no. 4
The rational homotopy type of configuration spaces of two points
[Le type d'homotopie rationnel des espaces de configuration de deux points]
Lambrechts, Pascal1 ; Stanley, Don
1 Université de Louvain, Institut Mathématique, 2 chemin du Cyclotron, 1348 Louvain-la-Neuve, (Belgique), University of Ottawa, Department of Mathematics and Statistics, 585 King Edward Ave., Ottawa, ON K1N 6N5 (CANADA)
Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1029-1052.

Nous démontrons que le type d’homotopie rationnelle de l’espace des configurations de deux points dans une variété fermée 2-connexe dépend uniquement du type d’homotopie rationnelle de cette variété et nous montrons comment construire un modèle de Sullivan de cet espace de configuration. Nous étudions aussi la formalité des espaces de configuration.

We prove that the rational homotopy type of the configuration space of two points in a 2-connected closed manifold depends only on the rational homotopy type of that manifold and we give a model in the sense of Sullivan of that configuration space. We also study the formality of configuration spaces.

DOI : 10.5802/aif.2042
Classification : 55P62
Keywords: configuration space, Sullivan model
Mot clés : espaces de configuration, modèles de Sullivan
Lambrechts, Pascal 1 ; Stanley, Don 

1 Université de Louvain, Institut Mathématique, 2 chemin du Cyclotron, 1348 Louvain-la-Neuve, (Belgique), University of Ottawa, Department of Mathematics and Statistics, 585 King Edward Ave., Ottawa, ON K1N 6N5 (CANADA) 
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Lambrechts, Pascal; Stanley, Don. The rational homotopy type of configuration spaces of two points. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1029-1052. doi : 10.5802/aif.2042. http://www.numdam.org/articles/10.5802/aif.2042/

[1] F. Cohen; L. Taylor Computations of Gelfand-Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, Geometric applications of homotopy theory I (Proc. Conf., Evanston 1977) (Lecture Notes in Math.), Volume 657 (1978), pp. 106-143 | Zbl

[2] P. Deligne; P. Griffiths; J. Morgan; D. Sullivan Real homotopy theory of Kähler manifolds, Invent. Math, Volume 29 (1975), pp. 245-274 | MR | Zbl

[3] Y. Félix; S. Halperin; J.-C. Thomas Rational homotopy theory, Graduate Text in Mathematics, vol. 210, Springer-Verlag, 2001 | MR | Zbl

[4] W. Fulton; R. Mac Pherson A compactification of configuration spaces, Annals of Math, Volume 139 (1994), pp. 183-225 | MR | Zbl

[5] J. Klein Poincaré duality embeddings and fiberwise homotopy theory, Topology, Volume 38 (1999), pp. 597-620 | MR | Zbl

[6] J. Klein Poincaré duality embeddings and fiberwise homotopy theory, II, Quart. Jour. Math. Oxford, Volume 53 (2002), pp. 319-335 | MR | Zbl

[7] I. Kriz On the rational homotopy type of configuration spaces, Annals of Math, Volume 139 (1994), pp. 227-237 | MR | Zbl

[8] P. Lambrechts Cochain model for thickenings and its application to rational LS-category, Manuscripta Math, Volume 103 (2000), pp. 143-160 | MR | Zbl

[9] P. Lambrechts; D. Stanley Algebraic models of the complement of a subpolyhedron in a closed manifold (submitted) (, http://gauss.math.ucl.ac.be/∼topalg/preprints/preprint-top)

[10] P. Lambrechts; D. Stanley DGmodule models for the configuration space of k points (in preparation)

[11] N. Levitt Spaces of arcs and configuration spaces of manifolds, Topology, Volume 34 (1995), pp. 217-230 | MR | Zbl

[12] J. Milnor; D. Husemoller Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, 1973 | MR | Zbl

[13] B. Totaro Configuration spaces of algebraic varieties, Topology, Volume 35 (1996) no. 4, pp. 1057-1067 | MR | Zbl

[14] J. Stasheff Rational Poincaré duality spaces, Illinois J. Math, Volume 27 (1983), pp. 104-109 | MR | Zbl

[15] D. Sullivan Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977) no. 47, pp. 269-331 | Numdam | MR | Zbl

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