Equivariant cohomology, homogeneous spaces and graphs

Abstract

The focus of this thesis is manifolds with group actions, in particular symplectic manifolds with Hamiltonian torus actions. We investigate the relationship between the equivariant cohomology of the manifold M and the fixed point data of the torus action. We are interested in understanding the topology of the space of T-orbits in M. In particular, we explore aspects of this topology which are determined by data from the image of a moment map [Phi] : M [right arrow] t* associated to the Hamiltonian action. To better understand the orbit space, we apply the algebraic techniques of equivariant cohomology to the study these systems further. Equivariant cohomology associates to a manifold with a G-action a ring H*G(M). Much of the topology of the orbit space is encoded in the equivariant cohomology ring H*G(M). In 1998, Goresky, Kottwitz and MacPherson provided a new method for computing this ring. Their method associates to this orbit space a graph [Gamma] whose vertices are the zero-dimensional orbits and edges the connected components of the set of one-dimensional orbits. The ring H*G(M) can then be computed combinatorially in terms of the data incorporated in [Gamma]. The strength of this construction is that it makes the computation of equivariant cohomology into a combinatorial computation, rather than a topological one. In the projects described herein, we apply the GKM theory to the case of homogeneous spaces by studying the combinatorics of their associated graphs. We exploit this theory to understand the geometry of homogeneous spaces with non-zero Euler characteristic.

(cont.) Next, we describe how to weaken the hypotheses of the GKM theorem. The spaces to which the GKM theorem applies must satisfy certain dimension conditions; however, there are many manifolds M with naturally arising T-actions that do not satisfy these conditions. We allow a more general situation, which includes some of these cases. Finally, we find a theory identical to the GKM theory in a setting suggested by work of Duistermaat. As in the GKM situation, this theory applies only when the spaces involved satisfy certain dimension conditions.