#### Volume 16, issue 3 (2012)

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Towards representation stability for the second homology of the Torelli group

### Søren K Boldsen and Mia Hauge Dollerup

Geometry & Topology 16 (2012) 1725–1765
##### Abstract

We show for $g\ge 7$ that the second homology group of the Torelli group, ${H}_{2}\left({\mathsc{ℐ}}_{g,1};ℚ\right)$, is generated as an $Sp\left(2g,ℤ\right)$–module by the image of ${H}_{2}\left({\mathsc{ℐ}}_{6,1};ℚ\right)$ under the stabilization map. In the process we also show that the quotient $B\left({F}_{g,i};i\right)∕{\mathsc{ℐ}}_{g,i}$ by the Torelli group of the complex of arcs with identity permutation is $\left(g-2\right)$–connected for $i=1,2$.

##### Keywords
representation stability, Torelli group
##### Mathematical Subject Classification 2010
Primary: 20C12, 20J06
##### Publication
Received: 4 September 2011
Accepted: 14 March 2012
Published: 3 August 2012
Proposed: Benson Farb
Seconded: Danny Calegari, Jesper Grodal
##### Authors
 Søren K Boldsen Mathematical Institute University of Bonn Endenicher Allee 60 D-53115 Bonn Germany Mia Hauge Dollerup Department of Mathematics Aarhus University Klokkerfaldet 109 DK-8210 Aarhus Denmark