$RO(C_3)$-graded Bredon Cohomology and $C_p$-surfaces

Abstract

Let $p$ be an odd prime, and let $C_p$ denote the cyclic group of order $p$. We use equivariant surgery methods to classify all closed, connected $2$-manifolds with an action of $C_p$. We then use this classification in the case $p=3$ to compute the $RO(C_3)$-graded Bredon cohomology of all $C_3$-surfaces in constant $\underline{\mathbb{Z}/3}$ coefficients as modules over the cohomology of a point. We show that the cohomology of a $C_3$-surface is completely determined by its genus, number of fixed points, and whether or not its underlying surface is orientable.