The algebra and topology of the mapping class group

William Marsh Rice University, Houston TX

Investigators

Abstract

The proposed research concerns the mapping class group of a surface and related groups. It contains four families of projects. The first concerns the cohomological properties of finite-index subgroups. Specific goals here include proving an analogue of the Borel stability theorem for the mapping class group and proving a sort of "equivariant homological stability" theorem for congruence subgroups of the special linear group. The second family of projects concerns the Picard groups of finite covers of the moduli space of curves. The goal here is to understand the divisibility properties of certain natural line bundles on these finite covers. The third family of projects concerns the Torelli subgroup of the mapping class group, which is the kernel of action of the mapping class group on the first homology group of the surface. The goal here is to clarify the basic cohomological and combinatorial properties of this group and its subgroups. The final family of projects concerns the analogue of the Torelli subgroup in the automorphism group of a free group. The goal here is to adapt tools that have been successful in studying the mapping class group to the setting of the automorphism group of a free group. In particular, analogues of the curve complex will be studied. The proposed projects concern mapping class groups, which play a key role in many fields of mathematics, ranging from algebraic geometry and low dimensional topology to mathematical physics. The problems which involve the cohomology groups of the mapping class group seek to measure one of the most basic invariants of these groups ? roughly, the k-dimensional cohomology groups count the k-dimensional "holes" in geometric models for the groups. These play an important role in the applications. Another set of problems concern the combinatorics of these groups. This should allow actual concrete calculations within them, facilitating the investigation of the diverse objects with which they interact.