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the cohomology algebra of unordered configuration spaces

Published online by Cambridge University Press:  04 October 2005

yves félix
Affiliation:
département de mathématiques, université catholique de louvain, 2 chemin du cyclotron, 1348 louvain-la-neuve, belgiumfelix@math.ucl.ac.be
daniel tanré
Affiliation:
département de mathématiques, umr 8524, université de lille 1, 59655 villeneuve d'ascq cedex, francedaniel.tanre@univ-lille1.fr
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Abstract

given an $n$-dimensional compact closed oriented manifold $m$ and a field $\mathbb k$, f. cohen and l. taylor have constructed a spectral sequence, ${\mathcal e}(m,n,\mathbb k)$, 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, an explicit description is provided of the invariants algebra $(e_1,d_1)^{\sigma_n}$ of the first term of $\mathcal e(m,n,\mathbb q)$. 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 $h^*(c_n(m);\mathbb q)$ of the space of unordered configurations of $n$ points in $m$ is obtained (the concrete example of $p^2(\mathbb c)$ is detailed).

(b) the degeneration of the spectral sequence formed of the $\sigma_n$-invariants $\mathcal e(m,n,\mathbb q)^{\sigma_n}$ 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 $\mathbb f_p$ with $p>n$.

Keywords

Type
notes and papers
Copyright
the london mathematical society 2005

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