Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Rational model of the configuration space of two points in a simply connected closed manifold


Author: Hector Cordova Bulens
Journal: Proc. Amer. Math. Soc. 143 (2015), 5437-5453
MSC (2010): Primary 55P62
DOI: https://doi.org/10.1090/proc/12666
Published electronically: April 14, 2015
MathSciNet review: 3411158
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a simply connected closed manifold of dimension $ n$. We study the rational homotopy type of the configuration space of two points in $ M$, $ F(M,2)$. When $ M$ is even dimensional, we prove that the rational homotopy type of $ F(M,2)$ depends only on the rational homotopy type of $ M$. When the dimension of $ M$ is odd, for every $ x\in H^{n-2} (M, \mathbb{Q})$, we construct a commutative differential graded algebra $ C(x)$. We prove that for some $ x \in H^{n-2} (M; \mathbb{Q})$, $ C(x)$ encodes completely the rational homotopy type of $ F(M,2)$. For some class of manifolds, we show that we can take $ x=0$.


References [Enhancements On Off] (What's this?)

  • [1] Hector Cordova Bulens, Modèle rationnel du complémentaire d'un sous-polyèdre dans une variété à bord et applications aux espaces des configurations, Ph.D. thesis, Université catholique de Louvain, 2013.
  • [2] Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847 (2002d:55014)
  • [3] Nathan Habegger, On the existence and classification of homotopy embeddings of a complex into a manifold, Ph.D. thesis, Université de Genève, 1981.
  • [4] Pascal Lambrechts and Don Stanley, The rational homotopy type of configuration spaces of two points, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 1029-1052 (English, with English and French summaries). MR 2111020 (2005i:55016)
  • [5] Pascal Lambrechts and Don Stanley, Poincaré duality and commutative differential graded algebras, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 4, 495-509 (English, with English and French summaries). MR 2489632 (2009k:55022)
  • [6] Pascal Lambrechts and Don Stanley, A remarkable DGmodule model for configuration spaces, Algebr. Geom. Topol. 8 (2008), no. 2, 1191-1222. MR 2443112 (2009g:55011), https://doi.org/10.2140/agt.2008.8.1191
  • [7] Riccardo Longoni and Paolo Salvatore, Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375-380. MR 2114713 (2005k:55024), https://doi.org/10.1016/j.top.2004.11.002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 55P62

Retrieve articles in all journals with MSC (2010): 55P62


Additional Information

Hector Cordova Bulens
Affiliation: Institut de Recherche en Mathémathique et Physique-IRMP; Université caltholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Email: hector.cordova@uclouvain.be

DOI: https://doi.org/10.1090/proc/12666
Received by editor(s): February 20, 2014
Received by editor(s) in revised form: October 27, 2014, and October 28, 2014
Published electronically: April 14, 2015
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society