2005/10/01 by Yves Félix, Daniel Tanré · 2 citations
Mathematics · #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models
paper · doi:10.1112/s0024610705006794
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 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 ( E 1 , d 1 ) ∑ 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*(Cn(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 Σn-invariants ε ( M , n , Σ ) ∑ 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 Fp with p > n.