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Celebratio Mathematica

Cameron McAllan Gordon

Cameron and the Cyclic Surgery Theorem:
a personal account

by Peter B. Shalen

Of all the math­em­at­ic­al pro­jects that I have been in­volved in, the proof of the Cyc­lic Sur­gery The­or­em [2], a joint pro­ject with Marc Cull­er, Camer­on Gor­don and John Luecke, is the one of which I am proudest. I would like to tell a little about the story of this pro­ject, be­cause it says a great deal about Camer­on. (It also provides an in­ter­est­ing ex­ample of what a wacky busi­ness do­ing math­em­at­ics is.)

The the­or­em is re­lated to the the­ory of knots. If N is an ori­ent­able 3-man­i­fold1 which is closed, i.e., com­pact without bound­ary,2 a knot in N is defined to be a sub­man­i­fold K of N that’s homeo­morph­ic to S1. Knot the­or­ists are in­ter­ested in study­ing knots in N up to iso­topy, which is of­ten called knot equi­val­ence.3 A knot is trivi­al if it is equi­val­ent to a geo­met­ric circle in S3. Many use­ful iso­topy in­vari­ants of a knot K are defined to be to­po­lo­gic­al in­vari­ants of the ex­ter­i­or of K in N, which is defined to be the com­ple­ment in N of an open tu­bu­lar neigh­bor­hood of K.

The pro­ced­ure by which the ex­ter­i­or of a knot K in N is ob­tained from N, which is well defined once the iso­topy class of K is spe­cified, is of­ten called Dehn drilling. This pro­ced­ure has a vague in­verse, called Dehn filling: one be­gins with a com­pact, ori­ent­able 3-man­i­fold M whose bound­ary is a 2-tor­us,4 and glues on the stand­ard sol­id tor­us S1×D2 to M via some homeo­morph­ism between M and S1×S1, the bound­ary of the sol­id tor­us.

What pre­vents Dehn filling, as I have defined it, from be­ing a pre­cise in­verse of Dehn drilling — and makes it an in­ter­est­ing pro­ced­ure! — is that it is not well defined for a giv­en man­i­fold M. By choos­ing dif­fer­ent homeo­morph­isms between M and S1×S1 one can get dif­fer­ent closed man­i­folds. Two closed 3-man­i­folds that are ob­tained by fillings of the same man­i­fold M are said to be re­lated by a Dehn sur­gery. So a Dehn sur­gery con­sists of a Dehn drilling fol­lowed by a Dehn filling: you re­move a sol­id tor­us and put it back via a pos­sibly dif­fer­ent homeo­morph­ism between bound­ar­ies. (Camer­on is fond of point­ing out that if the sur­gery con­sists of a drilling and a filling, it must be a dent­al sur­gery.) If you put back the sol­id tor­us via the same homeo­morph­ism, you of course re­cov­er the ori­gin­al man­i­fold. This is called the trivi­al sur­gery.

To say that two closed man­i­folds N and N are re­lated by a Dehn sur­gery is equi­val­ent to say­ing that there are knots in N and N whose ex­ter­i­ors are homeo­morph­ic, so Dehn sur­gery is a very nat­ur­al top­ic in knot the­ory. It also turns out to have im­port­ant con­nec­tions with oth­er top­ics in low-di­men­sion­al to­po­logy, in­clud­ing 4-man­i­fold the­ory and the the­ory of hy­per­bol­ic 3-man­i­folds.

The clas­sic­al prob­lems in the sub­ject con­cern the to­tal­ity of ways in which man­i­folds with pre­scribed prop­er­ties can be ob­tained by Dehn sur­gery on a giv­en knot, or Dehn filling of a giv­en 3-man­i­fold. Marc and I star­ted think­ing about Dehn filling in the early 1980s, when we real­ized that the meth­ods of our pa­per [e3] could be used to study the set of Dehn fillings or sur­ger­ies that can give a man­i­fold with a cyc­lic fun­da­ment­al group. I won’t say much about these meth­ods here, ex­cept that they in­volve sur­pris­ing in­ter­ac­tions among hy­per­bol­ic geo­metry, al­geb­ra­ic geo­metry, rep­res­ent­a­tion the­ory, the Bruhat–Tits build­ing for SL2, and clas­sic­al cut-and-paste 3-man­i­fold to­po­logy.

