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Celebratio Mathematica

Cameron McAllan Gordon

Working with Cameron: from exceptional surgeries to the L-space conjecture

by Steven Boyer

1. Introduction

I’ve been for­tu­nate in my col­lab­or­a­tions, many of which ex­ten­ded over a num­ber of years and pro­jects. My goal here is to give an over­view of my long run­ning col­lab­or­a­tion with Camer­on Gor­don, what led to it, how it evolved, the people who took part and, of course, the math­em­at­ics. The text re­flects a point of view drawn from my memor­ies, my read­ing of how the story un­fol­ded, as im­per­fect and idio­syn­crat­ic as that source may be. I’ve steered away from be­ing ex­haust­ive with re­spect to ref­er­ences, as my in­ten­tion is to tell a story.

2. First interactions

I first be­came aware of Camer­on through his writ­ings. As a postdoc­tor­al fel­low I de­voured his sur­vey Some as­pects of clas­sic­al knot the­ory [1], which in­tro­duced me to the more al­geb­ra­ic­ally soph­ist­ic­ated parts of the the­ory. This was in the mid 1980s, when Camer­on’s re­search in­terests were more and more fo­cused on ex­cep­tion­al sur­gery prob­lems. One of my early pa­pers [17] con­nec­ted with these in­terests and this promp­ted me to ap­proach him after a talk he gave at a con­fer­ence at Berke­ley in 1985. I would be very sur­prised if he re­mem­bers the en­counter, but it was im­port­ant to me that he listened pa­tiently as I de­scribed my res­ult and then took the time to pro­cess how it re­lated to his over­all pic­ture of things.

I re­turned to his sphere soon after when I solved a group the­or­et­ic prob­lem he had men­tioned in his talk: Show that a free product of non­trivi­al cyc­lic groups could not be nor­mally gen­er­ated by a prop­er power [e2]. As Camer­on had ob­served, a pos­it­ive solu­tion to the prob­lem would com­plete the proof that man­i­folds ob­tained by non­in­teg­ral sur­gery on a knot in the 3-sphere are ir­re­du­cible [3].

It would be an­oth­er twelve years be­fore we began to work to­geth­er. To de­scribe how that came about, let’s pause to dig a little deep­er in­to the area of ex­cep­tion­al sur­ger­ies.

2.1 Exceptional surgery problems
We be­gin by re­call­ing a few defin­i­tions.

A knot man­i­fold is a com­pact, con­nec­ted, ori­ent­able, 3-man­i­fold M whose bound­ary is an in­com­press­ible tor­us. A slope on the bound­ary of a knot man­i­fold M is a M-iso­topy class of (nonori­ented) es­sen­tial simple closed curves in M. Slopes cor­res­pond biject­ively with the set of prim­it­ive classes in αH1(M), con­sidered up to sign, and para­met­rise Dehn fillings M(α) of M.

A knot man­i­fold is hy­per­bol­ic if its in­teri­or sup­ports a com­plete, fi­nite volume hy­per­bol­ic met­ric. We call a slope on the bound­ary of a hy­per­bol­ic knot man­i­fold M ex­cep­tion­al if the as­so­ci­ated filling is not hy­per­bol­ic. The set of such slopes is de­noted by E(M)={α|M(α) is not hyperbolic}. By geo­met­risa­tion, a slope is con­tained in E(M) if and only if the as­so­ci­ated filled man­i­fold is re­du­cible, tor­oid­al, or an ir­re­du­cible Seifert fibred man­i­fold which con­tains no es­sen­tial tori. The lat­ter fam­ily is con­tained in the set of small Seifert man­i­folds and in­cludes all closed man­i­folds with cyc­lic or fi­nite fun­da­ment­al groups.

Un­der­stand­ing E(M) has been of long-stand­ing in­terest, mo­tiv­ated to a great ex­tent by Thur­ston’s hy­per­bol­ic Dehn filling the­or­em, which im­plies that it is al­ways fi­nite. Ba­sic ques­tions are:

  • How large can E(M) be and how are its ele­ments dis­trib­uted in the sur­gery plane H1(M;R)?
  • What is the to­po­logy of M when the car­din­al­ity of E(M) is re­l­at­ively large?

Res­ults on the first ques­tion are of­ten phrased in terms of the dis­tance between slopes α and β which, when they are rep­res­en­ted by prim­it­ive ele­ments of H1(M), is giv­en by the ab­so­lute value of the al­geb­ra­ic in­ter­sec­tion num­ber: Δ(α,β)=|αβ|. A pri­ori bounds on the dis­tances between ele­ments of a set of slopes S lead to bounds on the car­din­al­ity of S. For in­stance, a dis­tance bound of 1 im­plies that #S3.

