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 \( S^1 \). 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 \( S^3 \). 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 \( S^1\times D^2 \) to \( M \) via some homeomorphism between \( \partial M \) and \( S^1\times S^1 \), 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 \( \partial M \) and \( S^1\times S^1 \) 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 \( N^{\prime} \) are related by a Dehn surgery is equivalent to saying that there are knots in \( N \) and \( N^{\prime} \) 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 \( \mathrm{SL}_2 \), 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 \( \pi_1 \). 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 \( \partial M \) and \( S^1\times S^1 \); it is enough to know the isotopy class of the homotopically nontrivial simple closed curve in \( \partial M \) that is mapped under the homeomorphism to the curve \( \{\star\}\times S^1 \), where \( \star \) denotes an arbitrary point of \( S^1 \). 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 \( S^1\times S^1 \), which you may think of as a coordinate system. Under the covering map \( \mathbb{R}^2\to S^1\times S^1 \) defined by \( (x,y)\mapsto(e^{2\pi ix},e^{2\pi iy}) \), 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 \( \infty \). 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 \( \mathbb{Q}\cup\{\infty\} \) (“concrete slopes”). One reason why this is often useful is that in the context of Dehn surgery, the torus \( T=\partial M \) often comes equipped with a natural coordinate system. For example, if \( M \) is the exterior of a knot \( K \) in the three-sphere \( S^3 \), it is customary to choose the identification of \( \partial M \) with \( S^1\times S^1 \) in such a way that a curve of concrete slope \( \infty \) 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 \( \alpha \) is a(n abstract) slope on \( \partial M \), I’ll denote by \( M(\alpha) \) the closed manifold obtained from \( M \) by the Dehn filling determined by \( \alpha \).
If \( K \) is a knot in \( S^3 \), then for any \( r\in\mathbb{Q}\cup\{\infty\} \) I’ll set \( K(r)=M(\alpha) \), where \( M \) is the exterior of \( K \) and \( \alpha \) is the abstract slope corresponding to the concrete slope \( r \) in the standard coordinate system. With these conventions, \( K(\infty) \) is the result of the trivial surgery, and is therefore homeomorphic to \( S^3 \).
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 \( S^3 \) 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 \( \partial M \) 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 \( \alpha \) and \( \beta \) are slopes on a 2-torus \( T \), the distance between \( \alpha \) and \( \beta \), sometimes denoted \( \Delta(\alpha,\beta) \), is defined to be the minimum cardinality of \( A\cap B \), where \( A \) and \( B \) range over all nontrivial simple closed curves representing the slopes \( \alpha \) and \( \beta \) respectively. If the elements of \( \mathbb{Q}\cup\{\infty\} \) that give \( \alpha \) and \( \beta \) in some coordinate system for \( T \) are written in lowest terms as \( a/b \) and \( c/d \), then \( \Delta(\alpha,\beta)=|ad-bc| \). (Warning: this “distance” doesn’t satisfy a triangle inequality. However, it is 0 if and only if \( \alpha=\beta \).)
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 \( \alpha \) is a reducing slope, then by definition \( M(\alpha) \) contains a sphere \( S \) which does not bound a ball in \( M(\alpha) \). It is always possible to choose \( S \) in such a way that it meets the solid torus \( \overline{M(\alpha)-M} \) in a disjoint union of disks, and so that \( F:=S\cap M \) is a nonclosed, connected 2-manifold which is properly embedded in \( M \), in the sense that \( F\cap\partial M=\partial F \). In fact we can take \( F \) to be essential in \( M \), in the sense that
- \( F \) is properly embedded,
- \( F \) is \( \pi_1 \)-injective in \( M \), which means that the inclusion homomorphism \( \pi_1(F)\to\pi_1(M) \) is injective,
- \( F \) is not a sphere bounding a ball in \( M \), and
- \( F \) is not boundary-parallel in \( M \), which means that it’s not isotopic (in the nonambient sense) to a subsurface of \( \partial M \). Tautologically, this surface \( F \) is also planar in the sense that it’s homeomorphic to a subsurface of \( S^2 \).
