by Steven Boyer
1. Introduction
2. First interactions
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
A knot manifold is a compact, connected, orientable, 3-manifold
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
Understanding
- How large can
be and how are its elements distributed in the surgery plane ? - What is the topology of
when the cardinality of is relatively large?
Results on the first question are often phrased in terms of the distance between slopes
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.
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
2.2 First visit to Austin
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
In the late 1990s, Xingru and I had mapped out an approach to proving Conjecture 3.1 when
3.1 Reducible and finite filling slopes
At this point, Cameron, Xingru and I took on the challenge of showing that the distance of a reducible filling slope
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
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:
Our idea was to use
[e12]
to reduce Conjecture 3.2 to the case where
First we considered
Next up was
The wheels came off the programme when we turned to the case
The one case which remains to complete the proof of Conjecture 3.2
is when
As for Conjecture 3.1, Lackenby
and
Meyerhoff
verified its first claim
is a reducible filling slope and a nonstrict boundary slope, or is a toroidal filling slope and a nonstrict boundary slope, and (as above) is atleast 3,or is a small Seifert filling slope and 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
4. The L-space conjecture
Since 2009, our main focus has shifted from exceptional surgeries to the
4.1 The L-space conjecture
A cooriented taut foliation on a closed orientable 3-manifold
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
We say that a closed, connected, orientable 3-manifold
if is a nontrivial left-orderable group; if admits a cooriented taut foliation; if is not an -space.
The one implication known is
When the first Betti number of
4.2 The genesis of the L-space conjecture
Another problem on our list was to show that 2-fold cyclic branched covers
Thus motivated, Cameron, Liam and I turned to the problem of showing that rational homology sphere Sol manifolds were
We also verified the equivalence of
5. Recent work
In May 2016, Cameron and I got together with
Michel Boileau
at the ICTP Trieste. Michel pointed out that the canonical cyclic branched covers
In the first of these two papers, we conjectured that a prime fibred strongly quasipositive link has an
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
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
One consequence of this was that the 2-fold cyclic branched cover of a fibred hyperbolic strongly quasipositive link
As of this writing, we’ve verified most of the
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.