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Cohomology of generalized configuration spaces

Published online by Cambridge University Press:  20 December 2019

Dan Petersen
Affiliation:
Matematiska Institutionen, Stockholms Universitet, 106 91Stockholm, Sweden email dan.petersen@math.su.se
Corresponding
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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.

Type
Research Article
Copyright
© The Author 2019

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Footnotes

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

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