Poincaré duality and commutative differential graded algebras

Pascal Lambrechts; Don Stanley

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 4, page 497-511
  • ISSN: 0012-9593

Abstract

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We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has applications in rational homotopy, giving Poincaré duality at the cochain level, which is of interest in particular in the study of configuration spaces and in string topology.

How to cite

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Lambrechts, Pascal, and Stanley, Don. "Poincaré duality and commutative differential graded algebras." Annales scientifiques de l'École Normale Supérieure 41.4 (2008): 497-511. <http://eudml.org/doc/272242>.

@article{Lambrechts2008,
abstract = {We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has applications in rational homotopy, giving Poincaré duality at the cochain level, which is of interest in particular in the study of configuration spaces and in string topology.},
author = {Lambrechts, Pascal, Stanley, Don},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {poincaré duality; commutative differential graded algebra},
language = {eng},
number = {4},
pages = {497-511},
publisher = {Société mathématique de France},
title = {Poincaré duality and commutative differential graded algebras},
url = {http://eudml.org/doc/272242},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Lambrechts, Pascal
AU - Stanley, Don
TI - Poincaré duality and commutative differential graded algebras
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 4
SP - 497
EP - 511
AB - We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has applications in rational homotopy, giving Poincaré duality at the cochain level, which is of interest in particular in the study of configuration spaces and in string topology.
LA - eng
KW - poincaré duality; commutative differential graded algebra
UR - http://eudml.org/doc/272242
ER -

References

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  1. [1] M. Aubry, J.-M. Lemaire & S. Halperin, Poincaré duality models, preprint. 
  2. [2] H. J. Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, 1989. Zbl0688.55001MR985099
  3. [3] M. Chas & D. Sullivan, String topology, preprint arXiv:math/9911159, 1999. 
  4. [4] Y. Félix, S. Halperin & J.-C. Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer, 2001. Zbl0961.55002
  5. [5] Y. Félix & J.-C. Thomas, Rational BV-algebra in string topology, Bull. Soc. Math. France136 (2008), 311–327. Zbl1160.55006
  6. [6] Y. Félix, J.-C. Thomas & M. Vigué-Poirrier, Rational string topology, J. Eur. Math. Soc.9 (2007), 123–156. Zbl1200.55015
  7. [7] W. Fulton & R. MacPherson, A compactification of configuration spaces, Ann. of Math.139 (1994), 183–225. Zbl0820.14037
  8. [8] I. Kříž, On the rational homotopy type of configuration spaces, Ann. of Math.139 (1994), 227–237. Zbl0829.55008MR1274092
  9. [9] P. Lambrechts, Cochain model for thickenings and its application to rational LS-category, Manuscripta Math.103 (2000), 143–160. Zbl0964.55013MR1796311
  10. [10] P. Lambrechts & D. Stanley, The rational homotopy type of configuration spaces of two points, Ann. Inst. Fourier (Grenoble) 54 (2004), 1029–1052. Zbl1069.55006
  11. [11] P. Lambrechts & D. Stanley, A remarkable DG-module model for configuration spaces, Algebraic & Geometric Topology 8 (2008), 1191–1222. Zbl1152.55004
  12. [12] L. Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology, preprint arXiv:0711.1946, 2007. Zbl1180.16007MR2543477
  13. [13] J. Milnor & D. Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer, 1973. Zbl0292.10016
  14. [14] J. Neisendorfer & T. Miller, Formal and coformal spaces, Illinois J. Math.22 (1978), 565–580. Zbl0396.55011
  15. [15] J. Stasheff, Rational Poincaré duality spaces, Illinois J. Math.27 (1983), 104–109. Zbl0488.55010MR684544
  16. [16] D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.É.S. 47 (1977), 269–331. Zbl0374.57002MR646078
  17. [17] T. Tradler, The BV algebra on Hochschild cohomology induced by infinity inner products, preprint arXiv:math.QA/0210150. Zbl1218.16004
  18. [18] T. Yang, A Batalin-Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials, Mémoire, University of Regina, 2007. Zbl1282.55013

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