This project investigates collections of all matrices of certain specific types, viewing the collections as geometric objects. Specifically, it is concerned with the calculation of the cohomology rings of certain matrix groups over the finite field of two elements, with coefficients in this same finite field, and with the application of these results to the detection of torsion classes in the integral homology of the mapping class groups of surfaces. The mapping class groups of surfaces with a single fixed boundary component support a homology operation, or equivalently a double loop space structure, that is little understood. Maginnis plans to obtain information about this homology operation by detecting classes in the homology of matrix groups. These matrix groups will include the upper triangular matrices, a Sylow 2-subgroup of the general linear group, and symplectic matrices and their Sylow 2-subgroups. He also plans to study similar matrix groups over the integers.
