Mathematical Sciences: Cohomologies of Matrix Groups amd Mapping Class Groups

这个项目研究特定类型的所有矩阵的集合，将这些集合视为几何对象。具体地说，它涉及到两个元素的有限域上系数在同一有限域上的某些矩阵群的上同调环的计算，并将这些结果应用于曲面映射类群的积分同调中的扭类的判定。具有单个固定边界分量的曲面的映射类组支持同调运算，或者等价地支持双环空间结构，这是鲜为人知的。马金尼斯计划通过检测矩阵群的同调中的类来获得关于这种同调运算的信息。这些矩阵群将包括上三角矩阵、一般线性群的Sylow 2-子群、辛矩阵及其Sylow 2-子群。他还计划研究整数上类似的矩阵群。

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.