As of­ten hap­pens with new work, it took us quite a while to ar­rive at a good for­mu­la­tion of what we could prove about Dehn sur­gery and cyc­lic π1. Some form­al­ism that we learned from hear­ing Camer­on give talks on his work [1] with R. Lith­er­land, and on some re­lated work which is ap­par­ently un­pub­lished, was very help­ful in this con­nec­tion. (As this is a per­son­al ac­count, I will be em­phas­iz­ing where I learned about vari­ous ideas, rather than try­ing to track down their ori­gins.5 )

It turns out that to de­term­ine the to­po­lo­gic­al type of the man­i­fold ob­tained from a giv­en Dehn filling one does not need to know the en­tire homeo­morph­ism between M and S1×S1; it is enough to know the iso­topy class of the ho­mo­top­ic­ally non­trivi­al simple closed curve in M that is mapped un­der the homeo­morph­ism to the curve {}×S1, where de­notes an ar­bit­rary point of S1. In [1], an iso­topy class of non­trivi­al simple closed curves on a tor­us T is re­ferred to as a slope. The term makes sense if you fix in ad­vance a par­tic­u­lar homeo­morph­ic iden­ti­fic­a­tion of T with the stand­ard tor­us S1×S1, which you may think of as a co­ordin­ate sys­tem. Un­der the cov­er­ing map R2S1×S1 defined by (x,y)(e2πix,e2πiy), any iso­topy class has a unique rep­res­ent­at­ive which is the im­age of a line through the ori­gin whose slope is either a ra­tion­al num­ber or . This gives a biject­ive cor­res­pond­ence (de­pend­ing on the choice of co­ordin­ate sys­tem) between iso­topy classes of non­trivi­al simple closed curves on T (which may be thought of as “ab­stract slopes”) and ele­ments of Q{} (“con­crete slopes”). One reas­on why this is of­ten use­ful is that in the con­text of Dehn sur­gery, the tor­us T=M of­ten comes equipped with a nat­ur­al co­ordin­ate sys­tem. For ex­ample, if M is the ex­ter­i­or of a knot K in the three-sphere S3, it is cus­tom­ary to choose the iden­ti­fic­a­tion of M with S1×S1 in such a way that a curve of con­crete slope is the bound­ary of a disk in the tu­bu­lar neigh­bor­hood of K, and a curve of con­crete slope 0 is the bound­ary of a com­pact 2-di­men­sion­al sub­man­i­fold of the man­i­fold M.

If M is a com­pact, ori­ent­able 3-man­i­fold whose bound­ary is a tor­us, and α is a(n ab­stract) slope on M, I’ll de­note by M(α) the closed man­i­fold ob­tained from M by the Dehn filling de­term­ined by α.

If K is a knot in S3, then for any rQ{} I’ll set K(r)=M(α), where M is the ex­ter­i­or of K and α is the ab­stract slope cor­res­pond­ing to the con­crete slope r in the stand­ard co­ordin­ate sys­tem. With these con­ven­tions, K() is the res­ult of the trivi­al sur­gery, and is there­fore homeo­morph­ic to S3.

In 3-man­i­fold the­ory, a ba­sic ques­tion about a man­i­fold M turns out to be wheth­er it is re­du­cible in the sense that there is a (smooth) 2-sphere in M which is not the bound­ary of a 3-ball in M. The ex­ter­i­or of any knot in S3 is ir­re­du­cible. One of the main res­ults of [1] is that if M is an ir­re­du­cible, com­pact, ori­ent­able 3-man­i­fold bounded by a tor­us, then there are at most six slopes on M which are re­du­cing slopes in the sense that the fillings which they define give re­du­cible man­i­folds. (This res­ult was later greatly im­proved in Camer­on’s joint pa­per [5] with John Luecke, but in gen­er­al I won’t be talk­ing about up­dat­ing the res­ults I men­tion.)

The bound of 6 fol­lows from a res­ult about the so-called dis­tance between re­du­cing slopes. Geo­met­ric­ally, if α and β are slopes on a 2-tor­us T, the dis­tance between α and β, some­times de­noted Δ(α,β), is defined to be the min­im­um car­din­al­ity of AB, where A and B range over all non­trivi­al simple closed curves rep­res­ent­ing the slopes α and β re­spect­ively. If the ele­ments of Q{} that give α and β in some co­ordin­ate sys­tem for T are writ­ten in low­est terms as a/b and c/d, then Δ(α,β)=|adbc|. (Warn­ing: this “dis­tance” doesn’t sat­is­fy a tri­angle in­equal­ity. However, it is 0 if and only if α=β.)

