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Cohomology of generalized configuration spaces

Part of: Ordered sets Singularities Fiber spaces and bundles Homotopy theory Homology and cohomology theories

Published online by Cambridge University Press:  20 December 2019

Dan Petersen
Dan Petersen*
Affiliation:
Matematiska Institutionen, Stockholms Universitet, 106 91Stockholm, Sweden email dan.petersen@math.su.se
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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 ‘twisted commutative dg algebra models’ for the cochains on $X$. Suppose that $X$ is a ‘nice’ topological space, $R$ is any commutative ring, $H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$ is the zero map, and that $H_{c}^{\bullet }(X,R)$ is a projective $R$-module. We prove that 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.

Keywords

configuration spacesarrangementscohomology with compact support

MSC classification

Primary: 55R80: Discriminantal varieties, configuration spaces 32S60: Stratifications; constructible sheaves; intersection cohomology 55N30: Sheaf cohomology 06A07: Combinatorics of partially ordered sets 55P48: Loop space machines, operads
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 251 - 298
Copyright
© The Author 2019

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Footnotes

The author gratefully acknowledges support by ERC-2017-STG 759082.

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