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 is an
orientable 3-manifold1
which is closed, i.e., compact without boundary,2
a knot in is defined to be a submanifold of that’s
homeomorphic to . Knot theorists are interested in studying knots
in up to isotopy, which is often called knot equivalence.3
A knot is trivial if it is equivalent to a geometric circle in
. Many useful isotopy invariants of a knot are defined to be
topological invariants of the exterior of in , which is
defined to be the complement in of an open tubular neighborhood
of .
The procedure by which the exterior of a knot in is obtained
from , which is well defined once the isotopy class of is
specified, is often called Dehn drilling. This procedure has a
vague inverse, called Dehn filling: one begins with a compact,
orientable 3-manifold whose boundary is a 2-torus,4
and glues on the standard solid torus to via some homeomorphism between and , 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 . By choosing
different homeomorphisms between and one
can get different closed manifolds. Two closed 3-manifolds that are
obtained by fillings of the same manifold 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 and are related by a Dehn
surgery is equivalent to saying that there are knots in and
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 , 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 . 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 and ; it is
enough to know the isotopy class of the homotopically nontrivial
simple closed curve in that is mapped under the
homeomorphism to the curve , where
denotes an arbitrary point of . In
[1],
an isotopy
class of nontrivial simple closed curves on a torus is referred
to as a slope. The term makes sense if you fix in advance a
particular homeomorphic identification of with the standard torus
, which you may think of as a coordinate system. Under
the covering map defined by
, 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
(which may be thought of as “abstract slopes”) and elements of
(“concrete slopes”). One reason why this
is often useful is that in the context of Dehn surgery, the torus
often comes equipped with a natural coordinate system.
For example, if is the exterior of a knot in the three-sphere
, it is customary to choose the identification of
with in such a way that a curve of concrete slope
is the boundary of a disk in the tubular neighborhood of ,
and a curve of concrete slope 0 is the boundary of a compact
2-dimensional submanifold of the manifold .
If is a compact, orientable 3-manifold whose boundary is a
torus, and is a(n abstract) slope on , I’ll
denote by the closed manifold obtained from by the
Dehn filling determined by .
If is a knot in , then for any
I’ll set , where is the exterior of and
is the abstract slope corresponding to the concrete slope
in the standard coordinate system. With these conventions,
is the result of the trivial surgery, and is therefore homeomorphic to
.
In 3-manifold theory, a basic question about a manifold turns
out to be whether it is reducible in the sense that there is a
(smooth) 2-sphere in which is not the boundary of a 3-ball in
. The exterior of any knot in is irreducible. One of the main
results of
[1]
is that if is an irreducible, compact,
orientable 3-manifold bounded by a torus, then there are at most six
slopes on 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 and are
slopes on a 2-torus , the distance between and
, sometimes denoted , is defined to be
the minimum cardinality of , where and range over all
nontrivial simple closed curves representing the slopes and
respectively. If the elements of
that give and in some coordinate system for are
written in lowest terms as and , then
. (Warning: this “distance” doesn’t
satisfy a triangle inequality. However, it is 0 if and only if
.)
In
[1]
it is shown that if is irreducible, the distance
between two reducing slopes is at most 4. Now it’s an elementary
fact that if is a set of slopes on a torus, a finite upper bound
on the pairwise distances between slopes in gives a finite upper
bound on the cardinality of . 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 are strictly
bounded above by a prime , then has at most elements.
Taking , 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 is a reducing slope, then by definition
contains a sphere which does not bound a ball in
. It is always possible to choose in such a way that it
meets the solid torus in a disjoint union of
disks, and so that is a nonclosed, connected
2-manifold which is properly embedded in , in the sense
that . In fact we can take to be essential in , in the sense that
- is properly embedded,
- is -injective in , which means that the inclusion
homomorphism is injective,
- is not a
sphere bounding a ball in , and
- is not boundary-parallel
in , which means that it’s not isotopic (in the nonambient sense)
to a subsurface of . Tautologically, this surface is
also planar in the sense that it’s homeomorphic to a
subsurface of .
