Betti numbers and stability for configuration spaces via factorization homology

TL;DR

This paper uses factorization homology to connect the rational homology of configuration spaces on manifolds with Lie algebra homology, providing new proofs and explicit calculations for stability and homology results.

Contribution

It introduces a novel approach linking configuration space homology to Lie algebra homology via factorization homology, extending and providing new proofs of existing theorems.

Findings

Extended theorems of Bödigheimer-Cohen-Taylor and Félix-Thomas.

Provided a new combinatorial proof of homological stability.

Enabled explicit calculations of configuration space homology.

Abstract

Using factorization homology, we realize the rational homology of the unordered configuration spaces of an arbitrary manifold $M$, possibly with boundary, as the homology of a Lie algebra constructed from the compactly supported cohomology of $M$. By locating the homology of each configuration space within the Chevalley-Eilenberg complex of this Lie algebra, we extend theorems of B\"odigheimer-Cohen-Taylor and F\'elix-Thomas and give a new, combinatorial proof of the homological stability results of Church and Randal-Williams. Our method lends itself to explicit calculations, examples of which we include.