# The $RO(C_2)$-graded cohomology of $C_2$-surfaces in $\underline{\mathbb{Z}/2}$-coefficients

## TL;DR

This paper computes the $RO(C_2)$-graded Bredon cohomology of all $C_2$-surfaces with $ ext{Z}/2$ coefficients, revealing dependence on a small set of invariants based on surface classification.

## Contribution

It provides a complete calculation of the cohomology for $C_2$-surfaces, linking it to their classification invariants, and extends understanding of equivariant surface topology.

## Key findings

- Cohomology depends on three invariants for nonfree surfaces.
- Cohomology depends on two invariants for free surfaces.
- Explicit cohomology modules are computed for all $C_2$-surfaces.

## Abstract

A surface with an involution can be viewed as a $C_2$-space where $C_2$ is the cyclic group of order two. Using the classification of $C_2$-surfaces given by Dugger, we compute the $RO(C_2)$-graded Bredon cohomology of all $C_2$-surfaces in constant $\mathbb{Z}/2$ 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.