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Flux conjecture on symplectic submanifolds. (English) Zbl 1099.53056
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 7th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 2–10, 2005. Sofia: Bulgarian Academy of Sciences (ISBN 954-8495-30-9/pbk). 89-97 (2006).

Let (M,ω) be a compact symplectic 2n-dimensional manifold, and let F ω :π 1 (Symp 0 (M))H 1 (M,) be the flux homomorphism defined by (F ω (ϕ),α)=(ω,ϕ t α), where Symp 0 (M) is the identity component of the symplectomorphism group of M. Then, the flux group Γ M of M is defined to be the image of F ω . It is a discrete subgroup of H 1 (M,) if and only if the Hamiltonian diffeomorphism group Ham(M) is closed in Symp 0 (M). The flux conjecture is the statement that Γ M is discrete. In this paper, first by using results of F. Lalonde, D. McPuff and L. Polterovich [On the flux conjectures, CRM Proc. Lect. Notes 15, 69–85 (1998; Zbl 0974.53062)], and D. Gottlieb [Am. J. Math. 87, 840–856 (1965; Zbl 0148.17106)], Γ M is shown to be a subgroup of the kernel of the Lefschetz map Lef M if the Euler number of M is not equal to zero (Th. 1.1. proved in §2). Here Lef M :H 1 (M,)H 2n-1 (M,) is defined by Lef M (a)=aω n-1 .

Then, applying this result to Donaldson submanifolds (V 2m ,ω), 1mn-1, of (M,ω) [cf. S. Donaldson, J. Differ. Geom. 44, No. 4, 666–705 (1996; Zbl 0883.53032)], and use authors previous results [Acta Math. Sin., Engl. Ser. 22, No. 1, 115–122 (2006; Zbl 1227.53090)] and results of F. Lalonde, D. McPuff and L. Porterovich [Invent. Math. 135, No.2, 369–385 (1999; Zbl 0907.58004)], flux conjecture is shown to hold for all Donaldson submanifolds of M if the first Betti number of M is equal to one (Th. 1.5. proved in §3). Applications of these results to the symplectic fibration over S 2 with fiber (M,ω) associated with ϕπ 1 (Symp 0 (M)) are also studied (Cor. 1.8. proved in §4).


MSC:
53D35Global theory of symplectic and contact manifolds
57R17Symplectic and contact topology