The cohomology algebra of unordered configuration spaces
(2005) Journal of the London mathematical society — Vol. 72, p. 525-544 (2005)
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- Abstract
- Given an N-dimensional compact closed oriented manifold M and a field k, F. Cohen and L. Taylor have constructed a spectral sequence, epsilon(M, n, lk), converging to the cohomology of the space of ordered configurations of n points in M. The symmetric group E, acts on this spectral sequence giving a spectral sequence of E.-differential graded commutative algebras. Here, an explicit description is provided of the invariants algebra (E-1, d(1))(Sigma n) 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* (C-n (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 Sigma(n)-invariants epsilon(M, n, Q)(Sigma 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.
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Félix, Y., & Tanre, D. (2005). The cohomology algebra of unordered configuration spaces. Journal of the London mathematical society, 72, 525-544. https://doi.org/10.1112/S0024610705006794 (Original work published 2005)