Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology

TL;DR

This paper investigates the rational cohomology of configuration spaces on wedges of spheres, revealing their structure as polynomial functors and providing explicit computations for small cases, with implications for symmetric group actions.

Contribution

It introduces a new perspective on the cohomology representations as polynomial functors and computes their composition factors, including a complete analysis for up to 10 particles.

Findings

Cohomology representations form polynomial functors

Explicit composition factors computed for n ≤ 10

Established a super-exponential lower bound on symmetric group actions

Abstract

We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group $Out(F_g)$ of outer automorphisms of the free group. These representations show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups and Hochschild-Pirashvili cohomology. We show that these cohomology representations form a polynomial functor, and use various geometric models to compute many of its composition factors. We further compute the composition factors completely for all configurations of $n\leq 10$ particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight $0$ component of $H^*_c(M_{2,n})$.