Cohomology of generalized configuration spaces

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, $H_c^{\bullet }(X,R)\to H^{\bullet }(X,R)$ is the zero map, and that $H_c^{\bullet }(X,R)$ 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 $H_c^{\bullet }(X,R)$. This generalizes a theorem of Arabia.

47 pages. V2: significant revision, inaccuracies corrected. Final version