The cohomology algebra of unordered configuration spaces

Abstract

Given an $N$-dimensional compact manifold $M$ and a field $\bk$, F. Cohen and L. Taylor have constructed a spectral sequence, $\cE(M,n,\bk)$, converging to the cohomology of the space of ordered configurations of $n$ points in $M$. The symmetric group ${\Sigma }_n$ acts on this spectral sequence giving a spectral sequence of ${\Sigma }_n$ differential graded commutative algebras. Here, we provide an explicit description of the invariants algebra $(E_1,d_1{)}^{{\Sigma }_n}$ of the first term of $\cE(M,n,\Q)$. We apply this determination in two directions: -- in the case of a complex projective manifold or of an odd dimensional manifold $M$, we obtain the cohomology algebra $H^*(C_n(M);\Q)$ of the space of unordered configurations of $n$ points in $M$ (the concrete example of $P^2(\C)$ is detailed), -- we prove the degeneration of the spectral sequence formed of the ${\Sigma }_n$-invariants $\cE(M,n,\Q)^{Σ_n}$ at level 2, for any manifold $M$. These results use a transfer map and are also true with coefficients in a finite field $\F_p$ with $p>n$.