Riemann-Hurwitz formula for Morita-Mumford classes and surface symmetries

Abstract

Let a finite group G act on a compact Riemann surface C in a faithful and orientation preserving way. Then we describe the Morita-Mumford classes e_n(C_G)∈H²ⁿ(G; Z) of the homotopy quotient (or the Borel construction) C_G of the action in terms of fixed-point data. This fixed-point formula is deduced from a higher analogue of the classical Riemann-Hurwitz formula based on computations of Miller [Mi] and Morita [Mo].