Homological stability and densities of generalized configuration spaces

TL;DR

This paper develops a unified, conceptual framework for understanding the cohomology, homological stability, and densities of generalized configuration spaces using factorization homology and Koszul duality, extending previous work.

Contribution

It introduces a new approach linking factorization homology with cohomologies of configuration spaces, generalizing prior results and providing rational homotopy insights.

Findings

New expressions for cohomologies of generalized configuration spaces

A conceptual approach to homological stability and densities

Rational homotopy types answering open questions

Abstract

We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.