On the rational homotopy type of configuration spaces. (English) Zbl 0829.55008

Kříž, Igor

Ann. Math. (2) 139, No. 2, 227-237 (1994).

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 [W. Fulton and 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.

Reviewer: J.F.Oprea (Cleveland)

MSC:

55P62	Rational homotopy theory
55P15	Classification of homotopy type
20F34	Fundamental groups and their automorphisms (group-theoretic aspects)

Keywords:

ordered configuration space; smooth projective variety; cohomological model; orientation class

Citations:

Zbl 0829.55009; Zbl 0820.14037