Cohomology of configuration spaces of surfaces as mapping class group representations

TL;DR

This paper explores the rational cohomology of configuration spaces on surfaces, expressing it as a mapping class group representation and explicitly computing it for surfaces with boundary.

Contribution

It introduces a novel representation-theoretic framework for the cohomology of configuration spaces and computes explicit examples for surfaces with boundary.

Findings

The cohomology representation is not symplectic.

The second Johnson filtration subgroup acts trivially.

Explicit cohomology computations for surfaces with boundary.

Abstract

We express the rational cohomology of the unordered configuration space of a compact oriented manifold as a representation of its mapping class group in terms of a weight-decomposition of the rational cohomology of the mapping space from the manifold to a sphere. We apply this to the case of a compact oriented surface with one boundary component and explicitly compute the rational cohomology of its unordered configuration space as a representation of its mapping class group. In particular, this representation is not symplectic, but has trivial action of the second Johnson filtration subgroup of the mapping class group.