The natural action of the symmetric group on the configuration spaces F (X, n) induces an action on the Križ model E(X, n). The representation theory of this DGA is studied and a big acyclic subcomplex which is Sn-invariant is described. Recently Lambrechts and Stanley [LaSt] constructed a (quasi)-model for the configuration space of a topological space with Poincaré duality cohomology; if such a space is formal, the model of Lambrechts-Stanley is reduced to the Križ model and this is the case of Kähler manifolds, see [DGMS]. Therefore all the results of this paper could be applied to (simply connected) formal closed manifolds (with few changes for the odd-dimensional manifolds).Let us remind the construction of Križ. We denote by p * i : H * (X) → H * (X n ) and p * ij : H * (X 2 ) → H * (X n ) (for i = j) the pullbacks of the obvious projections and by m the complex dimension of X (for cohomology groups we use rational or complex coefficients). The model E(X, n) is defined as follows: as an algebra E(X, n) is isomorphic to the exterior algebra with generators G ij , 1 ≤ i, j ≤ n (of degree 2m − 1) and coefficients in H * (X) ⊗n modulo the relations G

We compute the Betti numbers and describe the cohomology algebras of the ordered and unordered configuration spaces of three points in complex projective spaces, including the infinite dimensional case. We also compute these invariants for the configuration spaces of three collinear and non-collinear points.

Abstract. The symmetric group S n acts on the power set P(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.