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Let be a regular covering of a surface of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group . We give necessary and sufficient conditions for an integral homology class on to admit a representative as a connected component of the preimage of a nonseparating simple closed curve on , possibly after passing to a “stabilization”, i.e. a -equivariant embedding of covering spaces .more » « less
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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.more » « less
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A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.more » « less
