The second rational homology of the Torelli group

TL;DR

This work computes the second rational cohomology $\mathrm{H}^2(\mathcal{I}_{g,p}^b;\mathbb{Q})$ of the Torelli group for $g\ge 6$, giving explicit decompositions into irreducible algebraic $\mathrm{Sp}_{2g}(\mathbb{Z})$-modules in the three basic cases $(p,b)\in\{(0,0),(1,0),(1,1)\}$. The authors develop a stability framework showing that $\mathrm{H}_2(\mathcal{I}_g^1;\mathbb{Q})$ forms a uniformly representation-stable sequence for $g\ge 6$, underpinning uniform control of these representations. The proof proceeds in three steps: (i) reduce the problem to curve stabilizers via Birman exact sequences and a “handle complex,” (ii) identify generators for the cokernel using equivariant homology of the Torelli action on a handle complex, and (iii) establish algebraicity of the cokernel by embedding into a finite, algebraic representation space and applying a refined unmixed-versus-mixed representation analysis. The algebraicity argument combines a finite-dimensionality criterion, a general algebraicity criterion, and new results on universally mixed subgroups of finite groups, culminating in a presentation of the symmetric kernel and the finiteness of the cokernel as a representation of the stabilizer subgroup. Overall, the paper advances understanding of Torelli homology, shows representation stability phenomena in low-degree homology, and provides a robust framework for future higher-degree investigations.

Abstract

We calculate the second rational homology group of the Torelli group for $g \geq 6$.

Theorems & Definitions (111)

• Theorem A
• Theorem B
• Remark 1.1
• Remark 1.2
• Corollary C
• Remark 1.3
• Example 1.4
• Example 1.5
• Theorem D: Stability Theorem
• Remark 1.6