The rational homotopy type of configuration spaces of two points

Pascal Lambrechts[1]; Don Stanley

  • [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 (2004)

  • Volume: 54, Issue: 4, page 1029-1052
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Lambrechts, Pascal, and Stanley, Don. "The rational homotopy type of configuration spaces of two points." Annales de l’institut Fourier 54.4 (2004): 1029-1052. <http://eudml.org/doc/116129>.

@article{Lambrechts2004,
abstract = {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.},
affiliation = {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)},
author = {Lambrechts, Pascal, Stanley, Don},
journal = {Annales de l’institut Fourier},
keywords = {configuration space; Sullivan model; rational homotopy type; Sullivan theory; CDGA-model; formal},
language = {eng},
number = {4},
pages = {1029-1052},
publisher = {Association des Annales de l'Institut Fourier},
title = {The rational homotopy type of configuration spaces of two points},
url = {http://eudml.org/doc/116129},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Lambrechts, Pascal
AU - Stanley, Don
TI - The rational homotopy type of configuration spaces of two points
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 4
SP - 1029
EP - 1052
AB - 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.
LA - eng
KW - configuration space; Sullivan model; rational homotopy type; Sullivan theory; CDGA-model; formal
UR - http://eudml.org/doc/116129
ER -

References

top
  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) 657 (1978), 106-143, Springer, Berlin Zbl0398.55004
  2. P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math 29 (1975), 245-274 Zbl0312.55011MR382702
  3. Y. Félix, S. Halperin, J.-C. Thomas, Rational homotopy theory, vol. 210 (2001), Springer-Verlag Zbl0961.55002MR1802847
  4. W. Fulton, R. Mac Pherson, A compactification of configuration spaces, Annals of Math 139 (1994), 183-225 Zbl0820.14037MR1259368
  5. J. Klein, Poincaré duality embeddings and fiberwise homotopy theory, Topology 38 (1999), 597-620 Zbl0928.57028MR1670412
  6. J. Klein, Poincaré duality embeddings and fiberwise homotopy theory, II, Quart. Jour. Math. Oxford 53 (2002), 319-335 Zbl1030.57036MR1930266
  7. I. Kriz, On the rational homotopy type of configuration spaces, Annals of Math 139 (1994), 227-237 Zbl0829.55008MR1274092
  8. P. Lambrechts, Cochain model for thickenings and its application to rational LS-category, Manuscripta Math 103 (2000), 143-160 Zbl0964.55013MR1796311
  9. P. Lambrechts, D. Stanley, Algebraic models of the complement of a subpolyhedron in a closed manifold (submitted) 
  10. P. Lambrechts, D. Stanley, D G module models for the configuration space of k points (in preparation) Zbl1069.55006
  11. N. Levitt, Spaces of arcs and configuration spaces of manifolds, Topology 34 (1995), 217-230 Zbl0845.57021MR1308497
  12. J. Milnor, D. Husemoller, Symmetric bilinear forms, Band 73 (1973), Springer-Verlag Zbl0292.10016MR506372
  13. B. Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), 1057-1067 Zbl0857.57025MR1404924
  14. J. Stasheff, Rational Poincaré duality spaces, Illinois J. Math 27 (1983), 104-109 Zbl0488.55010MR684544
  15. D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), 269-331 Zbl0374.57002MR646078

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.