Mapping Class Groups and Categorification (13w5108)

Arriving in Banff, Alberta Sunday, April 7 and departing Friday April 12, 2013

Organizers

(Lehman College, CUNY)

(Australian National University)

(Columbia University)

Objectives

The goal of this workshop is to bring together researchers in two related fields -- geometric group theory and categorification -- to improve each groups understanding of the tools and techniques of the other. Both fields are rapidly developing, and the relations between them are just starting to become apparent.

While quantum algebra has been working behind the scenes in some striking results in mapping class group theory -- notably, in the representation that lead to linearity, and Andersen's proof of the failure of property T -- techniques from the categorification of quantum groups have yet to yield results about mapping class groups. This seems likely to change in the near future. For example, it was proved six months ago that the mapping class group of a surface with boundary admits a faithful action on a category of modules over an algebra. This categorified representation, which comes from Heegaard Floer homology, carries deep information about the mapping class (for instance, the Thurston norm on its mapping torus), and appears to be related to variants of the pants complex and other geometric techniques used to study mapping class groups.

It is well known in group theory that faithful representations have consequences (for instance, the Tits alternative). Categorified representations ought to have even more consequences. For example, a categorified representation of the braid group typically also leads to actions by cobordisms (surfaces) between braids; this level of further structure does not make sense for a non-categorified representation. Similarly, cobordisms between mapping classes (for instance, Lefschetz fibrations) act in the categorifications of the braid group from Heegaard Floer homology.

This further structure has not yet been exploited for group-theoretic results, apparently because it is new and the right questions have not yet been asked. It seems likely that, by the time of the conference, some applications of this and related results to mapping class group theory will be known, and exchange of ideas should lead to further breakthroughs.

In the other direction, a key goal of the categorification program is to extend Khovanov / Khovanov-Rozansky homology theories of knots to give invariants of 3- and 4-manifolds. These knot invariants are most naturally defined in terms of categorical braid group actions. The natural analogues for closed 3-manifolds should be mapping class group actions; the relationship is via Heegaard splittings, say. While braid groups are typically familiar to students of quantum algebra, one hindrance to progress on closed 3-manifolds is a lack of knowledge in the categorification community about mapping class groups.

This conference should facilitate the flow of knowledge between these two subjects. In particular, two of the goals of this conference will be to make those working on categorification aware of the main questions of current interest in the study of mapping class groups, and for researchers on mapping class groups to learn what kinds of tools categorification makes available. BIRS, which is particularly well-suited to small group discussions, seems an ideal place for such an exchange of ideas to occur.