One of the first ma­jor res­ults in the area was the cyc­lic sur­gery the­or­em, con­tained in the sem­in­al pa­per [2] that Camer­on coau­thored with Marc Cull­er, John Luecke, and Peter Shalen.

The­or­em 2.1 (The cyc­lic sur­gery the­or­em): Let M be a com­pact, con­nec­ted, ori­ent­able, ir­re­du­cible, non-Seifert fibred 3-man­i­fold with tor­us bound­ary and α,β slopes on M such that π1(M(α)) and π1(M(β)) are cyc­lic. Then Δ(α,β)1. Hence there are at most three Dehn fillings of M with cyc­lic fun­da­ment­al groups.

This pa­per was im­port­ant as much for its meth­ods as its res­ults. Its first half con­tained Marc and Peter’s beau­ti­ful rep­res­ent­a­tion-the­or­et­ic meth­ods de­signed to deal with the case that M is hy­per­bol­ic, while its second half con­tained Camer­on and John’s power­ful de­vel­op­ments of the in­ter­sec­tion graph meth­od to deal with the non­hyper­bol­ic case. This is an overly simplist­ic as­sess­ment of the roles each half plays in the pa­per though. For in­stance, there are situ­ations where the graph the­or­et­ic meth­ods of Camer­on and John are needed when M is hy­per­bol­ic. See Peter Shalen’s art­icle Camer­on and the Cyc­lic Sur­gery The­or­em: a per­son­al ac­count on this site for a fuller dis­cus­sion of the pa­per.

2.2 First visit to Austin

I star­ted to work on a “fi­nite sur­gery the­or­em” with Xin­gru Zhang when he came to Montreal as a postdoc­tor­al fel­low in the fall of 1992. This was to be an ana­logue of The­or­em 2.1 for sur­ger­ies which yiel­ded man­i­folds with fi­nite fun­da­ment­al groups. Our ap­proach fol­lowed the same scheme and we ap­plied Camer­on and John’s work in much the same way. Our tech­nic­al con­tri­bu­tions lay in an en­hance­ment of Marc and Peter’s rep­res­ent­a­tion the­or­et­ic meth­ods, and this com­ple­ment­ar­ity with Camer­on’s meth­ods made our later col­lab­or­a­tions a nat­ur­al step.

After Xin­gru and I com­pleted our pa­per [e4], Camer­on in­vited me to Aus­tin in Feb­ru­ary 1995. The day I ar­rived, Camer­on and his wife Sue took me out for mar­gar­itas at a ter­raced res­taur­ant (The Oas­is) high over Lake Trav­is, where I rev­elled in the con­trast between the Ca­na­dian winter morn­ing that I had left and the warm, breezy sun­shine of an Aus­tin af­ter­noon.

I had a bril­liant time that week, as I have on each sub­sequent vis­it. There was plenty of math­em­at­ics to dis­cuss, not only with Camer­on, but also with John Luecke and Alan Re­id. And all of it was fun. The math­em­at­ics flowed back and forth with hanging out, listen­ing to mu­sic, and dis­cuss­ing mu­tu­al in­terests, all en­hanced by good food, drink and laughter. Sue was the ever-gra­cious host and friend, as was Alan’s wife Mara dur­ing later vis­its.

3. First collaborations

Giv­en a hy­per­bol­ic knot man­i­fold M, set Δ(M)=max{Δ(α,β)|α,βE(M)}. Here is a fam­ous con­jec­ture of Camer­on’s ([7], Con­jec­ture 3.4):
Con­jec­ture 3.1 (Gordon) For any hy­per­bol­ic knot man­i­fold M, #E(M)10 and Δ(M)8. Moreover, ex­clud­ing four ex­pli­cit pos­sib­il­it­ies for M, #E(M)7 and Δ(M)5.

In the late 1990s, Xin­gru and I had mapped out an ap­proach to prov­ing Con­jec­ture 3.1 when M had first Betti num­ber 2 or more. A case arose that we felt should be doable us­ing graph in­ter­sec­tion tech­niques, but as the de­tails were bey­ond us, we ap­proached Camer­on. His im­me­di­ate re­sponse was yes, his meth­ods did ap­ply, and after he worked out the finer points we had a sketch of a com­plete proof. There were some chal­len­ging gaps to over­come though, but in the end, the three of us com­pleted the pa­per [8].

3.1 Reducible and finite filling slopes

I saw a lot of Camer­on over the next few years, though Xin­gru and I didn’t start an­oth­er pro­ject with him till around 2006. By that time, vari­ous au­thors had used the in­ter­sec­tion graph meth­od to re­duce Con­jec­ture 3.1 to un­der­stand­ing the re­la­tion­ship between small Seifert filling slopes and oth­er ex­cep­tion­al filling slopes (see [5], [e5], [e6], [6]). Un­for­tu­nately, small Seifert man­i­folds slipped through the cracks of the tech­niques used up to that point. Rep­res­ent­a­tion the­or­et­ic meth­ods were well ad­ap­ted to study­ing Dehn fillings yield­ing small Seifert man­i­folds whose fun­da­ment­al groups ad­mit­ted re­l­at­ively few con­jugacy classes of ir­re­du­cible PSL(2,C) rep­res­ent­a­tions (e.g., those with cyc­lic or fi­nite fun­da­ment­al groups), but wer­en’t ob­vi­ously ap­plic­able in gen­er­al. Also, the in­ter­sec­tion graph meth­od was a non­starter, since most small Seifert man­i­folds do not con­tain es­sen­tial sur­faces. They do con­tain es­sen­tial im­mersed tori though, at least typ­ic­ally, a fact that Xin­gru and I were ex­ploit­ing with Marc Cull­er and Peter Shalen. (An early form of these ideas ori­gin­ated in un­pub­lished work of Camer­on, Marc and Peter.) We de­veloped an im­mer­sion graph meth­od and used it in [e12] to show that for hy­per­bol­ic knot man­i­folds, the dis­tance of a re­du­cible filling slope β to a small Seifert filling slope α was at most 4, at least in the gen­er­ic case that β was a strict bound­ary slope (e.g., when M(β) is neither S1×S2 nor P3#P3).

At this point, Camer­on, Xin­gru and I took on the chal­lenge of show­ing that the dis­tance of a re­du­cible filling slope β to a fi­nite filling slope α on the bound­ary of a hy­per­bol­ic knot man­i­fold M was 1, which was con­sist­ent with em­pir­ic­al evid­ence. The res­ults of [e12] had re­duced us to the case that M(α) and M(β) were very spe­cif­ic man­i­folds, while M ad­mit­ted an es­sen­tial planar sur­face with four bound­ary com­pon­ents which split M in­to two pieces with very simple to­po­logy.

We had ar­ranged to work on this prob­lem at a con­fer­ence in Beijing (June 2007). On his flight over, Camer­on real­ised that our con­straints on M im­plied that it ad­mit­ted an in­vol­u­tion with quo­tient the 3-ball and branch set a two strand tangle. The in­vol­u­tion ex­ten­ded over each Dehn filling M(γ) of M with quo­tient S3 and branch set a link LγS3. Know­ing the to­po­logy of M(α) and M(β) al­lowed us to de­term­ine Lα and Lβ, and com­par­ing them led us to the dis­tance bound of 1. Of course there were de­tails to be sor­ted out, but we over­came the fi­nal hurdles and the pa­per ap­peared a few years later [9].

3.2 Toroidal and small Seifert filling slopes

While still in Beijing, Camer­on, Xin­gru and I real­ised that we might be able to com­bine the tech­niques of [e12] with the in­vol­u­tion tech­nique we were dis­cuss­ing to veri­fy the case of Con­jec­ture 3.1 con­cerned with the dis­tance of a small Seifert filling slope to a tor­oid­al filling slope. More pre­cisely, we hoped to prove this:

Con­jec­ture 3.2 If M is a hy­per­bol­ic knot man­i­fold which ad­mits a tor­oid­al filling slope and a Seifert filling slope of dis­tance lar­ger than 5, then M is the fig­ure eight knot ex­ter­i­or.

Our idea was to use [e12] to re­duce Con­jec­ture 3.2 to the case where M ad­mit­ted an es­sen­tial genus 1 sur­face F which split it in­to two re­l­at­ively simple to­po­lo­gic­al pieces, ana­log­ous to what was done in [9]. Set­ting m1 to be the num­ber of bound­ary com­pon­ents of F, we knew that when m was 1 or 2 we could of­ten find an in­vol­u­tion on M, which should be use­ful. This ap­proach failed when m3, since F would not be a “mutat­ing” sur­face, but we hoped that the im­mer­sion graph meth­od could be re­fined to pick up the slack. I don’t think we real­ised how much work this pro­ject would in­volve, cer­tainly I didn’t, and in the end we had to throw everything but the kit­chen sink at it.

First we con­sidered m3. Though Camer­on’s ex­per­i­ence with in­ter­sec­tion graphs was es­sen­tial in al­low­ing us to wring the most out of im­mer­sion graphs, his con­tri­bu­tion which sticks out most in my mind oc­curred in the last case we had to con­sider. While dis­cuss­ing it dur­ing a vis­it to Aus­tin, we re­duced the prob­lem to un­der­stand­ing the prop­er­ties of a cer­tain group G. At some point, Camer­on went off to his of­fice to think and came back a little while later to say that we could ap­ply some Tiet­ze trans­form­a­tions to con­vert a nat­ur­al present­a­tion of G to one with a very par­tic­u­lar re­lat­or. The re­mark­able thing was that he had re­membered that Ed­jvet and How­ie had stud­ied groups with that ex­act re­lat­or some twenty years earli­er, and he thought that their con­clu­sions would be rel­ev­ant to our prob­lem. In short or­der we com­bined their con­clu­sions with some work of Jonath­an Hill­man and some of Xin­gru and mine to com­plete the proof when m3 and β was a strict bound­ary slope (e.g., M(β) was neither a tor­us bundle nor a tor­us semibundle). See [10].

Next up was m=1, where we were able to re­duce to the situ­ation to when M ad­mits an in­ter­est­ing in­vol­u­tion. It would be in­cor­rect to say that the re­duc­tion was simple; pro­du­cing the de­tails re­quired a lot of work. Nev­er­the­less, we com­pleted the pa­per in a pre­cise and sat­is­fy­ing way. See [12].

The wheels came off the pro­gramme when we turned to the case m=2. We re­duced the prob­lem to the situ­ation that F was a sep­ar­at­ing non­semi­fibre after a fair amount of work. This was what oc­curred for the fig­ure eight knot ex­ter­i­or and as such, we sus­pec­ted it would be the most chal­len­ging part of the pa­per. Nev­er­the­less, by late fall of 2012 we thought that we were done. Camer­on and I had ar­ranged a two week Re­search in Pairs vis­it to Ober­wolfach in March 2013 to go over the fi­nal de­tails. Our mood, which was tri­umphant and self-con­grat­u­lat­ory, in­stant­an­eously dis­in­teg­rated when we found an er­ror in the ar­gu­ment. (It didn’t help that we were the only hu­man be­ings on the MFO cam­pus over that dark wintry week­end.) Our ana­lys­is was para­met­erised by an in­teger d0 as­so­ci­ated to the knot man­i­fold M and we found a gap in the case that d=1, the case that oc­curred for the fig­ure eight ex­ter­i­or! We were stumped, and in spite of some fit­ful at­tempts over the next few years, we de­jec­tedly weaned ourselves off the prob­lem. It re­mained, however, the ever-present ele­phant in the room, fre­quently al­luded to with gal­lows hu­mour. Then out of the blue, some five years later, we real­ised that new work of Bruno Mar­telli based on ma­chine cal­cu­la­tion might help out. It did, and once we figured out the to­po­lo­gic­al side of things, Xin­gru im­ple­men­ted the com­puter cal­cu­la­tions which al­lowed us to fill the re­main­ing gap. At over 120 pages, the pa­per ap­peared in the Mem­oirs of the AMS [20].

The one case which re­mains to com­plete the proof of Con­jec­ture 3.2 is when m3 and β is not a strict bound­ary slope. This only arises when M(β) is either a tor­us bundle or a tor­us semibundle, so is highly non­gen­er­ic. It should be an “easy” case, so hope­fully we’ll clear it up at some point.

As for Con­jec­ture 3.1, Lack­enby and Mey­er­hoff veri­fied its first claim #E(M)10 and Δ(M)8 in [e13]. What re­mains to be es­tab­lished is that Δ(α,β)5 if α is a small Seifert filling slope and either

  1. β is a re­du­cible filling slope and a non­strict bound­ary slope, or
  2. β is a tor­oid­al filling slope and a non­strict bound­ary slope, and m (as above) is at least 3, or
  3. β is a small Seifert filling slope and M is not the fig­ure eight knot ex­ter­i­or.

Case (c) is widely open, though two in­ter­est­ing spe­cial cases are known. The first is the cyc­lic sur­gery the­or­em, which we men­tioned above (The­or­em 2.1). The second is the fi­nite sur­gery the­or­em, whose state­ment was con­jec­tured by Camer­on in his ICM ad­dress [4]: If M is a hy­per­bol­ic knot man­i­fold, then there are at most five slopes α on M such that π1(M(α)) is fi­nite. Fur­ther, the dis­tance between any two such slopes is at most 3. (These num­bers are real­ised when M is the ex­ter­i­or of the (2,3,7)-pret­zel knot.) Xin­gru and I veri­fied this con­jec­ture in [e7].

4. The L-space conjecture

Since 2009, our main fo­cus has shif­ted from ex­cep­tion­al sur­ger­ies to the L-space con­jec­ture.

4.1 The L-space conjecture

An L-space is a ra­tion­al ho­mo­logy 3-sphere whose re­duced Hee­gaard Flo­er ho­mo­logy M van­ishes. Ex­amples in­clude 3-man­i­folds with fi­nite fun­da­ment­al groups and 2-fold cyc­lic branched cov­ers of al­tern­at­ing knots.

A co­ori­ented taut fo­li­ation on a closed ori­ent­able 3-man­i­fold M is a codi­men­sion-1 fo­li­ation F on M with trivi­al nor­mal bundle for which we can find a trans­verse loop to F which passes through each of its leafs. Closed, ori­ent­able 3-man­i­folds ad­mit­ting co­ori­ented taut fo­li­ations are known to be prime and have in­fin­ite fun­da­ment­al groups.

A group is left-or­der­able if it is non­trivi­al and ad­mits a total or­der in­vari­ant un­der left mul­ti­plic­a­tion (a left-or­der). Left-or­der­able groups are tor­sion-free, so are in­fin­ite. An ob­vi­ous ex­ample of a left-or­der­able group is R un­der ad­di­tion. A less ob­vi­ous one is Homeo+(R). In fact, a non­trivi­al count­able group is left-or­der­able if and only if it is iso­morph­ic to a sub­group of Homeo+(R). A stronger res­ult holds for 3-man­i­fold groups: The fun­da­ment­al group of an ori­ent­able ir­re­du­cible 3-man­i­fold is left-or­der­able if and only if it ad­mits a ho­mo­morph­ism to Homeo+(R) with non­trivi­al im­age [e9].

We say that a closed, con­nec­ted, ori­ent­able 3-man­i­fold M is

  • LO if π1(M) is a non­trivi­al left-or­der­able group;
  • CTF if M ad­mits a co­ori­ented taut fo­li­ation;
  • NLS if M is not an L-space.
Con­jec­ture 4.1 (The L-space conjecture [11], [e14]) If M is a closed, con­nec­ted, ori­ent­able, ir­re­du­cible 3-man­i­fold, then M is LO if and only if it is NLS, and if and only if it is CTF.

The one im­plic­a­tion known is CTFNLS, which was proved by Oz­sváth and Szabó for C2 fo­li­ations in [e8] and later ex­ten­ded to C0 fo­li­ations by KazezRoberts [e17] and in­de­pend­ently by Bowden [e15].

When the first Betti num­ber of M is pos­it­ive, M is LO [e9], CTF [e1], and NLS (by defin­i­tion), so the in­ter­est­ing case arises when M is a ra­tion­al ho­mo­logy 3-sphere. Here, it is dif­fi­cult to know wheth­er any of the con­di­tions holds or not, though there are many fam­il­ies for which we know the status of at least one of them and this mo­tiv­ates de­vel­op­ing tech­niques to de­term­ine the status of the oth­ers.

4.2 The genesis of the L-space conjecture

Dale Rolf­sen, Bert Wi­est and I wrote a pa­per on the or­der­ab­il­ity of 3-man­i­fold groups, which ap­peared in 2005 [e9]. One con­sequence of it was that if M is a non­hyper­bol­ic geo­met­ric 3-man­i­fold (i.e., M is either Seifert fibred or a Sol man­i­fold), then M is LO if and only if it is CTF. Shortly af­ter­wards, Lis­ca and Stip­cisz proved that if M is a Seifert fibre space with base or­bi­fold a 2-sphere with cone points, then M is NLS if and only if it is CTF [e11]. Liam Wat­son and I dis­cussed this dur­ing the last year of his doc­tor­al pro­gramme at UQAM (2008-09), and wondered to what ex­tent the equi­val­ence of LO,CTF and NLS was more gen­er­ally true. As a first step, we made a hit-list of prob­lems whose solu­tions would tell us if we were bark­ing up the wrong tree. For in­stance, we knew from [e9] that Seifert fibre spaces with base space a pro­ject­ive plane were not LO, and that the same was true for ra­tion­al ho­mo­logy 3-sphere Sol man­i­folds. If we could show that they were L-spaces, then the three con­di­tions would be equi­val­ent for these fam­il­ies. We veri­fied the Seifert case im­me­di­ately, and with this in hand we knew that LONLSCTF for all Seifert fibre spaces. Show­ing that ra­tion­al ho­mo­logy 3-sphere Sol man­i­folds were L-spaces stumped us though, sur­pris­ingly to me since they have re­l­at­ively simple to­po­logy.

