  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.
