The $RO(C_3)$-graded Bredon cohomology of $C_3$-surfaces in $\underset{‾}{\mathbb{Z}\mathrm{/}3}$-coefficients

Using an equivariant surgery classification, we compute the $RO(C_3)$-graded Bredon cohomology of all $C_3$-surfaces in constant $\mathbb{Z}\mathrm{/}3$-coefficients as modules over the cohomology of a fixed point. We show that the cohomology of a given $C_3$-surface is determined by a handful of topological invariants and is directly determined by the construction of the surface via equivariant surgery.