An­oth­er prob­lem on our list was to show that 2-fold cyc­lic branched cov­ers Σ2(L) of non­split al­tern­at­ing links L were not LO. (They were known to be L-spaces [e10].) Serendip­ity played a role at this point. I was sched­uled to meet Camer­on at BIRS on a Re­search in Team vis­it whose goal was to make pro­gress on the case m3 of Con­jec­ture 3.2. Our first night there I asked him if he knew any in­ter­est­ing present­a­tions for the fun­da­ment­al group of the 2-fold cyc­lic branched cov­er of a link, ex­plain­ing that Liam and I sus­pec­ted that the group was non-left-or­der­able when L was non­split al­tern­at­ing. He replied that oddly enough, he had at­ten­ded a talk at the Tech­nion a few weeks earli­er in which Mi­chael Polyak de­scribed a link in­vari­ant he had dis­covered, which took the form of a group defined via a present­a­tion de­term­ined by a dia­gram for the link. Camer­on worked out that the group was ZZ/3 for the tre­foil and ZZ/5 for the fig­ure eight, from which he sur­mised that in gen­er­al it should be the free product of Z and the fun­da­ment­al group of the 2-fold cyc­lic branched cov­er of the link, a fact that he read­ily veri­fied.1 Dur­ing our week in Ban­ff, we dis­cussed how this present­a­tion could be used to study the non-left-or­der­ab­il­ity of π1(Σ2(L)) in our case of in­terest, and after re­lo­cat­ing to Montreal, Camer­on sug­ges­ted that the dual graph Γ to the black re­gions of a check­er­board sur­face might be use­ful. After a little thought, we saw that any left or­der on π1(Σ2(L)) in­duced a se­mi­ori­ent­a­tion on Γ re­lated to the dia­gram’s cross­ing pat­tern, and once that ob­ser­va­tion was made, the proof fol­lowed simply.

Thus mo­tiv­ated, Camer­on, Liam and I turned to the prob­lem of show­ing that ra­tion­al ho­mo­logy sphere Sol man­i­folds were L-spaces. Camer­on and I suc­ceeded in show­ing that roughly 75% of them were us­ing the tri­ad con­di­tion (me) and Némethi’s graph ho­mo­logy ap­proach (Camer­on), but frus­trat­ingly, we couldn’t see how to get all of them. Liam ap­proached the prob­lem via the re­cently de­veloped bordered Hee­gaard Flo­er the­ory. He as­sim­il­ated enough of the the­ory to be able to es­tab­lish the res­ult for a fam­ily of Sol man­i­folds that we then used as the base case of an in­duct­ive ar­gu­ment us­ing tri­ads, thus com­plet­ing the proof.

We also veri­fied the equi­val­ence of LO,CTF and NLS in oth­er cases, in­clud­ing in­fin­ite fam­il­ies of hy­per­bol­ic man­i­folds. At this point we felt that LONLSCTF might just al­ways hold. This would be rather mys­ter­i­ous since the con­di­tions were of a quite dif­fer­ent nature and we knew no strong heur­ist­ic con­nect­ing them. It was known that CTFNLS, and Oz­sváth and Szabó had asked if the con­verse held. As such, and giv­en the evid­ence we’d com­piled, the equi­val­ence of NLS and CTF seemed reas­on­able to con­jec­ture. The pa­per that Camer­on, Liam and I wrote [11] was con­cerned with the con­nec­tion between LO and NLS, and we con­jec­tured the equi­val­ence of these prop­er­ties in it. An­dras Juhász stated the full con­jec­ture in his sur­vey art­icle [e14]. Though there was a lot of scep­ti­cism ini­tially, it has been a pleas­ure watch­ing the evid­ence mount over the last dec­ade.

5. Recent work

Camer­on, Liam and I pur­sued dif­fer­ent dir­ec­tions after we fin­ished our pa­per [11]. Adam Clay and I stud­ied the L-space con­jec­ture in the con­text of graph man­i­folds [e16], while Camer­on and Tye Lid­man, and later Ying Hu and I, in­vest­ig­ated it in the con­text of cyc­lic branched cov­ers [13], [16]. Liam worked with vari­ous sub­sets of Jonath­an Hansel­man, Jake Rasmussen and Sarah Rasmussen in a series of pa­pers on Hee­gaard Flo­er the­ory, and their pa­per [e19] com­bined with [e16] to prove that the L-space con­jec­ture held for graph man­i­folds.

In May 2016, Camer­on and I got to­geth­er with Michel Boileau at the ICTP Trieste. Michel poin­ted out that the ca­non­ic­al cyc­lic branched cov­ers Σn(L) of a qua­si­pos­it­ive link L bound Stein do­mains, and if one of these do­mains has an in­def­in­ite in­ter­sec­tion pair­ing, the as­so­ci­ated cyc­lic branched cov­er is not an L-space. Giv­en that, he wondered what con­sequences there were for qua­si­pos­it­ive links L for which some Σn(L) is an L-space. Quite a few in fact, two longish pa­pers worth [15], [14], which we worked out in what I like to think of as a Tale of Four In­sti­tutes, since they were com­pleted over the next year dur­ing vis­its to the CRM in Montreal, the New­ton In­sti­tute in Cam­bridge, and fi­nally at the Casa Matemática Oax­aca.

