Generating the homology of covers of surfaces

Abstract Putman and Wieland conjectured that if is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of under the action of lifts to of mapping classes on are infinite. We prove that this holds if is generated by the homology classes of lifts of simple closed curves on . We also prove that the subspace of spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2‐holed spheres, and we prove that is generated by the homology classes of lifts of loops on lying on subsurfaces homeomorphic to 3‐holed spheres.