A remark on the homology of finite coverings of a surface

TL;DR

The paper proves that for genus g ≥ 3, the rational homology of finite covers of surfaces is generated by cycles supported on simple closed curves with images contained in a 3-punctured sphere, answering a question by Autumn Kent.

Contribution

It establishes a new generating set for the homology of surface covers using simple closed curves with specific properties, for genus g ≥ 3.

Findings

Homology groups are generated by cycles supported on simple closed curves.

Curves' images are contained in 3-punctured spheres in the base surface.

Answers positively a question by Autumn Kent for g ≥ 3.

Abstract

Let $p: S\to S_g$ be a finite covering of an orientable closed surface of genus $g$. We prove that, for $g\geq 3$, the rational homology group $H_1(S;{\mathbb Q})$ is generated by cycles supported on simple closed curves $\gamma\subset S$ such that $p(\gamma)$ is contained in a $3$-punctured, genus $0$ subsurface of $S_g$. In particular, this answers positively, for $g\geq 3$ and rational coefficients, a question by Autumn Kent.