Generating the homology of covers of surfaces

Putman and Wieland conjectured that if sigma similar to ->sigma$\widetilde{\Sigma }\rightarrow \Sigma$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ under the action of lifts to sigma similar to$\widetilde{\Sigma }$ of mapping classes on sigma$\Sigma$ are infinite. We prove that this holds if H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ is generated by the homology classes of lifts of simple closed curves on sigma$\Sigma$. We also prove that the subspace of H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ is generated by the homology classes of lifts of loops on sigma$\Sigma$ lying on subsurfaces homeomorphic to 3-holed spheres.