On the rational homotopy type of configuration spaces
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.
scientific article; zbMATH DE number 591412