In the first of these two pa­pers, we con­jec­tured that a prime fibred strongly qua­si­pos­it­ive link L has an L-space ca­non­ic­al cyc­lic branched cov­er if and only if it was what we now call an ADE link: i.e., a link ob­tained by plumb­ing pos­it­ive Hopf bands ac­cord­ing to one the ADE Dynkin dia­grams. Re­pla­cing “L-space” by “non-LO” or “non-CTF” yields two sis­ter con­jec­tures, all three of which I began to dis­cuss with Ying and Camer­on dur­ing the year be­fore the pan­dem­ic hit. ( John Bald­win pitched in dur­ing the ini­tial stages.)

Our am­bi­tions evolved once we un­der­stood what the tech­niques we were de­vel­op­ing could do and what star­ted out as one pa­per be­came two, then three, then four [17], [18], [19], [21]. One of my fa­vour­ite res­ults from our first pa­per was that ir­re­du­cible, tor­oid­al, in­teger ho­mo­logy 3-spheres were LO. This was a lovely ap­plic­a­tion of our res­ults on the slope de­tec­tion and glu­ing tech­no­logy ori­gin­at­ing in [e16], though it had noth­ing to do with our ini­tial study. ( Eftekhary had already shown that such man­i­folds were NLS.) The meth­od also de­term­ined an ap­proach for show­ing that they were CTF and led to a proof of this un­der an ad­ded fib­ring as­sump­tion.

One of the motive forces in the evol­u­tion of our ideas was Camer­on’s habit of in­cess­antly ques­tion­ing what we really had done and what we could do. Here is a typ­ic­al ex­ample.

We knew in the fall of 2019 how to show that the ca­non­ic­al cyc­lic branched cov­ers of hy­per­bol­ic fibred strongly qua­si­pos­it­ive knots were LO, but we wanted to ex­tend the res­ult to links, and as our fo­cus was else­where, the pro­ject lan­guished. We were us­ing Fen­ley’s asymp­tot­ic circle con­struc­tion to build Homeo+(S1) rep­res­ent­a­tions of the fun­da­ment­al groups of these branched cov­ers and needed to show that their Euler classes van­ish. Dur­ing a chance dis­cus­sion with Adam Clay a few years later, I real­ised that some ideas from a pa­per of his with Id­rissa Ba should do what we wanted, which turned out to be the case.

One con­sequence of this was that the 2-fold cyc­lic branched cov­er of a fibred hy­per­bol­ic strongly qua­si­pos­it­ive link L was LO. Em­pir­ic­al evid­ence then sug­ges­ted that no mat­ter how the ori­ent­a­tion of L was altered, the ca­non­ic­al cyc­lic branched cov­ers of the res­ult­ing link were LO. Meet­ing after meet­ing after meet­ing Camer­on would ask how we could prove this. At some point, Ying re­spon­ded “Isn’t chan­ging the ori­ent­a­tion of a link com­pon­ent just the flip­ping op­er­a­tion de­scribed by Calegari and Dun­field?”. It wasn’t long be­fore we saw that not only was it so, but there was a much more gen­er­al op­er­a­tion we called re­cal­ib­ra­tion (and in­form­ally stir-fry­ing) which had much more gen­er­al con­sequences. For in­stance, we could show that all of the cyc­lic branched cov­ers of prime, fibred, strongly qua­si­pos­it­ive, hy­per­bol­ic links were LO, even those where the branch­ing in­dex var­ied with the com­pon­ent of L [18]. Ow­ing to Camer­on’s per­sist­ent curi­os­ity, the ADE link con­jec­ture ori­gin­at­ing in [15] now be­came:

Con­jec­ture 5.1 (The ADE link conjecture) If L is a prime, fibred, strongly qua­si­pos­it­ive link that is not an ADE link, then all cyc­lic branched cov­ers of L are NLS, LO, and CTF.

As of this writ­ing, we’ve veri­fied most of the LO case of Con­jec­ture 5.1 [17], [18], [21]. though much of the NLS and CTF cases re­main open.

