Torelli group action on the configuration space of a surface

TL;DR

This paper demonstrates that the Torelli group for a genus ≥ 3 surface acts nontrivially on the rational cohomology of the configuration space of three points, revealing new insights into surface group actions.

Contribution

It establishes a nontrivial action of the Torelli group on the rational cohomology of a specific configuration space, a novel result in surface topology.

Findings

Torelli group acts nontrivially on the cohomology.

Focus on the configuration space of 3 points.

Results apply to surfaces of genus ≥ 3.

Abstract

We show that the Torelli group of a closed surface of genus $\ge 3$ acts nontrivially on the rational cohomology of its space of $3$-element subsets.