Towards representation stability for the second homology of the Torelli group

TL;DR

This paper proves that for genus g > 6, the second homology of the Torelli group stabilizes and is generated by a specific base case, revealing new insights into the group's structure.

Contribution

It establishes representation stability for the second homology of the Torelli group for g > 6, linking it to the homology at genus 6 and analyzing the connectivity of arc complexes.

Findings

Second homology group is generated by the genus 6 case for g > 6.

The quotient of the arc complex by the Torelli group is (g-2)-connected.

Provides a stabilization map linking homology across genera.

Abstract

We show for g > 6 that the second homology group of the Torelli group of a surface of genus g and 1 boundary component is generated as an Sp(2g,Z)-module by the image under the stabilization map of the second homology group of the Torelli group of a surface of genus 6 and 1 boundary component. In the process we also show that the quotient of the complex of arcs with identity permutation by the Torelli group is (g-2)-connected, for one or two boundary components.