Equivariant fundamental classes in $RO(C_2)$-graded cohomology in $\underline{\mathbb{Z}/2}$-coefficients

TL;DR

This paper develops a theory of equivariant fundamental classes in $RO(C_2)$-graded cohomology with $Z/2$ coefficients, enabling explicit computations of cohomology rings for $C_2$-surfaces and related spaces.

Contribution

It introduces a new framework for fundamental classes in $RO(C_2)$-graded cohomology, including the study of Thom spaces and their cohomology, facilitating computations in equivariant topology.

Findings

Cohomology of any $C_2$-surface is generated by fundamental classes.

Fundamental classes enable straightforward computation of cohomology ring structures.

Thom classes exist in equivariant Thom spaces, allowing isomorphisms via cupping within certain ranges.

Abstract

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 paper, we develop a theory of fundamental classes for equivariant submanifolds in $RO(C_2)$-graded cohomology in constant $\mathbb{Z}/2$ coefficients. We show the cohomology of any $C_2$-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.