In [1] it is shown that if M is ir­re­du­cible, the dis­tance between two re­du­cing slopes is at most 4. Now it’s an ele­ment­ary fact that if S is a set of slopes on a tor­us, a fi­nite up­per bound on the pair­wise dis­tances between slopes in S gives a fi­nite up­per bound on the car­din­al­ity of S. In fact, this was later put in a very el­eg­ant form by Ian Agol, who used a neat al­geb­ra­ic ar­gu­ment to show that if the pair­wise dis­tances between slopes in S are strictly bounded above by a prime p, then S has at most p+1 ele­ments. Tak­ing p=5, it fol­lows that the bound of 4 on dis­tances between re­du­cing slopes gives the bound of 6 on the num­ber of re­du­cing slopes.

Gor­don and Lith­er­land’s bound of 4 for the dis­tance between re­du­cing slopes is in turn a con­sequence of a res­ult about so-called bound­ary slopes. If α is a re­du­cing slope, then by defin­i­tion M(α) con­tains a sphere S which does not bound a ball in M(α). It is al­ways pos­sible to choose S in such a way that it meets the sol­id tor­us M(α)M in a dis­joint uni­on of disks, and so that F:=SM is a non­closed, con­nec­ted 2-man­i­fold which is prop­erly em­bed­ded in M, in the sense that FM=F. In fact we can take F to be es­sen­tial in M, in the sense that

  1. F is prop­erly em­bed­ded,
  2. F is π1-in­ject­ive in M, which means that the in­clu­sion ho­mo­morph­ism π1(F)π1(M) is in­ject­ive,
  3. F is not a sphere bound­ing a ball in M, and
  4. F is not bound­ary-par­al­lel in M, which means that it’s not iso­top­ic (in the no­nam­bi­ent sense) to a sub­sur­face of M. Tau­to­lo­gic­ally, this sur­face F is also planar in the sense that it’s homeo­morph­ic to a sub­sur­face of S2.

If F is any non­closed, es­sen­tial sur­face in M (where M still de­notes an ir­re­du­cible, com­pact, ori­ent­able 3-man­i­fold whose bound­ary is a tor­us), then the com­pon­ents of F are all ho­mo­top­ic­ally non­trivi­al closed curves on M, and they all have the same slope. A slope on M that is de­term­ined in this way by some non­closed, es­sen­tial sur­face F in M is called a bound­ary slope, and I’ll say that the sur­face F real­izes the bound­ary slope in ques­tion. A bound­ary slope that is real­ized by a planar sur­face may be called a planar bound­ary slope. So every re­du­cing slope is a planar bound­ary slope. The “real” res­ult that is proved in [1] and un­der­lies the ones I have stated is that the dis­tance between any two planar bound­ary slopes is at most 4. Camer­on also gave a bound on the dis­tance between the bound­ary slopes real­ized by two es­sen­tial sur­faces in M in terms of the gen­era of the sur­faces; this was the sub­stance of one of the talks that I heard him give, al­though I don’t be­lieve he ever pub­lished it.

The tech­niques that Camer­on used for bound­ing the dis­tances between two bound­ary slopes, both in [1] and in his un­pub­lished work, are graph-the­or­et­ic­al. Let’s say α1 and α2 are bound­ary slopes, real­ized by es­sen­tial sur­faces F1 and F2 in M. After an iso­topy we can ar­range that the com­pon­ents of F1F2 are non­trivi­al simple closed curves, and arcs that are prop­erly em­bed­ded and non-bound­ary-par­al­lel both in F1 and in F2. For i=1,2 we can form a to­po­lo­gic­al quo­tient space F^i of Fi by identi­fy­ing each com­pon­ent of Fi to a point. Then F^i is a 2-sphere if Fi is planar, and in gen­er­al it’s a closed sur­face of the same genus as Fi. Fur­ther­more, F^i con­tains a graph Γi, whose ver­tices are the im­ages un­der the quo­tient map of com­pon­ents of Fi, and whose edges are the im­ages un­der the quo­tient map of those com­pon­ents of F1F2 which are arcs. Com­bin­at­or­i­al ar­gu­ments com­par­ing the graphs Γ1 and Γ2 give the re­quired bound on Δ(α1,α2).

