A remark on the homology of finite coverings of a surface

Abstract

Let $p:S\to S_g$ be a finite covering of an orientable closed surface of genus $g$. We prove that, for $g\ge 3$, 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 $g\ge 3$ and rational coefficients, a question by Autumn Kent.

Superseded by arXiv:2305.13109

References in corpus (2)

• Quotients of surface groups and homology of finite covers via quantum representations
• Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_n)