2003/11/19 by Yves Félix, Yves Felix, Daniel Tanré +2
Mathematics · #55P60 #55T99 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55P60 #msc:55T99
paper · pdf · doi:10.48550/arxiv.math/0311323
arxiv created 2003/11/19 · openalex publication_date 2003/11/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Σn acts on this spectral sequence giving a spectral sequence of Σn differential graded commutative algebras. Here, we provide an explicit description of the invariants algebra (E1,d1)Σn 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^*(Cn(M);\Q) of the space of unordered configurations of n points in M (the concrete example of P2(\C) is detailed), -- we prove the degeneration of the spectral sequence formed of the Σn-invariants \cE(M,n,\Q)Σn at level 2, for any manifold M. These results use a transfer map and are also true with coefficients in a finite field \Fp with p>n.