On the second homology group of the Torelli subgroup of Aut(F_n)

Abstract

Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for H_2(IA_n), the vanishing of the GL_n(Z)-coinvariants of H_2(IA_n), and the vanishing of the second rational homology group of the level l congruence subgroup of Aut(F_n). Our generating set is derived from a new group presentation for IA_n which is infinite but which has a simple recursive form.

39 pages; minor revision; to appear in Geom. Topol

References in corpus (4)

• Dimension of the Torelli group for Out(F_n)
• Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1
• The codimension-one cohomology of SL_n Z
• A Birman exact sequence for the Torelli subgroup of Aut(F_n)

Cited by in corpus (3)

• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• Representation stability for filtrations of Torelli groups
• Equivariant group presentations and the second homology group of the Torelli group