Cohomology of generalized configuration spaces

TL;DR

This paper develops a systematic framework using tcdga models to study the cohomology of generalized configuration spaces on topological spaces, showing that under certain conditions, their cohomology depends only on the compact support cohomology of the base space.

Contribution

It introduces tcdga models for cochains and proves that the cohomology of generalized configuration spaces depends solely on the compact support cohomology of the underlying space, generalizing Arabia's theorem.

Findings

Cohomology depends only on compact support cohomology under specified conditions.

Framework applies to a broad class of topological spaces.

Generalizes previous results by Arabia.

Abstract

Let $X$ be a topological space. We consider certain generalized configuration spaces of points on $X$, obtained from the cartesian product $X^n$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call "tcdga models" for the cochains on $X$. We prove the following theorem: suppose that $X$ is a "nice" topological space, $R$ is any commutative ring, $H^\bullet_c(X,R)\to H^\bullet(X,R)$ is the zero map, and that $H^\bullet_c(X,R)$ is a projective $R$-module. Then the compact support cohomology of any generalized configuration space of points on $X$ depends only on the graded $R$-module $H^\bullet_c(X,R)$. This generalizes a theorem of Arabia.