The cohomology representation of an action of 𝐶_{𝑝} on a surface

Abstract

When a finite group G G acts on a surface S S , then H 1 ( S ; Z ) {H^1}(S;\,{\mathbf {Z}}) posseses naturally the structure of a Z G {\mathbf {Z}}G -module with invariant symplectic inner product. In the case of a cyclic group of odd prime order we describe explicitly this symplectic inner product space in terms of the fixed-point data of the action.