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Abelian covers of surfaces and the homology of the Torelli group

Daniel Minahan, Andrew Putman

TL;DR

The paper investigates the first homology of the mapping class group and the Torelli group with coefficients in the first rational homology of the universal abelian cover, revealing a contrast between cases with boundary components and punctures: the twisted homology is finite-dimensional for boundary cases but infinite-dimensional for punctures. Central to the analysis is the Reidemeister pairing, a twisted intersection form that becomes a connecting homomorphism in long exact sequences, and its coinvariant version under the Torelli action; the authors reduce finiteness questions to understanding the kernel and image of this pairing. The work develops a detailed algebraic framework, constructs generators for the kernel, and derives relations to show the kernel is a finite-dimensional, algebraic representation of the symplectic group $ ext{Sp}_{2g}(Z)$, with the cokernel also finite-dimensional. These results underpin the authors’ broader program, including calculations of $H_2( ext{I}_{g,p}^b;Q)$, and illuminate how boundary deco­rations influence twisted homology in comparison to puncture cases. The methodology blends Birman exact sequences, coinvariants, symplectic representation theory, and intricate combinatorial-generator–relation arguments around separating curves and genus-1 subsurfaces, yielding a robust description of the twisted Torelli-homology landscape in high genus. The findings offer precise finite- vs infinite-dimensional dichotomies and provide the necessary algebraic scaffolding for further study of Torelli group homology with twisted coefficients.

Abstract

We study the first homology group of the mapping class group and Torelli group with coefficients in the first rational homology group of the universal abelian cover of the surface. We prove two contrasting results: for surfaces with one boundary component these twisted homology groups are finite-dimensional, but for surfaces with one puncture they are infinite-dimensional. These results play an important role in a recent paper of the authors calculating the second rational homology group of the Torelli group.

Abelian covers of surfaces and the homology of the Torelli group

TL;DR

The paper investigates the first homology of the mapping class group and the Torelli group with coefficients in the first rational homology of the universal abelian cover, revealing a contrast between cases with boundary components and punctures: the twisted homology is finite-dimensional for boundary cases but infinite-dimensional for punctures. Central to the analysis is the Reidemeister pairing, a twisted intersection form that becomes a connecting homomorphism in long exact sequences, and its coinvariant version under the Torelli action; the authors reduce finiteness questions to understanding the kernel and image of this pairing. The work develops a detailed algebraic framework, constructs generators for the kernel, and derives relations to show the kernel is a finite-dimensional, algebraic representation of the symplectic group , with the cokernel also finite-dimensional. These results underpin the authors’ broader program, including calculations of , and illuminate how boundary deco­rations influence twisted homology in comparison to puncture cases. The methodology blends Birman exact sequences, coinvariants, symplectic representation theory, and intricate combinatorial-generator–relation arguments around separating curves and genus-1 subsurfaces, yielding a robust description of the twisted Torelli-homology landscape in high genus. The findings offer precise finite- vs infinite-dimensional dichotomies and provide the necessary algebraic scaffolding for further study of Torelli group homology with twisted coefficients.

Abstract

We study the first homology group of the mapping class group and Torelli group with coefficients in the first rational homology group of the universal abelian cover of the surface. We prove two contrasting results: for surfaces with one boundary component these twisted homology groups are finite-dimensional, but for surfaces with one puncture they are infinite-dimensional. These results play an important role in a recent paper of the authors calculating the second rational homology group of the Torelli group.

Paper Structure

This paper contains 84 sections, 55 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Homology of mapping class group
  3. Low degree homology of Torelli
  4. Action on fundamental group
  5. Main theorems
  6. Linear congruence subgroups
  7. Reidemeister pairing
  8. Outline
  9. Notation and conventions
  10. Standing assumption
  11. Warning
  12. The Reidemeister pairing and the homology of the Torelli group
  13. Boundary components, punctures, and the Birman exact sequence
  14. Boundary component vs puncture: mapping class groups
  15. Factoring the representation
  16. ...and 69 more sections

Key Result

Theorem A

For $g \geq 4$, both $\mathop{\mathrm{H}}\nolimits_1(\operatorname{Mod}_g^1;\mathcal{C}_g^1)$ and $\mathop{\mathrm{H}}\nolimits_1(\mathcal{I}_g^1;\mathcal{C}_g^1)$ are finite-dimensional. Moreover, $\mathop{\mathrm{H}}\nolimits_1(\mathcal{I}_g^1;\mathcal{C}_g^1)$ is an algebraic representation of $\

Theorems & Definitions (129)

  • Remark 1.1
  • Remark 1.2
  • Theorem A
  • Remark 1.3
  • Theorem B
  • Remark 1.4
  • Lemma 2.1: FarbMargalitPrimer
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • ...and 119 more