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 \( \partial F \) are all homotopically nontrivial closed curves on \( \partial M \), and they all have the same slope. A slope on \( \partial M \) 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 \( \alpha_1 \) and \( \alpha_2 \) are boundary slopes, realized by essential surfaces \( F_1 \) and \( F_2 \) in \( M \). After an isotopy we can arrange that the components of \( F_1\cap F_2 \) are nontrivial simple closed curves, and arcs that are properly embedded and non-boundary-parallel both in \( F_1 \) and in \( F_2 \). For \( i=1,2 \) we can form a topological quotient space \( \hat{F}_i \) of \( F_i \) by identifying each component of \( \partial F_i \) to a point. Then \( \hat{F}_i \) is a 2-sphere if \( F_i \) is planar, and in general it’s a closed surface of the same genus as \( F_i \). Furthermore, \( \hat{F}_i \) contains a graph \( \Gamma_i \), whose vertices are the images under the quotient map of components of \( \partial F_i \), and whose edges are the images under the quotient map of those components of \( F_1\cap F_2 \) which are arcs. Combinatorial arguments comparing the graphs \( \Gamma_1 \) and \( \Gamma_2 \) give the required bound on \( \Delta(\alpha_1,\alpha_2) \).
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 \( S^3 \), i.e., a knot that is contained in a standard torus in \( S^3 \), then there are many surgeries that give 3-manifolds with cyclic \( \pi_1 \), 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 \( \partial M \) 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 \( \Delta(\alpha,\beta)\le1 \) for any two cyclic slopes \( \alpha \) and \( \beta \) on \( \partial M \). This easily implies that if \( K \) is a knot in \( S^3 \) whose exterior is hyperbolic, and if \( \pi_1(K(r)) \) is cyclic for some given \( r\in\mathbb{Q} \), then \( r \) must be an integer; and furthermore, that if \( \pi_1(K(r)) \) and \( \pi_1(K(s)) \) 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:
A corollary to the theorem is that, under the hypotheses, there are at most three cyclic slopes on \( \partial M \). Again this is sharp, because for the \( (-2,3,7) \)-pretzel knot, the concrete slopes 18, 19 and \( \infty \) 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 \( \alpha \) or \( \beta \) 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 \( \alpha \) and \( \beta \) 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 \( \alpha \) or \( \beta \) was a strict boundary slope actually struck us as a very positive thing. This is because if \( \alpha \) (say) is a boundary slope, there is a simple approach to trying to prove that \( \alpha \) is not a cyclic slope. In fact, if \( F \) is an essential surface in \( M \) realizing the boundary slope \( \alpha \), then the boundary components of \( F \) bound disks in the solid torus \( \overline{M(\alpha)-M} \), and the union of \( F \) with these disks is a closed surface \( F(\alpha)\subset M(\alpha) \). Our idea was that if we were lucky, \( F(\alpha) \) would have positive genus and would be \( \pi_1 \)-injective in \( M(\alpha) \). This would mean that \( \pi_1(M(\alpha)) \) 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 \( F(\alpha) \) is \( \pi_1 \)-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 \( F_0 \) having the minimal number of boundary curves among all essential surfaces realizing the boundary slope \( \alpha \), and to use some fancy logic based on the handle addition lemma to prove that \( F_0(\alpha) \) is still \( \pi_1 \)-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 \( \operatorname{int} M \) instead of \( \partial M \)). 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 \( F_0 \) is a surface having the minimal number of boundary curves among all essential surfaces realizing a given boundary slope \( \alpha \), then \( F_0(\alpha) \) is \( \pi_1 \)-injective in \( M(\alpha) \), so that if \( F_0 \) has positive genus then \( \pi_1(M(\alpha)) \) 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 \( F_0 \) is planar, then, basically, the sphere \( F_0(\alpha) \) divides \( M(\alpha) \) into two non-simply connected pieces. This implies that \( \pi_1(M(\alpha)) \) 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 \( \partial M \) 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 \( \partial M \) is at most 1. The underlying idea is simple. If \( \alpha_1 \) and \( \alpha_2 \) are cyclic slopes, then \( E \) is a closed orientable surface of positive genus in \( M(\alpha_i)\supset M \) for \( i=1,2 \). The cyclicity of \( \pi_1(M(\alpha_i)) \) implies that \( E \) is not \( \pi_1 \)-injective in either \( M(\alpha_i) \). Classical 3-manifold topology then gives a compressing disk for \( E \) in each \( M(\alpha_i) \), i.e., a disk \( D_i\subset M(\alpha_i) \) such that \( D_i\cap E=\partial D_i \), but \( \partial D_i \) bounds no disk in \( E \). One can choose the \( D_i \) so that \( D_1\cap M \) and \( D_2\cap M \) are \( \pi_1 \)-injective planar surfaces in \( M \). These surfaces define an intersection graph, and one can hope to use graph-theoretical methods to show that \( \Delta(\alpha_1,\alpha_2)\le1 \).
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 \( \alpha \) is a cyclic slope which is also a boundary slope, then the distance from \( \alpha \) 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 \( \alpha \) such that \( \pi_1(M(\alpha)) \) 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 \( S^3 \) can never give \( S^3 \). This implies that if two knots in \( S^3 \) 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.
