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Discreteness of flux groups. (English) Zbl 1227.53090

Summary: Let \((M,\omega)\) be a closed symplectic \(2n\)-dimensional manifold. S. K. Donaldson [J. Differ. Geom. 44, No. 4, 666–705 (1996; Zbl 0883.53032)] showed that there exist \(2m\)-dimensional symplectic submanifolds \((V^{2m},\omega)\) of \((M,\omega)\), \(1\leq m\leq n-1\), with \((m-1)\)-equivalent inclusions. On the basis of this fact, we obtain isomorphic relations between the kernel of the Lefschetz map of \(M\) and kernels of Lefschetz maps of Donaldson submanifolds \(V^{2m}\), \(2\leq m\leq n-1\). Then, using this relation, we show that the flux group of \(M\) is discrete if the action of \(\pi_1(M)\) on \(\pi_2(M)\) is trivial and there exists a retraction \(r\colon M\to V\), where \(V\) is a 4-dimensional Donaldson submanifold. Also, in the symplectically aspherical case, we investigate the flux groups of the manifolds.

MSC:

53D35 Global theory of symplectic and contact manifolds
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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