If is any nonclosed, essential surface in (where still
denotes an irreducible, compact, orientable 3-manifold whose
boundary is a torus), then the components of are all
homotopically nontrivial closed curves on , and they all
have the same slope. A slope on that is determined in
this way by some nonclosed, essential surface in is called a
boundary slope, and I’ll say that the surface
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 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 and are
boundary slopes, realized by essential surfaces and in
. After an isotopy we can arrange that the components of are nontrivial simple closed curves, and arcs that are properly
embedded and non-boundary-parallel both in and in . For
we can form a topological quotient space of
by identifying each component of to a point. Then
is a 2-sphere if is planar, and in general it’s a
closed surface of the same genus as . Furthermore,
contains a graph , whose vertices are the images under the
quotient map of components of , and whose edges are the
images under the quotient map of those components of
which are arcs. Combinatorial arguments comparing the graphs
and give the required bound on
.
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
is a torus knot in , i.e., a knot that is contained
in a standard torus in , then there are many surgeries that give
3-manifolds with cyclic , 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 whose boundary is a torus,
and look at the slopes on 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 either is hyperbolic in the sense that
its interior6
admits a complete Riemannian metric with constant curvature 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 is hyperbolic, we have
for any two cyclic slopes and
on . This easily implies that if is a knot in
whose exterior is hyperbolic, and if is cyclic for
some given , then must be an integer; and furthermore,
that if and are cyclic for some given
distinct integers and , then and must be consecutive
integers. We knew this result would be sharp, because
Fintushel
and
Stern
had shown that for a certain hyperbolic knot , called the
-pretzel knot, both and 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 is a compact, orientable 3-manifold which is irreducible
and not Seifert-fibered, and is a torus, then the
distance between any two cyclic slopes on is at most
1.
A corollary to the theorem is that, under the hypotheses, there are at
most three cyclic slopes on . Again this is sharp, because
for the -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 or is a “strict boundary
slope”: this means a boundary slope in the sense that I defined
above, given by an essential surface 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 and are not strict boundary slopes,
but also the hypothesis that 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 or was a strict boundary
slope actually struck us as a very positive thing. This is because if
(say) is a boundary slope, there is a simple approach to
trying to prove that is not a cyclic slope. In fact, if
is an essential surface in realizing the boundary slope ,
then the boundary components of bound disks in the solid torus
, and the union of with these disks is a
closed surface . Our idea was that if we
were lucky, would have positive genus and would be
-injective in . This would mean that
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 contains
no closed essential surface, and that separates and has
exactly two boundary components, then one can use a cool result called
the handle addition lemma to prove that is
-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 has just two boundary components. The idea was
supposed to be to take a surface having the minimal number of
boundary curves among all essential surfaces realizing the
boundary slope , and to use some fancy logic based on the handle
addition lemma to prove that is still -injective.
This would have brought us close to proving our conjecture in the case
where 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 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
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 instead of ). 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 contains no closed essential surface, they had
succeeded in using the handle addition lemma to show,
basically,8
that if is a
surface having the minimal number of boundary curves among all
essential surfaces realizing a given boundary slope , then is
-injective in , so that if has positive genus
then 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 is planar, then, basically,
the sphere divides into two
non-simply connected pieces. This implies that is a
nontrivial free product, hence noncyclic.
These results, combined with my earlier work with Marc, basically
implied that if is not Seifert-fibered and contains no closed
essential surface, then the distance between any two cyclic slopes on
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 does contain a closed essential surface
, they conceived a plan for showing that the distance between any
two cyclic slopes on is at most 1. The underlying idea
is simple. If and are cyclic slopes, then is
a closed orientable surface of positive genus in
for . The cyclicity of
implies that is not -injective in
either . Classical 3-manifold topology then gives a
compressing disk for in each , i.e., a disk
such that , but
bounds no disk in . One can choose the so that
and are -injective planar surfaces in
. These surfaces define an intersection graph, and one can hope to
use graph-theoretical methods to show that
.
This idea doesn’t always work for an arbitrary closed essential
surface in , 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 is a cyclic slope which is also a boundary slope,
then the distance from 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 was neither
hyperbolic nor Seifert fibered turned out to come out of Cameron and
John’s work. If 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
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 such
that 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 can never give . This
implies that if two knots in 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.