I have men­tioned that Marc and I were study­ing the fillings of a giv­en man­i­fold, or sur­ger­ies on a giv­en knot, that give man­i­folds with cyc­lic fun­da­ment­al group. It had been known for a long time that if K is a tor­us knot in S3, i.e., a knot that is con­tained in a stand­ard tor­us in S3, then there are many sur­ger­ies that give 3-man­i­folds with cyc­lic π1, and the slopes cor­res­pond­ing to such sur­ger­ies are clas­si­fied. One there­fore fo­cuses on knots that are not (equi­val­ent to) tor­us knots. From the view­point of 3-man­i­fold the­ory, tor­us knots are the knots whose ex­ter­i­ors are so-called Seifert fibered spaces. These are defined to be 3-man­i­folds that map to sur­faces in such a way that all fibers are 1-spheres, and the map sat­is­fies a con­di­tion some­what weak­er than that of be­ing a loc­ally trivi­al fibra­tion. So we wanted to con­sider a non-Seifert-fibered, com­pact, ir­re­du­cible, ori­ent­able 3-man­i­fold M whose bound­ary is a tor­us, and look at the slopes on M that are “cyc­lic” in the sense that the cor­res­pond­ing fillings give man­i­folds with cyc­lic fun­da­ment­al group. It fol­lows from Thur­ston’s geo­met­riz­a­tion the­or­em that such a man­i­fold M either is hy­per­bol­ic in the sense that its in­teri­or6 ad­mits a com­plete Rieman­ni­an met­ric with con­stant curvature 1 and fi­nite volume, or con­tains a tor­us which is es­sen­tial (in the sense that I defined above). We fo­cused on the hy­per­bol­ic case, be­cause that is the case to which the meth­ods of [e3] ap­ply most dir­ectly, and be­cause we knew that ques­tions about man­i­folds that con­tain es­sen­tial tori can of­ten be re­duced to ques­tions about man­i­folds that do not.

Thanks to the form­al­ism that Marc and I had learned from Camer­on, we were able to or­gan­ize our thoughts about the prob­lem in the form of a con­jec­ture: that in the case where M is hy­per­bol­ic, we have Δ(α,β)1 for any two cyc­lic slopes α and β on M. This eas­ily im­plies that if K is a knot in S3 whose ex­ter­i­or is hy­per­bol­ic, and if π1(K(r)) is cyc­lic for some giv­en rQ, then r must be an in­teger; and fur­ther­more, that if π1(K(r)) and π1(K(s)) are cyc­lic for some giv­en dis­tinct in­tegers r and s, then r and s must be con­sec­ut­ive in­tegers. We knew this res­ult would be sharp, be­cause Fin­tushel and Stern had shown that for a cer­tain hy­per­bol­ic knot K, called the (2,3,7)-pret­zel knot, both K(18) and K(19) are lens spaces — 3-man­i­folds of a very clas­sic­al kind that have fi­nite cyc­lic fun­da­ment­al groups.

Our con­jec­ture was ac­tu­ally very close to what would later be­come the Cyc­lic Sur­gery The­or­em, which I will state now so as not to keep you in sus­pense any longer:

Cyc­lic Sur­gery The­or­em: If M is a com­pact, ori­ent­able 3-man­i­fold which is ir­re­du­cible and not Seifert-fibered, and M is a tor­us, then the dis­tance between any two cyc­lic slopes on M is at most 1.

A co­rol­lary to the the­or­em is that, un­der the hy­po­theses, there are at most three cyc­lic slopes on M. Again this is sharp, be­cause for the (2,3,7)-pret­zel knot, the con­crete slopes 18, 19 and are all cyc­lic.

Our plan for ap­ply­ing the meth­ods of [e3] to the proof of our con­jec­ture was in­geni­ous, if I say so my­self, but we knew that it could not give the full con­jec­ture. In par­tic­u­lar we knew that it could not work when either α or β is a “strict bound­ary slope”: this means a bound­ary slope in the sense that I defined above, giv­en by an es­sen­tial sur­face F sat­is­fy­ing a mild nonde­gen­er­acy con­di­tion.7 It is not at all ob­vi­ous why bound­ary slopes (strict or oth­er­wise) should come up here. I’ll just say that the reas­on has to do with the the­ory de­veloped in [e3], and has noth­ing to do with the reas­on why bound­ary slopes come up (for ex­ample) in [1].

