Lifts of simple curves in finite regular coverings of closed surfaces

TL;DR

This paper investigates whether lifts of simple curves in finite regular covers of closed surfaces generate the first homology group, providing examples where they do not, thus addressing a question in geometric topology.

Contribution

The paper presents explicit examples showing that lifts of simple curves do not always generate the first homology group in finite regular covers of surfaces.

Findings

Lifts of simple curves do not always generate H_1 in certain covers.

Counterexamples to a previously posed question.

Insights into the structure of surface coverings and their homology.

Abstract

Suppose $S$ is a closed orientable surface and $\tilde{S}$ is a finite sheeted regular cover of $S$. The following question was posed by Juli\'{e}n March\'{e} in Mathoverflow: Do the lifts of simple curves from $S$ generate $H_{1}(\tilde{S},\mathbb{Z})$? A family of examples is given for which the answer is "no".