by Rob Kirby
The following is the transcription of a Zoom conversation with Joel Hass about his collaboration with Peter Scott, recorded on Friday, August 27, 2021. The text has been lightly edited for clarity.
Rob Kirby: Joel, tell me about your early work with Peter Scott and Mike Freedman. There were two papers that appeared out of this.
Joel Hass: I was a graduate student at Berkeley (advised by you) and studying 3-manifolds. Bill Meeks came by to visit Berkeley for a semester or year, and gave some lectures on minimal surfaces. He talked about his work with Yau on embedding problems for least area disks and spheres, and the techniques they used are very much 3-manifold techniques, based on Dehn’s Lemma and the work of Papakyriakopoulos and so on, and I was quite familiar with them. So I started working on the problem of generalizing that to incompressible surfaces, to show that least area incompressible surfaces were embedded when possible. And I made some progress with a few cases. One case I managed to do was when the surface was a fiber of a 3-manifold that fibers over the circle. This case was not that rare, but very much a special case.
And then Mike Freedman came by to give a seminar talk — I think in your seminar, in the weekly topology seminar — in which he talked about some ideas that he and Karen Uhlenbeck were working on to do the general case. But there were still some gaps. We talked for a while. I should say at that time he very generously offered to make whatever we got joint work even though initially we thought he had done much more. Eventually we did solve the problem. It turned out that the missing part of the argument was a requirement that lifts of least area surfaces also be least area, which was not at all obvious.
RK: This was a lift to a finite cover of the 3-manifold?
JH: Not necessarily, the lift could be an infinite cover. Some cover of the surface will lift to a cover of the 3-manifold, but it is not at all obvious that the lift is a least area surface, meaning each compact subsurface is least area relative to its boundary. In fact in the nonorientable case, it’s not true, which leads to some interesting still open problems for nonorientable incompressible surfaces.
Then that summer (1980), which was my last summer as a Berkeley graduate student, there was a conference on Thurston’s work at Bowdoin College, which was a very dynamic conference. I happened to be put in the same dormitory at Bowdoin College as Peter Scott. So we spent a lot of time talking. There was also some related work of Peter Shalen. He had made some progress on the torus case of this least area problem with Rick Schoen. In my conversations with Peter Scott when we discussed this problems of lifts of surfaces, I learned about his theorem that surface groups are LERF, locally extendable, residually finite. That property was exactly what was needed to show that the covers of least area surfaces are least area. And then the whole thing came together. There were quite a few other moving parts in it. We needed to generalize the Papakyriakopolous tower construction from disks and spheres to general incompressible surfaces.
Rob: You did that with Peter?
JH: Well, we had some approach, but the actual method we used was written by Mike. Mike created a very nice argument for that. So by the end of that Bowdoin conference, we realized that we had all the pieces that we needed for this theorem. This paper [2] has had many applications in the theory of 3-manifolds, playing a part, for example, in Bonahon’s work on the ends of hyperbolic 3-manifolds and in Pardon’s work on the Hilbert–Smith conjecture.
Later on, Mike realized that if we drop a dimension to the much easier case of curves on surfaces, we had solved an open problem about geodesics, which was that if you took shortest geodesics, they had the minimal number of intersections.
Perhaps the key new idea was that intersections are counted in an appropriate covering space rather than directly. This was known for hyperbolic metrics, but in fact it holds for any Riemannian metric at all with no curvature conditions.
RK: So then you went on continuing to work with Peter Scott using minimal surfaces. Tell me more about that. What was next?
JH: So, we were looking at the open problems in 3-manifolds, a lot of which have now been subsumed by geometrization. For example, Waldhausen had proved that homotopy equivalent, Haken 3-manifolds were homeomorphic. And that the universal covers of Haken 3-manifolds were homeomorphic to \( R^3 \) Also some results about homotopic diffeomorphisms being isotopic. So in each case, for each of those three theorems, we were able to greatly extend the context in which they held from non-Haken manifolds. So for example, in a paper that was joint with Peter Scott and Hyam Rubinstein [5], we showed that just having a surface subgroup was enough to show that the universal cover of a 3-manifold is homeomorphic to \( R^3 \). So if you had an irreducible 3-manifold with a surface group in it,
RK: A subgroup of the 3-manifold group isomorphic to a surface group?
JH: Yes. It didn’t have to be normal, just an arbitrary orientable
surface subgroup. Not the 2-sphere, but any other surface. That’s saying there’s some sort of immersed,
\( \pi_1 \)-injective surface inside the 3-manifold. And if the manifold
was irreducible and had such a surface in it, then its universal cover
was \( R^3 \).
Waldhausen had proved that for the case where you had an
embedded incompressible surface, and we were able to extend it to drop
the embedded condition. In a similar way, we were able to extend the
homotopy equivalence implies homeomorphism result
[7].
