Ryan Budney

Postdoctoral position at the University of Rochester

Representations of Mapping Class Groups via Topological Constructions [abstract]

Allen Hatcher

Low Dimensional Topology

Abstract: This dissertation is an investigation into the action of mapping class groups on objects one would naturally associate with such groups, such as the homology of configuration spaces, homology of covering spaces of configuration spaces and generalized homology theories associated to the underlying surface. The main results are explicit CW decompositions of configuration spaces, constructed using Morse electrostatic potential functions. In Chapter 3, these are applied to give insight into the Lawrence-Krammer representation, showing that it is a unitary representation and giving some insight into the conjugacy problem for braid groups. Chapter 4 is concerned with the construction of an analogue of the Lawrence-Krammer representation. Chapter 2 is concerned with the action of mapping class groups on the homology of configuration spaces, the main result being that these representations are largely not faithful. In Chapter 5 generalized homology theories are shown to be very similar to standard, singular homology from the point of view of representations of mapping class groups.

PhD

PhD