On the rational homotopy type of configuration spaces
type: Item
instance of: scholarly article
title: On the rational homotopy type of configuration spaces
zbMATH Open document ID: 0829.55008
DOI: 10.2307/2946581
published in: Annals of Mathematics. Second Series
publication date: 1996-01-15T00:00:00Z
review text: The ordered configuration space of \(n\) points in a space \(X\) is defined to be the set \(F(X,n) = \{(x_1, \ldots, x_n)\mid x_i \neq x_j\) for \(i \neq j\}\). In [\textit{W. Fulton} and \textit{R. MacPherson}, ibid. 139, No 1, 183-225 (1994; Zbl 0820.14037)], it was shown that the configuration space of a smooth projective variety \(X\) possesses a cohomological differential graded algebra model \(F(n)\) depending on \(H^*(X)\), the orientation class corresponding to the diagonal in \(X \times X\) and \(X\)'s Chern classes. In this paper, the author does away with the dependence on Chern classes by describing another cohomological model for \(F(X,n)\), \(E(n)\), which depends only on \(H^* (X)\) and the orientation class. Explicitly, \(E(n)\) is defined by adjoining certain generators to \(H^* (X^n)\), where \(X^n = X \times \dots \times X\) (\(n\)-times), then quotienting by relations in the new generators. Finally, a differential is defined in terms of the orientation class. The author shows that \(E(n)\) is a model for \(F(X,n)\) by describing an explicit map \(\phi: E(n) \to F(n)\) inducing an isomorphism in cohomology.
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 591412
zbMATH Keywords: ordered configuration space
zbMATH Keywords: smooth projective variety
zbMATH Keywords: cohomological model
zbMATH Keywords: orientation class
author: Igor Kriz
reviewed by: John Oprea
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946581
OpenAlex ID: W1971051468
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description: scientific article; zbMATH DE number 591412