Celebratio Mathematica

Robion C. Kirby

My professional life

1. Early work

I be­lieve that I first learned of the an­nu­lus con­jec­ture in 1962 by see­ing a pre­print of the prob­lem list [e5], p. 579, con­tain­ing Mil­nor’s fam­ous list of the sev­en most im­port­ant prob­lems in geo­met­ric to­po­logy.

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.


[1]R. C. Kirby: “On the an­nu­lus con­jec­ture,” Proc. Amer. Math. Soc. 17 (1966), pp. 178–​185. MR 0192481 Zbl 0151.​32902

[2]R. Kirby: Ex­ten­sion to stable homeo­morph­isms. Single-page typescript (un­pub­lished), 1968. techreport

[3]R. C. Kirby: “On the set of non-loc­ally flat points of a sub­man­i­fold of codi­men­sion one,” Ann. of Math. (2) 88 (1968), pp. 281–​290. MR 0236900

[4] R. Kirby and L. Sieben­mann: “For man­i­folds the Hauptver­mu­tung and the tri­an­gu­la­tion con­jec­ture are false,” No­tices Amer. Math. Soc. 16 : 4 (1969), pp. 695. Ab­stract 69T-G90. article

[5]R. C. Kirby and L. C. Sieben­mann: Found­a­tion­al es­says on to­po­lo­gic­al man­i­folds, smooth­ings, and tri­an­gu­la­tions. An­nals of Math­em­at­ics Stud­ies 88. Prin­ceton Uni­versity Press, 1977. With notes by John Mil­nor and Mi­chael Atiyah. MR 0645390 Zbl 0361.​57004 book

[6]R. Kirby: “A cal­cu­lus for framed links in \( S^{3} \),” In­vent. Math. 45 : 1 (1978), pp. 35–​56. MR 0467753 Zbl 0377.​55001

[7]R. Kirby: “Prob­lems in low di­men­sion­al man­i­fold the­ory,” pp. 273–​312 in Al­geb­ra­ic and geo­met­ric to­po­logy (Stan­ford Univ., CA, 1976), part 2. Edi­ted by R. J. Mil­gram. Proc. Sym­pos. Pure Math. XXXII. Amer. Math. Soc. (Provid­ence, R.I.), 1978. MR 520548 Zbl 0394.​57002

[8]S. Ak­bu­lut and R. Kirby: “An exot­ic in­vol­u­tion of \( S^{4} \),” To­po­logy 18 : 1 (1979), pp. 75–​81. MR 528237 Zbl 0465.​57013

[9]S. Ak­bu­lut and R. Kirby: “Mazur man­i­folds,” Michigan Math. J. 26 : 3 (1979), pp. 259–​284. MR 544597 Zbl 0443.​57011

[10]R. C. Kirby and M. G. Schar­le­mann: “Eight faces of the Poin­caré ho­mo­logy 3-sphere,” pp. 113–​146 in Geo­met­ric to­po­logy (Athens, GA, 1977). Edi­ted by J. C. Cantrell. Aca­dem­ic Press (New York), 1979. MR 537730 Zbl 0469.​57006

[11]R. C. Kirby and W. B. R. Lick­or­ish: “Prime knots and con­cord­ance,” Math. Proc. Cam­bridge Philos. Soc. 86 : 3 (1979), pp. 437–​441. MR 542689 Zbl 0426.​57001

[12]S. Ak­bu­lut and R. Kirby: “Branched cov­ers of sur­faces in 4-man­i­folds,” Math. Ann. 252 : 2 (1979/80), pp. 111–​131. MR 593626 Zbl 0421.​57002

[13]S. Ak­bu­lut and R. Kirby: “A po­ten­tial smooth counter­example in di­men­sion 4 to the Poin­caré con­jec­ture, the Schoen­flies con­jec­ture, and the An­drews–Curtis con­jec­ture,” To­po­logy 24 : 4 (1985), pp. 375–​390. MR 816520 Zbl 0584.​57009

[14]J. Harer, A. Kas, and R. Kirby: “Handle­body de­com­pos­i­tions of com­plex sur­faces,” Mem. Amer. Math. Soc. 62 : 350 (1986), pp. iv+102. MR 849942

[15]R. C. Kirby: The to­po­logy of 4-man­i­folds. Lec­ture Notes in Math­em­at­ics 1374. Spring­er (Ber­lin), 1989. MR 1001966 Zbl 0668.​57001

[16]R. C. Kirby and L. R. Taylor: “\( \mathit{Pin} \) struc­tures on low-di­men­sion­al man­i­folds,” pp. 177–​242 in Geo­metry of low-di­men­sion­al man­i­folds (Durham, 1989), vol. 2. Edi­ted by S. K. Don­ald­son and C. B. Thomas. Lon­don Math. Soc. Lec­ture Note Ser. 151. Cam­bridge Univ. Press, 1990. MR 1171915 Zbl 0754.​57020

[17]R. Kirby and P. Melvin: “The 3-man­i­fold in­vari­ants of Wit­ten and Resh­et­ikh­in–Tur­aev for \( \mathrm{sl}(2,\mathbf{C}) \),” In­vent. Math. 105 : 3 (1991), pp. 473–​545. MR 1117149 Zbl 0745.​57006

[18]R. Kirby, P. Melvin, and X. Zhang: “Quantum in­vari­ants at the sixth root of unity,” Comm. Math. Phys. 151 : 3 (1993), pp. 607–​617. MR 1207268 Zbl 0779.​57007

[19]R. Kirby: “Prob­lems in low-di­men­sion­al to­po­logy,” pp. 35–​473 in Geo­met­ric to­po­logy (Athens, GA, 1993). Edi­ted by W. H. Kazez. AMS/IP Stud. Adv. Math. 2. Amer. Math. Soc. (Provid­ence, RI), 1997. MR 1470751

[20]D. T. Gay and R. Kirby: “Con­struct­ing sym­plect­ic forms on 4-man­i­folds which van­ish on circles,” Geom. To­pol. 8 (2004), pp. 743–​777. MR 2057780 Zbl 1054.​57027

[21]D. T. Gay and R. Kirby: “Con­struct­ing Lef­schetz-type fibra­tions on four-man­i­folds,” Geom. To­pol. 11 (2007), pp. 2075–​2115. MR 2350472 Zbl 1135.​57009

[22] D. T. Gay and R. Kirby: “In­def­in­ite Morse 2-func­tions: Broken fibra­tions and gen­er­al­iz­a­tions,” Geom. To­pol. 19 : 5 (2015), pp. 2465–​2534. MR 3416108 Zbl 1328.​57019 article

[23] D. Gay and R. Kirby: “Tri­sect­ing 4-man­i­folds,” Geom. To­pol. 20 : 6 (2016), pp. 3097–​3132. MR 3590351 Zbl 1372.​57033 article

[24] A. Ab­rams, D. Gay, and R. Kirby: “Group tri­sec­tions and smooth 4-man­i­folds,” Geom. To­pol. 22 : 3 (2018), pp. 1537–​1545. MR 3780440 ArXiv 1605.​06731 article