All closed surfaces with a C_p-action where p is an odd prime were classified in [Pohl] using equivariant surgery methods. Using this classification in the case p=3, we compute the RO(C₃)-graded Bredon cohomology of all C₃-surfaces in constant Z/3-coefficients as modules over the cohomology of a fixed point. We show that the cohomology of a given C₃-surface is determined by a handful of topological invariants and is directly determined by the construction of the surface via equivariant surgery.

The work in this paper was a portion of the author's thesis project at the University of Oregon. The author would first like to thank her doctoral advisor Dan Dugger for his invaluable guidance and support. The author would also like to thank Christy Hazel and Clover May for countless helpful conversations. This research was partially supported by NSF grant DMS-2039316.