Generating the homology of covers of surfaces (2305.13109v2)

Abstract: Putman and Wieland conjectured that if $\tilde{\Sigma} \rightarrow \Sigma$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of $H_1(\tilde{\Sigma};\mathbb{Q})$ under the action of lifts to $\tilde{\Sigma}$ of mapping classes on $\Sigma$ are infinite. We prove that this holds if $H_1(\tilde{\Sigma};\mathbb{Q})$ is generated by the homology classes of lifts of simple closed curves on $\Sigma$. We also prove that the subspace of $H_1(\tilde{\Sigma};\mathbb{Q})$ spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that $H_1(\tilde{\Sigma};\mathbb{Q})$ is generated by the homology classes of lifts of loops on $\Sigma$ lying on subsurfaces homeomorphic to 3-holed spheres.