Multiplicities and equivariant cohomology
Date
2010
Authors
Lynn, Rebecca E., author
Duflot, Jeanne, advisor
Miranda, Rick, committee member
Hulpke, Alexander, committee member
Iyer, Hariharan K., committee member
Journal Title
Journal ISSN
Volume Title
Abstract
The aim of this paper is to address the following problem: how to relate the algebraic definitions and computations of multiplicity from commutative algebra to computations done in the cohomology theory of group actions on manifolds. Specifically, this paper is concerned with applications of commutative algebra to the study of cohomology rings arising from group actions on manifolds, in the way that Quillen initiated. This paper synthesizes two distinct areas of pure mathematics (commutative algebra and cohomology theory) and two ways of computing multiplicities in order to link them. In order to accomplish this task, a discussion of commutative algebra will be followed by a discussion of cohomology theory. A link between commutative algebra and cohomology theory will be presented, followed by its application to a significant example. In commutative algebra, we discuss graded rings, Pioncaré Series, dimension, and multiplicities. Whereas the theory for multiplicities has been developed for local rings, we give an exposition of the theory for graded rings. Several definitions for dimension will be presented, and it will be proven that all of these distinct definitions are equal. The basic properties of multiplicities will be introduced, and a brief discussion of a classical multiplicity in commutative algebra, the Samuel multiplicity, will be presented. Then, Maiorana's C-multiplicity will be defined, and a relationship between all of these multiplicities will be observed. In cohomology theory, we address smooth actions of finite groups on manifolds. As a part of this study in cohomology theory, we will consider group actions on topological spaces and the Borel construction (equivariant cohomology), completing this part of the paper with a discussion of smooth (or differentiable) actions, setting some notation necessary for our discussion of Maiorana's results, which inspire some of our main theorems, but on which we do not rely in this dissertation. Following the treatments of commutative algebra and cohomology theory, we present one of Quillen's main results without proof, linking these two distinct areas of pure mathematics. Quillen's work results in a formula for finding the multiplicity of the equivariant cohomology of a compact G-manifold with G a compact Lie group. We apply these results to the compact G-manifold U/S, where G (a compact Lie group) is embedded in a unitary group U=U(n) and S=S(n) is the diagonal p-torus of rank n in U(n), resulting in a nice topological formula for computing multiplicities. Finally, we end the paper with a proposal for future research.
Description
Department Head: Gerhard Dangelmayr.