Abstract
We say that a simply connected space X is pre-c-symplectic if it is the fibre of a rational fibration \({X \to Y \to \mathbb{C}P^{\infty}}\) where Y is cohomologically symplectic in the sense that there is a degree 2 cohomology class which cups to a top class. It is a rational homotopical property but not a cohomological one. By using Sullivan’s minimal models (Félix et al. in Rational homotopy theory. Graduate Texts in Mathematics, vol. 205. Springer, Berlin, 2001), we give the necessary and sufficient condition that the product of odd-spheres \({X=S^{k_1}\times \cdots \times S^{k_n}}\) is pre-c-symplectic and see some related topics. Also we give a charactarization of the Hasse diagram of rational toral ranks for a space X (Yamaguchi in Bull Belg Math Soc Simon Stevin 18:493–508, 2011) as a necessary condition to be pre-c-symplectic and see some examples in the cases of finite-oddly generated rational homotopy groups.
Article PDF
Similar content being viewed by others
References
Arkowitz, M.: Introduction to Homotopy Theory. Springer Universitext (2011)
Allday C., Halperin S.: Lie group actions on spaces of finite rank. Quart. J. Math. Oxford 29(2), 63–76 (1978)
Allday C., Puppe V.: Cohomological Methods in Transformation Groups. Cambridge University Press, Cambridge (1993)
Audin M.: Examples de variétés presque complexes. Enseign Math. 37, 175–190 (1991)
Bazzoni, G., Fernández, M., Muñoz, V.: Non-formal co-symplectic manifolds. arXiv:1203.6422
Bazzoni G., Muñoz V.: Classification of minimal algebras over any field up to dimension 6. Trans. AMS 364, 1007–1028 (2012)
Félix Y., Halperin S., Lemaire J.M.: The rational LS category of products and of Poincaré duality complexes. Topology 37, 749–756 (1998)
Félix, Y., Halperin, S., Thomas, J.C.: Rational homotopy theory. Graduate Texts in Mathematics, vol. 205. Springer, Berlin (2001)
Félix, Y. Oprea, J., Tanré, D.: Algebraic models in geometry. GTM, vol. 17. Oxford (2008)
Halperin, S.: Rational homotopy and torus actions. Aspects of topology, pp. 293–306. Cambridge University Press, Cambridge (1985)
Hilton, P., Mislin, G., Roitberg, J.: Localization of nilpotent groups and spaces. North-Holland Mathematical Studies, vol. 15 (1975)
Hajduk, B., Walczak, R.: Presymplectic manifolds. arXiv:0912.2297v2
Jessup B., Lupton G.: Free torus actions and two-stage spaces. Math. Proc. Camb. Philos. Soc. 137(1), 191–207 (2004)
Karshon Y., Tolman S.: The moment map and line bundles over presymplectic toric manifolds. J. Differ. Geometry 38, 465–484 (1993)
Kedra J.: KS-models and symplectic structures on total spaces of bundles. Bull. Belg. Math. Soc. Simon Stevin 7, 377–385 (2000)
Kedra J., McDuff D.: Homotopy properties of Hamiltonian group actions. Geom. Topol. 9, 121–162 (2005)
Kotani Y., Yamaguchi T.: Rational toral ranks in certain algebras. IJMMS 69, 3765–3774 (2004)
Kuribayashi K.: On extensions of a symplectic class. Differ. Geom. Appl. 29, 801–815 (2011)
Lalonde F., McDuff D.: Symplectic structures on fibre bundles. Topology 42, 309–347 (2003)
Lupton G., Oprea J.: Symplectic manifolds and formality. JPAA 91, 193–207 (1994)
Lupton G., Oprea J.: Cohomologically symplectic spaces: toral actions and the Gottlieb group. Trans AMS 347, 261–288 (1995)
McDuff, D., Salamon, D.: Introduction to symplectic toplogy. Oxford Mathematical Monographs (1995)
Mimura, M.: Homotopy theory of Lie groups. Handbook of Algebraic Topology, Chap. 19, pp. 951–991 (1995)
Mimura M., Shiga H.: On the classification of rational homotopy types of elliptic spaces with homotopy Euler characteristic zero for dim < 8. Bull. Belg. Math. Soc. Simon Stevin 18, 925–939 (2011)
Nakamura O., Yamaguchi T.: Lower bounds of Betti numbers of elliptic spaces with certain formal dimensions. Kochi J. Math. 6, 9–28 (2011)
Oda S.: On bounding problems in totally ordered commutative semi-groups. J. Algebra Numb. Theory Acad. 2, 301–311 (2012)
Shiga H., Yamaguchi T.: Principal bundle maps via rational homotopy theory. Publ. Res. Inst. Math. Sci. 39(1), 49–57 (2003)
Sullivan D.: Infinitesimal computations in topology. Publ. IHES 47, 269–331 (1977)
Tralle, A., Oprea, J.: Symplectic manifolds with no Kähler structure. LNM, vol. 1661. Springer, Berlin (1997)
Thurston W.P.: Some simple examples of symplectic manifolds. Proc. AMS 55, 467–468 (1976)
Yamaguchi T.: A Hasse diagram for rational toral ranks. Bull. Belg. Math. Soc. Simon Stevin 18, 493–508 (2011)
Yamaguchi T.: Examples of a Hasse diagram of free circle actions in rational homotopy. JP J. Geom. Topol. 11(3), 181–191 (2011)
Yamaguchi, T.: Examples of rational toral rank complex. IJMMS 2012, Article ID 867247 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sato, J., Yamaguchi, T. Pre-c-symplectic condition for the product of odd-spheres. J. Homotopy Relat. Struct. 8, 13–34 (2013). https://doi.org/10.1007/s40062-012-0011-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40062-012-0011-6
Keywords
- Symplectic
- c-Symplectic
- Pre-c-symplectic
- Sullivan model
- Rational homotopy type
- Almost free toral action
- Rational toral rank
- Hasse diagram of rational toral ranks
- KS-model
- Elliptic
- Formal