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On the second homology group of the Torelli subgroup of $\mathrm{Aut}(F_n)$

Matthew Day and Andrew Putman

Geometry & Topology 21 (2017) 2851–2896

Let IAn be the Torelli subgroup of Aut(Fn). We give an explicit finite set of generators for H2(IAn) as a GLn()–module. Corollaries include a version of surjective representation stability for H2(IAn), the vanishing of the GLn()–coinvariants of H2(IAn), and the vanishing of the second rational homology group of the level congruence subgroup of Aut(Fn). Our generating set is derived from a new group presentation for IAn which is infinite but which has a simple recursive form.

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automorphism group of free group, Torelli group
Mathematical Subject Classification 2010
Primary: 20E05, 20E36, 20F05, 20J06
Supplementary material

GAP code for the calculations

Received: 22 October 2015
Revised: 8 November 2016
Accepted: 23 December 2016
Published: 15 August 2017
Proposed: Shigeyuki Morita
Seconded: Benson Farb, Mark Behrens
Matthew Day
Department of Mathematical Sciences
University of Arkansas
Fayetteville, AR
United States
Andrew Putman
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States