Torsion in the homology of the Torelli group and the Birman-Craggs-Johnson homomorphism

Abstract

The Birman-Craggs-Johnson homomorphism is a homomorphism $\sigma :{\mathcal{I}}_g\to {\mathbb{B}}_3^{\prime }$ from the Torelli group to a certain $\mathbb{Z}/2\mathbb{Z}$-vector space of Boolean polynomials. In 1983, Johnson computed $H_1({\mathcal{I}}_g)$ for $g\ge 3$ and showed, in particular, that the induced homomorphism on $H_1({\mathcal{I}}_g)$ is injective when restricted to the subgroup generated by Dehn twists about separating simple closed curves. In this paper, we extend Johnson's result to higher homology groups. Given any collection of pairwise disjoint separating simple closed curves on ${\Sigma }_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k({\mathcal{I}}_g)$ called an abelian cycle. We prove that the pushforward homomorphism restricted to the subgroup of $H_k({\mathcal{I}}_g)$ generated by such abelian cycles is injective for $k\le g-2$.

21 pages, 3 figures