A remark on the homology of finite coverings of a surface

Abstract

Let p : S → S g be a finite covering of an orientable closed surface of genus g. We prove that, for g ≥ 3, the rational homology group H 1 (S; Q) is generated by cycles supported on simple closed curves γ ⊂ S such that p(γ) is contained in a 3-punctured, genus 0 subsurface of S g . In particular, this answers positively, for g ≥ 3 and rational coefficients, a question by Autumn Kent (cf. [4]).