2023/05/22 by Marco Boggi, Boggi, Marco, Andrew Putman +3
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2305.13109
openalex publication_date 2023/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Putman and Wieland conjectured that if Σ → Σ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of H1(Σ;ℚ) under the action of lifts to Σ of mapping classes on Σ are infinite. We prove that this holds if H1(Σ;ℚ) is generated by the homology classes of lifts of simple closed curves on Σ. We also prove that the subspace of H1(Σ;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(Σ;ℚ) is generated by the homology classes of lifts of loops on Σ lying on subsurfaces homeomorphic to 3-holed spheres.