It’s now known to
be true for any irreducible 3-manifold with infinite \( \pi_1 \). We
were able to show that this result was true if there was a surface in
the 3-manifold that had a condition called the four-plane one-line
property. If you have an immersed surface in a 3-manifold, it can
intersect in crazy ways. There’s all kinds of triple points. It’s very
hard to understand how it self-intersects. But if you go to the
universal cover, you see a collection of planes intersecting. And this
condition says that any pair of those planes intersect in at most one
line, and any four of them contain a disjoint pair.
RK: Just topological \( R^2 \)s right?
JH: Yes, embedded topological planes. So if you have any four planes, at least one pair is disjoint. Like planes parallel to the coordinate planes in \( R^3 \). What you don’t see is four planes bounding a little tetrahedron.
RK: Okay. So that was a condition to prove the second of the three thing, that homotopy equivalent manifolds are actually homeomorphic.
JH: A lot of those results are now known as part of geometrization, from the topological view. But there’s also information about the properties of least area surfaces, which remain interesting from the differential geometry point of view.
RK: That was early, but you continued to collaborate with Peter Scott after that.
JH: That’s right. We wrote a series of papers on the topic of minimal surfaces in 3-manifolds and their applications in topology [5], [6], [7]. Some ventured into geometric analysis, giving new proofs of the existence theorems for minimal surfaces [4]. Actually if you look at citations, we have two papers on curves on surfaces, that are quite highly cited in computer science [3], [8]. We actually wrote several others as well. But on curves on surfaces there are some interesting problems. One problem, for example, was to show when a given curve on a surface has the minimal number of self intersections possible in its homotopy class. There’s some obvious obstructions to that. If you see a little monogon or bigon, it’s clearly not minimizing the self intersections. We were able to essentially show the converse that if you don’t see any of these simple obstructions, then it does minimize the self-intersection number [3].
RK: So you had a finite list of simple ones?
JH: Just two, monogons and bigons, except that they can be immersed rather than embedded. So you might think of a bigon, which is sort of stretched out so that the two end points of the bigon go around some path and the bigon starts intersecting itself. We call that an immersed bigon.
RK: So if you know that there are no immersed bigons, and no immersed monogons, then it’s a minimally self-intersecting curve.
JH: Yes. Computer scientists create algorithms to simplify curves and their intersections to be used for algorithms in computer graphics. This gives an algorithmic way to test whether your curve is as simple as possible.
And then Peter and I proved another theorem in a different paper about curves on surfaces. Suppose you have two curves on a surface that are homotopic and each has \( k \) crossings. The question is, can you always find a homotopy in which each of the intermediate curves has at most \( k \) crossings. Or do you sometimes need to go up? So we proved that you never needed to increase the number of crossings during a homotopy [8]. And that’s useful for algorithms to move one curve to another.
RK: Anything else with you and Peter that you want to mention?
JH: More recently we did some work on discrete harmonic mappings, about five years ago, There are thousands of papers on harmonic mappings in various settings, which are really solutions of various forms of the Laplacian. Many of them are for the planar case, the classic Laplacian on planar regions. There’s also a lot of work in differential geometry on the Laplacian for general manifolds. But for the discrete theory, even for the harmonic maps between surfaces, it turns out there’s just some very basic open questions. It’s not straightforward to discretize harmonic maps between surfaces. So Peter and I wrote a paper [9] giving a topological or combinatorial argument for the existence of families of finite area mappings, avoiding the need for deep theories from analysis. That was our original motivation, which we carried out. One application, for example, is to try and understand the space of diffeomorphisms from a surface to surface, related to the homotopy type of that space.
RK: So is this not known?
JH: Some things are known. Smale showed that the diffeomorphism group of the 2-sphere is homotopy equivalent to \( \operatorname{SO}(3) \) And for the higher genus surfaces it was known by the work of Earle and Eells that each component is contractible for genus 2 and above. What was not known in general was the homotopy type of the space of triangulations of a surface.
Suppose you triangulate a surface. So take, for example, a triangulation of a genus-2 surface. And now look at all homotopic triangulations. So look at all triangulations that can be connected to a base triangulation via a homotopy of the surface isotopic to the identity. We can think of taking one fixed triangulation, as a base point, move it around and while it moves, it’s allowed to cross itself and fold over, to do all kinds of crazy things. At the end it all turns into a nice triangulation, with the same combinatorics as initially, but moved around on the surface in some unknowable way. And then the question is, if we look at all such triangulations, is that space even connected? You could ask more generally, what is the homotopy type of that space? This problem has just recently been solved in general (by Yanwen Luo, Tianqi Wu and Xiaoping Zhu), following our approach.1 But Peter and I solved it for the case of one vertex triangulations using this idea of discrete harmonic maps [9]. So it’s of interest again to applied mathematicians, to computer scientists and applied mathematicians because in practice, when you want to compute harmonic maps between surfaces, which you do for many reasons, you need to discretize them.
RK: Is there anything else about your joint work that you want to mention?
JH: That pretty much covers it. Those are the highlights. I should say that for me Peter is one of the most inspiring lecturers I’ve encountered, that he always presents things in close to an ideal way. Also he is quite modest about his achievements, which are of great significance in topology and group theory. There is little in current 3-manifold theory that does not rely on his contributions.
