Towards representation stability for the second homology of the Torelli group

Boldsen, Søren K¹ ; Dollerup, Mia Hauge²

¹ Mathematical Institute, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
² Department of Mathematics, Aarhus University, Klokkerfaldet 109, DK-8210 Aarhus, Denmark

Geometry & topology, Tome 16 (2012) no. 3, pp. 1725-1765.

Résumé

We show for $g\ge 7$ that the second homology group of the Torelli group, $H_2\left({\mathcal{ℐ}}_{g,1};\mathbb{Q}\right)$, is generated as an $Sp\left(2g,\mathbb{Z}\right)$–module by the image of $H_2\left({\mathcal{ℐ}}_{6,1};\mathbb{Q}\right)$ under the stabilization map. In the process we also show that the quotient $B\left(F_{g,i};i\right)∕{\mathcal{ℐ}}_{g,i}$ by the Torelli group of the complex of arcs with identity permutation is $\left(g-2\right)$–connected for $i=1,2$.

Détail

DOI : 10.2140/gt.2012.16.1725

Classification : 20C12, 20J06
Keywords: representation stability, Torelli group

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