Our ori­gin­al par­tial proof of our con­jec­ture in­volved not only the hy­po­thes­is that α and β are not strict bound­ary slopes, but also the hy­po­thes­is that M con­tains no closed es­sen­tial sur­face. However, we even­tu­ally re­fined our meth­ods so as to dis­pense with the lat­ter hy­po­thes­is.

The fact that the case in which our meth­od seemed use­less was pre­cisely the case in which α or β was a strict bound­ary slope ac­tu­ally struck us as a very pos­it­ive thing. This is be­cause if α (say) is a bound­ary slope, there is a simple ap­proach to try­ing to prove that α is not a cyc­lic slope. In fact, if F is an es­sen­tial sur­face in M real­iz­ing the bound­ary slope α, then the bound­ary com­pon­ents of F bound disks in the sol­id tor­us M(α)M, and the uni­on of F with these disks is a closed sur­face F(α)M(α). Our idea was that if we were lucky, F(α) would have pos­it­ive genus and would be π1-in­ject­ive in M(α). This would mean that π1(M(α)) had a sub­group iso­morph­ic to a pos­it­ive-genus sur­face group, so it couldn’t be cyc­lic.

We were en­cour­aged by a par­tial res­ult. If we as­sume that M con­tains no closed es­sen­tial sur­face, and that F sep­ar­ates M and has ex­actly two bound­ary com­pon­ents, then one can use a cool res­ult called the handle ad­di­tion lemma to prove that F(α) is π1-in­ject­ive. The handle ad­di­tion lemma was first proved by Bus Jaco [e4]. The most el­eg­ant proof is the one later giv­en in a joint pa­per of Camer­on’s with An­drew Cas­son [3].

For a while, Marc and I hoped to ex­tend the ar­gu­ment so as to avoid the as­sump­tion that F has just two bound­ary com­pon­ents. The idea was sup­posed to be to take a sur­face F0 hav­ing the min­im­al num­ber of bound­ary curves among all es­sen­tial sur­faces real­iz­ing the bound­ary slope α, and to use some fancy lo­gic based on the handle ad­di­tion lemma to prove that F0(α) is still π1-in­ject­ive. This would have brought us close to prov­ing our con­jec­ture in the case where M con­tains no closed es­sen­tial sur­face. (Pos­it­ive genus would be a bridge to cross when we came to it.) After think­ing about this for a long time we de­cided it was hope­less.

I spent the cal­en­dar year 1984 in France, vis­it­ing Nantes in the spring semester and Or­say in the fall semester. Dur­ing the sum­mer I crossed the chan­nel to spend a month vis­it­ing the Uni­versity of War­wick, and to at­tend a week-long con­fer­ence at the Uni­versity of Durham.

Dur­ing my vis­it to Nantes, I thought hard about my pro­ject with Marc, and came up with a new ap­proach to the proof of Marc’s and my con­jec­ture in the case where M con­tains no closed es­sen­tial sur­face. I no­ticed that the par­tic­u­lar strict bound­ary slopes which ap­peared as ex­cep­tions to our ar­gu­ment for this case are of a spe­cial kind, in that they were real­ized by es­sen­tial sur­faces with a cer­tain tech­nic­al prop­erty, and I had what looked like an ar­gu­ment for giv­ing a bound on the dis­tance from a strict bound­ary slope of this spe­cial kind to a cyc­lic slope. This in it­self would not prove the con­jec­ture, but it would give a bound on the dis­tance between two cyc­lic slopes when M con­tains no closed es­sen­tial sur­face, and the hope was that the ar­gu­ment might be re­fined to give a bound of 1.

The ar­gu­ment, which was rather in­volved, used some fancy 3-man­i­fold the­ory, in­clud­ing the the­ory of the char­ac­ter­ist­ic sub­man­i­fold [e1], [e2]. It also used a vari­ant on the meth­ods that I had learned about from Camer­on’s talks. Where­as those meth­ods give bounds on the dis­tance between two bound­ary slopes, my ar­gu­ment in­volved bound­ing the dis­tance between a cer­tain kind of strict bound­ary slope and a cer­tain kind of “sin­gu­lar bound­ary slope” which is defined by a (planar) sur­face that is im­mersed rather than be­ing em­bed­ded (and has its bound­ary in intM in­stead of M). This led to the study of com­bin­at­or­i­al con­fig­ur­a­tions that were re­mark­ably sim­il­ar to the ones that Camer­on had used, and some ele­ment­ary graph the­ory sug­ges­ted by Camer­on’s ar­gu­ments gave me my bounds on dis­tance.

