refereeing process for Mike’s *J. Diff.
Geom.* article;2
the person in charge of refereeing the paper
asked me to check over the section on the MBS result. I
remembered finding what I thought was a small, easily rectified
technical error. I later asked Mike if he had corrected it; and he
told me I was wrong. There was no error. Maybe I was.

In retrospect, Mike’s proof is a monumental structure in which the MBS result is a sine qua non stroke of genius. The speculation that without Mike’s proof the 4-dimensional Poincaré conjecture might still be unresolved is reinforced by the novelty of his proof of the MBS theorem which is unlike anything I’ve seen before or since.

**— Ric**

On reading the above letter, Rob Kirby remarked to Ric Ancel (in an email):

I have said for years that Mike’s proof is the single most impressive piece of mathematics that I am aware of. Not only the proof, but the results are so unexpected, so unlikely. Who would have guessed that the structure of topological 4-manifolds is so much simpler than the smooth case, even if we still don’t know which fundamental groups are “good”.