In the previous post, we discussed MIST presentations for mapping class groups of closed oriented surfaces of genus 1 and 2, and (in a weaker sense) also for genus 3. How might we set about obtaining good presentations for mapping class groups of surfaces of higher genus? A-priori, this looks as though it might be a difficult problem.
The breakthrough was a paper by Hatcher and Thurston. The background for their idea was a result of Brown about how to deduce a presentation of the group from a finite description of its action on a simply-connected simplicial complex . The mapping class group acts on a surface rather than on a simply-connected complex; but simply-connected complexes can be built out of a choice of curves on the surface. Hatcher and Thurston make such a choice, and construct the cut-system complex, which they show to be simply-connected using Morse-Cerf theory. This gives an algorithm which in principle constructs a finite presentation for a mapping class group of a surface of arbitrarily high genus. All papers about presentations of mapping class groups for surfaces of arbitrary genus seem to factor through these ideas of Hatcher and Thurston.
Unfortunately, the finite presentation which one would obtain from running Hatcher and Thurston’s algorithm is huge and unwieldy, and the relations don’t seem conceptually meaningful. So the next step, taken by Harer, was to trim down the complex, to obtain a shorter presentation (which is still huge and unwieldy). Finally, Wajnryb succeeded in obtaining a relatively compact presentation, and in showing here that the Hatcher-Thurston complex is simply connected by elementary combinatorial methods.
Everyone seems to use Wajnryb’s presentation in practice. I suppose that the main reason for this is that the relations are relatively short, and are sort-of-meaningful I suppose, as explained in Farb and Margalit’s book. But it’s not a presentation I would ever want to memorize.
The second famous presentation is the one by Gervais. Its advantages are that it’s memorable, and it works for a surface with an arbitrary number of boundary components. It’s main disadvantage is that it lacks simplicity, in the sense that it uses an infinite number of generators, and I’ve never found it particularly easy to work with. Again, it’s explained in Farb-Margalit. Gervais’s star-relations were simplified by Feng Luo to an even simpler set of relations, all of which turn out to have been mentioned by Dehn in 1938!
Gervais proves his presentation on the basis of Wajnryb’s. Wajnryb needed to make some involved calculations, and Gervais made some involved calculations, so taken as a unit this makes Gervais’s proof a bit of a monster. So the next step was to simplify Gervais’s proof. I know two papers which do this, by Hirose and by Benvenuti. Both choose different simply-connected complexes on which acts. Hirose uses a complex of curves in which the curves are non-separating. This complex is unique up to homeomorphism over the surface, which argues that it is somehow a canonical choice. Benvenuti uses an ordered complex of curves, which is good for calculation because all of its 2-cells are triangular. Both approaches have advantages, and I don’t know which one is better in total. Benvenuti’s method was used by Szepietowski to give an algorithm to present the mapping class group of a non-orientable surface. But it looks to me as though the presentation one were to obtain by running his algorithm would be quite complicated.
In a different direction, there is Makoto Matsumoto’s exciting paper, which was written originally in Japanese. Working on the strength of an analogy with deformation spaces of singularities, Matsumoto conjectured, and proved by a computer calculation based on Wajryb’s presentation, that the mapping class group is presented by Humphries generators, plus the relations:
The relations are between fundamental elements of Dynkin diagrams inside the Coxeter graph associated to the Humphries generators, so this presentation looks a bit like Looijenga’s, and is certainly memorable. It’s also somewhat informative, because it’s easy to see why these relations should hold by drawing pictures (although not why they should be a complete set of relations). But it’s different from Looijenga’s. I am guessing that, if we really understood where this presentation comes from conceptually, then it (or a small perturbation of it) might be the best of all. Labruere and Paris generalize it to surfaces with punctures in a fairly straightforward way, so it is typical in some sense.
I suppose the ultimate point of these posts is that I wish I knew more about presentations of mapping class groups of surfaces. I wish I knew a MIST presentation for mapping class groups of all compact surfaces, with or without punctures, oriented or non-oriented, and for 3-dimensional handlebodies and compression bodies.