Celebratio Mathematica

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.

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

1. \( S(f) ={y \in S^4: \text{ diameter } f^{-1}(y) > 0} \) is nowhere dense, that is, its clos­ure has empty in­teri­or, and
2. 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:

1. \( \!\!\!^{\prime} \) \( S(f) \) is not dense in \( S^4 \), and
2. \( \!\!\!^{\prime} \) \( S(f) \) 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.

Ric An­cel sent 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”.