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Generating the homology of covers of surfaces

Marco Boggi, Andrew Putman, Nick Salter

Abstract

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

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.
Paper Structure (24 sections, 15 theorems, 53 equations)

This paper contains 24 sections, 15 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Putman--Wieland conjecture
  3. Simple closed curve homology
  4. Relationship to previous work
  5. Symplectic subspace
  6. Pants homology
  7. Outline
  8. Acknowledgments
  9. Simple closed curve homology and Dehn twists
  10. Notation
  11. Lifting Dehn twists
  12. Fixed set of lifted twists
  13. Putman--Wieland conjecture
  14. The symplectic nature of simple closed curves
  15. Notation
  16. ...and 9 more sections

Key Result

Theorem A

Let $\pi\colon \widetilde{\Sigma} \rightarrow \Sigma$ be a finite branched cover between closed oriented surfaces. Consider some nonzero $\vec{v} \in \mathop{\mathrm{H}}\nolimits^{\mathop{\mathrm{scc}}\nolimits}_1(\widetilde{\Sigma};\mathbb{Q})$. Then the $\mathop{\mathrm{Mod}}\nolimits(\Sigma,\wide

Theorems & Definitions (32)

  • Conjecture 1.1: PutmanWieland
  • Theorem A
  • Remark 1.2
  • Theorem B
  • Theorem B$'$
  • Theorem C
  • Remark 1.4
  • Theorem D
  • Remark 1.5
  • Remark 1.6
  • ...and 22 more