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.

arXiv comment47 pages. V2: significant revision, inaccuracies corrected. Final version

References in corpus (14)

• Stability patterns in representation theory
• Gröbner bases for operads
• Diagram spaces and symmetric spectra
• Homological perturbation theory for algebras over operads
• Cochains and Homotopy Type
• Bar constructions for topological operads and the Goodwillie derivatives of the identity
• An algebraic model for commutative HZ-algebras
• A spectral sequence for stratified spaces and configuration spaces of points
• Higher enveloping algebras
• Betti numbers and stability for configuration spaces via factorization homology
• On the homotopy invariance of configuration spaces
• Symmetric operads in abstract symmetric spectra
• Non-commutative formality implies commutative and Lie formality
• Free factorization algebras and homology of configuration spaces in algebraic geometry

Cited by in corpus (8)

• Etale cohomology, purity and formality with torsion coefficients
• Homological stability and densities of generalized configuration spaces
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• A strictly commutative model for the cochain algebra of a space
• Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
• Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces
• The Frobenius character of the Orlik-Terao algebra of type A
• Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras