Proves finite generation of H_k(I_g; Z) for k ≤ g-2 and that rational homology is an algebraic Sp(2g,Z)-representation, turning conditional cohomology computations into theorems and proving Morita's conjecture.
On torsion in the homology of the Torelli group
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abstract
Let $S_g$ be a closed, oriented surface of genus $g$, and let $\operatorname{Mod}(S_g)$ denote its mapping class group. The Torelli group $\mathcal{I}_g$ is the subgroup of $\operatorname{Mod}(S_g)$ consisting of mapping classes that act trivially on $H_1(S_g)$. For any collection of pairwise disjoint, separating simple closed curves on $S_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k(\mathcal{I}_g)$, which is called an abelian cycle. We prove that the subgroup of $H_k(\mathcal{I}_g)$ generated by such abelian cycles is a $\mathbb{Z}/2\mathbb{Z}$-vector space for all $k$, and that it is finite-dimensional for $k = 2$ and $g \geq 4$.
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math.GT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
The Birman-Craggs-Johnson homomorphism is injective on the subgroup of H_k(I_g) generated by abelian cycles from disjoint separating Dehn twists, for k ≤ g-2.
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Finite generation, algebraicity, and representation stability for homology of Torelli groups
Proves finite generation of H_k(I_g; Z) for k ≤ g-2 and that rational homology is an algebraic Sp(2g,Z)-representation, turning conditional cohomology computations into theorems and proving Morita's conjecture.
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Torsion in the homology of the Torelli group and the Birman-Craggs-Johnson homomorphism
The Birman-Craggs-Johnson homomorphism is injective on the subgroup of H_k(I_g) generated by abelian cycles from disjoint separating Dehn twists, for k ≤ g-2.