Generalized symmetric spaces and minimal models

Anna Dumańska-Małyszko; Zofia Stępień; Aleksy Tralle

Annales Polonici Mathematici (1996)

  • Volume: 64, Issue: 1, page 17-35
  • ISSN: 0066-2216

Abstract

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We prove that any compact simply connected manifold carrying a structure of Riemannian 3- or 4-symmetric space is formal in the sense of Sullivan. This result generalizes Sullivan's classical theorem on the formality of symmetric spaces, but the proof is of a different nature, since for generalized symmetric spaces techniques based on the Hodge theory do not work. We use the Thomas theory of minimal models of fibrations and the classification of 3- and 4-symmetric spaces.

How to cite

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Anna Dumańska-Małyszko, Zofia Stępień, and Aleksy Tralle. "Generalized symmetric spaces and minimal models." Annales Polonici Mathematici 64.1 (1996): 17-35. <http://eudml.org/doc/269981>.

@article{AnnaDumańska1996,
abstract = {We prove that any compact simply connected manifold carrying a structure of Riemannian 3- or 4-symmetric space is formal in the sense of Sullivan. This result generalizes Sullivan's classical theorem on the formality of symmetric spaces, but the proof is of a different nature, since for generalized symmetric spaces techniques based on the Hodge theory do not work. We use the Thomas theory of minimal models of fibrations and the classification of 3- and 4-symmetric spaces.},
author = {Anna Dumańska-Małyszko, Zofia Stępień, Aleksy Tralle},
journal = {Annales Polonici Mathematici},
keywords = {minimal model; Koszul complex; generalized symmetric space; formality in the sense of Sullivan},
language = {eng},
number = {1},
pages = {17-35},
title = {Generalized symmetric spaces and minimal models},
url = {http://eudml.org/doc/269981},
volume = {64},
year = {1996},
}

TY - JOUR
AU - Anna Dumańska-Małyszko
AU - Zofia Stępień
AU - Aleksy Tralle
TI - Generalized symmetric spaces and minimal models
JO - Annales Polonici Mathematici
PY - 1996
VL - 64
IS - 1
SP - 17
EP - 35
AB - We prove that any compact simply connected manifold carrying a structure of Riemannian 3- or 4-symmetric space is formal in the sense of Sullivan. This result generalizes Sullivan's classical theorem on the formality of symmetric spaces, but the proof is of a different nature, since for generalized symmetric spaces techniques based on the Hodge theory do not work. We use the Thomas theory of minimal models of fibrations and the classification of 3- and 4-symmetric spaces.
LA - eng
KW - minimal model; Koszul complex; generalized symmetric space; formality in the sense of Sullivan
UR - http://eudml.org/doc/269981
ER -

References

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  1. [1] C. Allday and V. Puppe, Cohomology Theory of Transformation Groups, Cambridge Univ. Press, 1993. 
  2. [2] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274. Zbl0312.55011
  3. [3] L. Flatto, Invariants of reflection groups, Enseign. Math. 28 (1978), 237-293. Zbl0401.20041
  4. [4] A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geom. 7 (1972), 343-369. Zbl0275.53026
  5. [5] V. Greub, S. Halperin and R. Vanstone, Curvature, Connections and Cohomology, Vol. 3, Acad. Press, 1976. Zbl0372.57001
  6. [6] S. Halperin, Lectures on Minimal Models, Hermann, 1982. 
  7. [7] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Acad. Press, 1978. 
  8. [8] J. A. Jiménez, Riemannian 4-symmetric spaces, Trans. Amer. Math. Soc. 306 (1988), 715-734. Zbl0647.53039
  9. [9] O. Kowalski, Classification of generalized Riemannian symmetric spaces of dimension ≤ 5, Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd 85 (1975). 
  10. [10] O. Kowalski, Generalized Symmetric Spaces, Springer, 1980. Zbl0431.53042
  11. [11] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, 1985. Zbl0563.13001
  12. [12] D. Lehmann, Théorie homotopique des formes différentielles (d'après D. Sullivan), Astérisque 45 (1977). 
  13. [13] G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), 193-207. Zbl0789.55010
  14. [14] T. Miller and J. Neisendorfer, Formal and coformal spaces, Illinois J. Math. 22 (1978), 565-580. Zbl0396.55011
  15. [15] D. Sullivan, Infinitesimal computations in topology, Publ. IHES 47 (1977), 269-331. Zbl0374.57002
  16. [16] M. Takeuchi, On Pontrjagin classes of compact symmetric spaces, J. Fac. Sci. Univ. Tokyo Sect. I 9 (1962), 313-328. Zbl0108.35802
  17. [17] D. Tanré, Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan, Springer, 1988. 
  18. [18] J.-C. Thomas, Homotopie rationnelle des fibrés de Serre, Ph.D. Thesis, Université des Sciences et Techniques de Lille 1, 1980. 
  19. [19] J.-C. Thomas, Rational homotopy of Serre fibrations, Ann. Inst. Fourier (Grenoble) 31 (3) (1981), 71-90. Zbl0446.55009
  20. [20] M. Vigué-Poirrier and D. Sullivan, Cohomology theory of the closed geodesic problem, J. Differential Geom. 11 (1976), 633-644. Zbl0361.53058
  21. [21] R. O. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall, 1973. Zbl0262.32005

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