Equivariant Fundamental Classes in RO(C_2)-Graded Cohomology

Let C_2 denote the cyclic group of order two. Given a manifold with a C_2-action, we can consider its equivariant Bredon RO(C_2)-graded cohomology. In this talk, we give an overview of RO(C_2)-graded cohomology in constant Z/2 coefficients, and then explain how a version of the Thom isomorphism theorem in this setting can be used to develop a theory of fundamental classes for equivariant submanifolds. We illustrate how these classes can be used to understand the cohomology of any C_2-surface in constant Z/2 coefficients, including the ring structure.