THE COHOMOLOGY ALGEBRA OF UNORDERED CONFIGURATION SPACES

Given an N-dimensional compact closed oriented manifold M and a field lk, F. Cohen and L. Taylor have constructed a spectral sequence, E(M, n, lk), converging to the cohomology of the space of ordered configurations of n points in M. The symmetric group Σn acts on this spectral sequence giving a spectral sequence of Σn -differential graded commutative algebras. Here, an explicit description is provided of the invariants algebra (E1, d1)Σn of the first term of E(M, n,Q). This determination is applied in two directions. (a) In the case of a complex projective manifold or of an odd-dimensional manifold M, the cohomology algebra H∗(Cn (M);Q) of the space of unordered configurations of n points in M is obtained (the concrete example of P 2(C) is detailed). (b) The degeneration of the spectral sequence formed of the Σn -invariants E(M, n,Q)Σn at level 2 is proved for any manifold M. These results use a transfer map and are also true with coefficients in a finite field Fp with p > n.