Abstract
In this paper, we introduce a new characterisation of closed symplectically aspherical manifolds which can be understood as the symplectic analogue of the Shafarevich conjecture.
Similar content being viewed by others
References
M. Adachi, Embeddings and Immersions. Transl. Math. Monogr. 124, Amer. Math. Soc., Providence, RI, 1993.
Y. Eliashberg and N. Mishachev, Introduction to the h-Principle. Grad. Stud. Math. 48, American Mathematical Society, Providence, RI, 2002.
M. Fernández, V. Muñoz and J. A. Santisteban, Symplectically aspherical manifolds with nontrivial \({\pi_{2}}\) and with no Kähler metrics. In: Contemporary Geometry and Related Topics (Belgrade, Yugoslavia, 2002), World Scientific, River Edge, NJ, 2004, 187–200,
A. Floer, Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120 (1989), 575–611.
S. Gal and J. Kędra, A two-cocycle on the group of symplectic diffeomorophisms. Math. Z. 271 (2012), 693–706.
R. E. Gompf, A new construction of symplectic manifolds. Ann. of Math. (2) 142 (1995), 527–595.
R. E. Gompf, Symplectically aspherical manifolds with nontrivial \({\pi_{2}}\) . Math. Res. Lett. 5 (1998), 599–603.
M. Gromov, Partial Differential Relations. Springer-Verlag, Berlin, 1986.
A. Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002.
H. Hofer, Lusternik-Schnirelmann theory for Lagrangian intersections. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), 465–499.
R. Ibáñez, J. Kȩdra, Yu. Rudyak and A. Tralle, On fundamental groups of symplectically aspherical manifolds. Math. Z. 248 (2004), 805–826.
J. Kȩdra, Yu. Rudyak and A. Tralle, On fundamental groups of symplectically aspherical manifolds. II. Abelian groups. Math. Z. 256 (2007), 825–835.
J. Kollár, Shafarevich Maps and Automorphic Forms. Princeton University Press, Princeton, NJ, 1995.
G. Lupton and J. Oprea, Cohomologically Symplectic Spaces: Toral Actions and the Gottlieb Group. Trans. Amer. Math. Soc. 347 (1995), 261–288.
D. McDuff and D. Salamon, Introduction to Symplectic Topology. 2nd ed., The Clarendon Press, Oxford University Press, New York, 1998.
D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology. Amer. Math. Soc., Providence, RI, 2004.
Y. Rudyak and J. Oprea, On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture. Math. Z. 230 (1999), 673–678.
Y. Rudyak and A. Tralle, On symplectic manifolds with aspherical symplectic form. Topol. Methods Nonlinear Anal. 14 (1999), 353–361.
R. I. Shafarevich, Basic Algebraic Geometry. Nauka, Moscow, 1972 (in Russian).
H. Whitney, Differentiable manifolds. Ann. of Math. (2) 37 (1936), 645–680.
H. Whitney, The self-intersections of a smooth n-manifold in 2n-space. Ann. of Math. (2) 45 (1944), 220–246.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Borat, A. A characterisation of symplectically aspherical manifolds. J. Fixed Point Theory Appl. 17, 477–482 (2015). https://doi.org/10.1007/s11784-014-0194-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-014-0194-z