Betti numbers and stability for configuration spaces via factorization 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 Bödigheimer-Cohen-Taylor and Félix-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.

To appear in Algebraic & Geometric Topology. May vary slightly from published version

References in corpus (2)

• A spectral sequence for stratified spaces and configuration spaces of points
• Betti numbers of configuration spaces of surfaces

Cited by in corpus (9)

• Higher enveloping algebras
• Cohomology of generalized configuration spaces
• The Lambrechts-Stanley Model of Configuration Spaces
• Splitting of the homology of the punctured mapping class group
• The factorization theory of Thom spectra and twisted non-abelian Poincaré duality
• Homological stability and densities of generalized configuration spaces
• Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces
• A remark on extremal stability of configuration spaces
• The cohomology rings of the unordered configuration spaces of the torus