I figured the best course was to tell Camer­on about my ideas. I knew he was go­ing to be at War­wick at the same time as I was, which seemed like a stroke of luck. I had not worked with him be­fore, but I knew him well enough to know that he is a hell of a good guy, and I was sure he would make a great col­lab­or­at­or if I suc­ceeded in in­ter­est­ing him in the pro­ject.

I was not dis­ap­poin­ted. When I told Camer­on about my ideas, he re­acted with a de­gree of en­thu­si­asm that I have sel­dom seen in a math­em­atician. (I’m not sure I have ever seen it in a math­em­atician who was not Camer­on.) We star­ted talk­ing about de­tails the same day, and he showed me how to or­gan­ize my ideas, which as usu­al had been a bit chaot­ic. With­in a few hours it be­came clear that the bound on the dis­tance giv­en by the ar­gu­ment, which I had thought was 10 or 11, was in fact equal to 5.

Camer­on and I worked very in­tensely on the pro­ject over the next sev­er­al weeks, and we con­tin­ued to work on it, with Marc, dur­ing the Durham con­fer­ence. Much of the pro­gress came from Camer­on’s side, as he showed us how to im­prove my fairly naïve graph-the­or­et­ic­al ar­gu­ments by sup­ple­ment­ing them with far subtler ones. This in­volved the use of so-called Schar­le­mann cycles, named after Marty Schar­le­mann who, I be­lieve, had first used them in his own proof of the handle ad­di­tion lemma [e5]. Dur­ing the time that we were work­ing to­geth­er, the bound on the dis­tance os­cil­lated a good deal, but by the end we had a pretty sol­id ar­gu­ment giv­ing a bound of 4, and prom­ising ideas for im­prov­ing the bound to 3.

I really can­not ima­gine a bet­ter col­lab­or­at­or than Camer­on. People who are fa­mil­i­ar with his pa­pers don’t have to be told about his cre­ativ­ity, his tech­nic­al skill, and his know­ledge. What may be less ob­vi­ous from the out­side is the tre­mend­ous pleas­ure that he takes in do­ing re­search. His eyes light up every time a new idea ap­pears, wheth­er it is gen­er­ated by him­self or by someone else. He has a nat­ur­al gift for team­work and for show­ing ap­pre­ci­ation. Work­ing with him was, quite simply, a joy.

Dur­ing our stay at War­wick, both Camer­on and I were both ac­com­pan­ied by our fam­il­ies, and we hap­pen to have been housed next door to each oth­er. Our chil­dren spent much of the time play­ing to­geth­er. We of­ten all had din­ner to­geth­er, and we all went on one or two sight­see­ing trips to­geth­er on the week­ends. It was a happy time.

When I re­turned to France after the Durham con­fer­ence for my semester at Or­say, I con­tin­ued work­ing very hard with the ideas that Camer­on, Marc and I had de­veloped. I suc­ceeded, more or less, in im­prov­ing the bound on the dis­tance from 4 to 3. You prob­ably think you know where this is go­ing, and you are prob­ably wrong.

Late in the fall of 1984, I got a let­ter from Camer­on about some re­mark­able work that he had done with John Luecke. First of all, for the case where M con­tains no closed es­sen­tial sur­face, they had suc­ceeded in us­ing the handle ad­di­tion lemma to show, ba­sic­ally,8 that if F0 is a sur­face hav­ing the min­im­al num­ber of bound­ary curves among all es­sen­tial sur­faces real­iz­ing a giv­en bound­ary slope α, then F0(α) is π1-in­ject­ive in M(α), so that if F0 has pos­it­ive genus then π1(M(α)) can’t be cyc­lic. If that sounds fa­mil­i­ar, it’s be­cause it’s ex­actly what Marc and I had tried to do, but had de­cided was hope­less. The de­tails of the lo­gic in Camer­on and John’s ap­plic­a­tion of the handle ad­di­tion lemma were subtler than any­thing that we had been able to ima­gine.

Camer­on and John also showed that if F0 is planar, then, ba­sic­ally, the sphere F0(α) di­vides M(α) in­to two non-simply con­nec­ted pieces. This im­plies that π1(M(α)) is a non­trivi­al free product, hence non­cyc­lic.

