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

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 $\mathbb{Z}\mathrm{/}2$ 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.