An equivariant surgery classification of $C_p$-surfaces

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 additionally provide a way to construct representatives of each isomorphism class using a series of equivariant surgery operations. The results in this paper serve as an odd prime analogue to a similar classification proved by Dan Dugger.