These res­ults, com­bined with my earli­er work with Marc, ba­sic­ally im­plied that if M is not Seifert-fibered and con­tains no closed es­sen­tial sur­face, then the dis­tance between any two cyc­lic slopes on M is at most 1. They also made the en­tire Nantes–War­wick–Durham–Or­say pro­ject ob­sol­ete. In fact, they im­plied that es­sen­tial sur­faces with the prop­er­ties that we were us­ing for this pro­ject do not ex­ist!

However, this was only the be­gin­ning of what Camer­on and John had done. For the case where M does con­tain a closed es­sen­tial sur­face E, they con­ceived a plan for show­ing that the dis­tance between any two cyc­lic slopes on M is at most 1. The un­der­ly­ing idea is simple. If α1 and α2 are cyc­lic slopes, then E is a closed ori­ent­able sur­face of pos­it­ive genus in M(αi)M for i=1,2. The cyc­li­city of π1(M(αi)) im­plies that E is not π1-in­ject­ive in either M(αi). Clas­sic­al 3-man­i­fold to­po­logy then gives a com­press­ing disk for E in each M(αi), i.e., a disk DiM(αi) such that DiE=Di, but Di bounds no disk in E. One can choose the Di so that D1M and D2M are π1-in­ject­ive planar sur­faces in M. These sur­faces define an in­ter­sec­tion graph, and one can hope to use graph-the­or­et­ic­al meth­ods to show that Δ(α1,α2)1.

This idea doesn’t al­ways work for an ar­bit­rary closed es­sen­tial sur­face in M, but by com­bin­ing it with a strong form of their res­ult based on the handle ad­di­tion lemma, Camer­on and John were ba­sic­ally able to show that the idea works of­ten enough to al­low one to prove that if α is a cyc­lic slope which is also a bound­ary slope, then the dis­tance from α to any oth­er cyc­lic slope is at most 1.

The graph-the­or­et­ic­al ar­gu­ments that Camer­on and John used to do this were based on Schar­le­mann cycles, but these ar­gu­ments in­volved such dazzling in­genu­ity that they made all pre­vi­ous graph-the­or­et­ic­al ar­gu­ments in the sub­ject look like kids’ stuff.

This all made it seem very likely that a proof of the Cyc­lic Sur­gery The­or­em was with­in reach. The proof in the case where M was neither hy­per­bol­ic nor Seifert fibered turned out to come out of Camer­on and John’s work. If M is hy­per­bol­ic and neither of the giv­en slopes is a strict bound­ary slope, it looked as if the proof should fol­low from my work with Marc, but we needed to re­move the hy­po­thes­is that M con­tained no closed es­sen­tial sur­faces. For the case where one of the giv­en slopes is a strict bound­ary slope, it looked as if the proof should fol­low from Camer­on and John’s work, but as I’ve hin­ted, there were a few loose ends in that part as well. These is­sues were all worked out at MSRI in the spring of 1985, when Camer­on, Marc and I were vis­it­ing for the semester and John made some short­er vis­its.

In the months and years fol­low­ing the proof of the Cyc­lic Sur­gery The­or­em, a good many in­ter­est­ing con­sequences were proved by a num­ber of re­search­ers. I per­son­ally am fond of the ap­plic­a­tions of the res­ult, by De Witt Sum­ners and his col­lab­or­at­ors, to the study of the struc­ture of re­com­bin­ant DNA — not be­cause I know what re­com­bin­ant DNA is, but be­cause it’s the only case that I know of in which my work has been ap­plied out­side of pure math­em­at­ics. For a nice sur­vey on these ap­plic­a­tions, see De Witt’s In­tel­li­gen­cer art­icle [e6].

I also have a par­tic­u­lar fond­ness for Steve Boy­er and Xin­gru Zhang’s pa­per [e7]. The main res­ult of this pa­per is not an ap­plic­a­tion of [2]; rather, it builds on the meth­ods of [2] to give ana­log­ous res­ults, some of them sharp, for slopes α such that π1(M(α)) is fi­nite (but not ne­ces­sar­ily cyc­lic). The au­thors needed to de­vel­op some very new, deep, and sur­pris­ing ideas in or­der to ad­apt the meth­ods of Chapter I of [2] to this con­text. On the oth­er hand, the ar­gu­ments of Chapter II of [2] — the part writ­ten by Camer­on and John — were power­ful enough that Steve and Xin­gru were able to quote the res­ults of that chapter without change.

