An equivariant surgery classification of $C_p$-surfaces

TL;DR

This paper classifies all closed, connected 2-manifolds with an action of a cyclic group of odd prime order using equivariant surgery, providing explicit construction methods for each class.

Contribution

It extends the classification of $C_p$-surfaces to odd primes, offering a systematic construction approach and an analogue to Dugger's previous work.

Findings

Complete classification of $C_p$-surfaces for odd primes

Explicit construction methods for each isomorphism class

Extension of known classifications to new prime cases

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 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.