Configuration spaces on a wedge of spheres and HochschildPirashvili homology
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 HochschildPirashvili 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 superexponential lower bound on the symmetric group action on the weight $0$ component of $H^*_c(M_{2,n})$.
 Publication:

arXiv eprints
 Pub Date:
 February 2022
 DOI:
 10.48550/arXiv.2202.12494
 arXiv:
 arXiv:2202.12494
 Bibcode:
 2022arXiv220212494G
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Representation Theory;
 55R80;
 14H10;
 14Q05;
 13D03
 EPrint:
 44 pages + 8 appendix pages