In our pa­per [2], it was ex­tremely sat­is­fy­ing to see how the bril­liant ar­gu­ments de­veloped by Camer­on and John, to which the second chapter is de­voted, com­ple­ment the rad­ic­ally dif­fer­ent ar­gu­ments de­veloped by Marc and me, which oc­cupy the first chapter. This is why I am so proud of my role in the pa­per. At the same time, I found it hard to let go of the ideas that I had de­veloped, partly in col­lab­or­a­tion with Camer­on and Marc, that were ec­lipsed by the second chapter. Of course this was neither the first nor the last time that I spent a year de­vel­op­ing ideas that would be­come ir­rel­ev­ant to the prob­lems that they were de­signed to solve; I am a math­em­atician, after all. But in this case I couldn’t help feel­ing that the ideas should be good for something. I talked about this to Camer­on dur­ing a vis­it to Aus­tin a few years after [2] was pub­lished. Once again his sig­na­ture en­thu­si­asm was hugely help­ful. We dis­cussed the situ­ation and came to the con­clu­sion that the ideas that I had de­veloped were likely to ap­ply to oth­er prob­lems in Dehn sur­gery. I came home in­spired, thought the idea over a num­ber of times, and fi­nally dis­cussed it with Steve Boy­er when he was vis­it­ing the Uni­versity of Illinois at Chica­go. This led to a col­lab­or­at­ive ef­fort by Steve, Marc, Xin­gru and me that pro­duced the pa­per [e8]. Al­though Camer­on was not dir­ectly in­volved in this col­lab­or­a­tion, it would nev­er have happened without him.

In the mean time, the amaz­ing graph-the­or­et­ic­al tech­niques de­veloped by Camer­on and John and used in [2] led to a tre­mend­ous blos­som­ing of Dehn sur­gery as a re­search area. Of the huge num­ber of pa­pers, by many people, that ex­ploited these tech­niques, the most fam­ous is surely Camer­on and John’s own pa­per [4]. The main res­ult of this pa­per (which over­laps with the Cyc­lic Sur­gery The­or­em, al­though neither im­plies the oth­er) as­serts that a non­trivi­al Dehn sur­gery on a non­trivi­al knot in S3 can nev­er give S3. This im­plies that if two knots in S3 have homeo­morph­ic ex­ter­i­ors, they are equi­val­ent. This was one of the most clas­sic­al con­jec­tures in knot the­ory.

One thing that was re­volu­tion­ary about Camer­on and John’s pa­per [4] was that the graph-the­or­et­ic­al tech­niques were com­bined with Cerf the­ory. But that’s a story for a dif­fer­ent day.

Peter Shalen is Pro­fess­or Emer­it­us in the De­part­ment of Math­em­at­ics, Stat­ist­ics, and Com­puter Sci­ence at the Uni­versity of Illinois at Chica­go. He is a to­po­lo­gist by train­ing, and has re­cently been study­ing hy­per­bol­ic 3-man­i­folds.

Works

[1] C. M. Gor­don and R. A. Lith­er­land: “In­com­press­ible planar sur­faces in 3-man­i­folds,” To­po­logy Ap­pl. 18 : 2–​3 (December 1984), pp. 121–​144. MR 769286 Zbl 0554.​57010 article

[2] M. Cull­er, C. M. Gor­don, J. Luecke, and P. B. Shalen: “Dehn sur­gery on knots,” Ann. of Math. (2) 125 : 2 (1987), pp. 237–​300. MR 881270 Zbl 0633.​57006 article

[3] A. J. Cas­son and C. M. Gor­don: “Re­du­cing Hee­gaard split­tings,” To­po­logy Ap­pl. 27 : 3 (December 1987), pp. 275–​283. MR 918537 Zbl 0632.​57010 article

[4] C. M. Gor­don and J. Luecke: “Knots are de­term­ined by their com­ple­ments,” Bull. Am. Math. Soc. (N.S.) 20 : 1 (January 1989), pp. 83–​87. A much longer ver­sion of this art­icle was pub­lished in J. Am. Math. Soc. 2:2 (1989). MR 972070 Zbl 0672.​57009 article

[5] C. M. Gor­don and J. Luecke: “Re­du­cible man­i­folds and Dehn sur­gery,” To­po­logy 35 : 2 (April 1996), pp. 385–​409. MR 1380506 Zbl 0859.​57016 article