6. Final thoughts

Look­ing back over my ca­reer, it is clear that happy col­lab­or­a­tions fol­low the Anna Karen­ina prin­ciple: many factors are needed to main­tain them; the ab­sence of only one can spell doom. My col­lab­or­a­tion with Camer­on has been a happy one. The mech­an­ics have evolved of course, from e-mails in the nought­ies, to weekly Fri­day af­ter­noon phone calls 2010–20. Ying sug­ges­ted that we switch to zoom meet­ings after the cov­id lock­down hit in early 2020, and these meet­ings helped me main­tain a health­i­er, more bal­anced per­spect­ive dur­ing the shut­down. They re­main a high­light of my week.

A key pleas­ure in col­lab­or­at­ing is ar­riv­ing some­where that you couldn’t have found your­self. I have ex­per­i­enced this pleas­ure in all my work with Camer­on (and my oth­er col­lab­or­at­ors). But I’d like to stress that our col­lab­or­a­tion re­flects much more than a tech­nic­al com­pat­ib­il­ity. There is a shared need to un­der­stand, a shared am­bi­tion to work on chal­len­ging prob­lems, a shared en­joy­ment in do­ing the math­em­at­ics. And I will al­ways be grate­ful for the ever-present ca­marader­ie, mani­fes­ted as much in fail­ure as in suc­cess.

Works

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[10] S. Boy­er, C. M. Gor­don, and X. Zhang: “Char­ac­ter­ist­ic sub­man­i­fold the­ory and tor­oid­al Dehn filling,” Adv. Math. 230 : 4–​6 (July–August 2012), pp. 1673–​1737. MR 2927352 Zbl 1248.​57004 ArXiv 1104.​3321 article

[11] S. Boy­er, C. M. Gor­don, and L. Wat­son: “On L-spaces and left-or­der­able fun­da­ment­al groups,” Math. Ann. 356 : 4 (2013), pp. 1213–​1245. MR 3072799 Zbl 1279.​57008 ArXiv 1107.​5016 article

[12] S. Boy­er, C. M. Gor­don, and X. Zhang: “Dehn fillings of knot man­i­folds con­tain­ing es­sen­tial once-punc­tured tori,” Trans. Am. Math. Soc. 366 : 1 (2014), pp. 341–​393. MR 3118399 Zbl 1290.​57005 ArXiv 1109.​5151 article

[13] C. Gor­don and T. Lid­man: “Taut fo­li­ations, left-or­der­ab­il­ity, and cyc­lic branched cov­ers,” Acta Math. Vi­et­nam. 39 : 4 (December 2014), pp. 599–​635. A cor­ri­gendum to this art­icle was pub­lished in Acta Math. Vi­et­nam. 42:4 (2014). MR 3292587 Zbl 1310.​57023 ArXiv 1406.​6718 article

[14] M. Boileau, S. Boy­er, and C. M. Gor­don: “On def­in­ite strongly qua­si­pos­it­ive links and L-space branched cov­ers,” Adv. Math. 357 (2019), pp. 106828, 63. MR 4016557 Zbl 1432.​57005 article

[15] M. Boileau, S. Boy­er, and C. M. Gor­don: “Branched cov­ers of quasi-pos­it­ive links and L-spaces,” J. To­pol. 12 : 2 (2019), pp. 536–​576. MR 4072174 Zbl 1422.​57012 article

[16] S. Boy­er and Y. Hu: “Taut fo­li­ations in branched cyc­lic cov­ers and left-or­der­able groups,” Trans. Amer. Math. Soc. 372 : 11 (2019), pp. 7921–​7957. MR 4029686 Zbl 1445.​57017 article

[17] S. Boy­er, C. M. Gor­don, and Y. Hu: Slope de­tec­tion and tor­oid­al 3-man­i­folds. Pre­print, 2021. Zbl 0587.​57008 ArXiv 2106.​14378 techreport

[18] S. Boy­er, C. M. Gor­don, and Y. Hu: Re­cal­ib­rat­ing R-or­der trees and Homeo+(S1)-rep­res­ent­a­tions of link groups. Pre­print, 2023. ArXiv 2306.​10357 techreport

[19] S. Boy­er, C. M. Gor­don, and Y. Hu: JSJ de­com­pos­i­tions of knot ex­ter­i­ors, Dehn sur­gery and the L-space con­jec­ture. Pre­print, 2023. Zbl 1058.​57004 ArXiv 2307.​06815 techreport

[20] S. Boy­er, C. M. Gor­don, and X. Zhang: Dehn fillings of knot man­i­folds con­tain­ing es­sen­tial twice-punc­tured tori. Mem. Amer. Math. Soc. 295. 2024. ArXiv 2004.​04219 book

[21] S. Boy­er, C. M. Gor­don, and Y. Hu: Cyc­lic branched cov­ers of Seifert links and prop­er­ties re­lated to the ADE link con­jec­ture. Pre­print, 2024. ArXiv 2402.​15914 techreport