Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
Nir Gadish, Louis Hainaut

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 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 particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight component of .
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
