The RO(C2)-graded Cohomology of C2-Surfaces and Equivariant Fundamental Classes

Abstract

Let C2 denote the cyclic group of order two. Given a manifold with a C2-action, we can consider its equivariant Bredon RO(C2)-graded cohomology. We first use a classification due to Dugger to compute the Bredon cohomology of all C2-surfaces in constant Z/2 coefficients as modules over the cohomology of a point. We show the cohomology depends only on three numerical invariants in the nonfree case, and only on two numerical invariants in the free case. We next develop a theory of fundamental classes for equivariant submanifolds of any dimension in RO(C2)-graded cohomology in constant Z/2-coefficients. We connect these classes back to our initial computations by showing the cohomology of any C2-surface is generated by fundamental classes, and these classes can be used to easily compute the ring structure. To define fundamental classes we are led to study the cohomology of Thom spaces of equivariant vector bundles. In general the cohomology of the Thom space is not just a shift of the cohomology of the base space, but we show there are still elements that act as Thom classes, and cupping with these classes gives an isomorphism within a certain range.