The Cohomology Algebra of Unordered Configuration Spaces

Abstract

Given an N‐dimensional compact closed oriented manifold M and a field lk, F. Cohen and L. Taylor have constructed a spectral sequence, ε(M, n, k), converging to the cohomology of the space of ordered configurations of n points in M. The symmetric group Σ_n acts on this spectral sequence giving a spectral sequence of Σ_n‐differential graded commutative algebras. Here, an explicit description is provided of the invariants algebra ${(E_1,d_1)}^{{\sum }_n}$ of the first term of ɛ(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^*(C_n(M);Q) of the space of unordered configurations of n points in M is obtained (the concrete example of P²(C) is detailed).

(b) The degeneration of the spectral sequence formed of the Σ_n‐invariants ${\epsilon (M,n,\Sigma )}^{{\sum }_n}$ 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 F_p with p > n.