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

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