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

Day, Matthew; Putman, Andrew

doi:10.2140/gt.2017.21.2851

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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

DOI: 10.2140/gt.2017.21.2851

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.

Keywords

automorphism group of free group, Torelli group

Mathematical Subject Classification 2010

Primary: 20E05, 20E36, 20F05, 20J06

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References

Publication

Received: 22 October 2015

Revised: 8 November 2016

Accepted: 23 December 2016

Published: 15 August 2017

Proposed: Shigeyuki Morita

Seconded: Benson Farb, Mark Behrens

Authors

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

Volume 21, issue 5 (2017)