Discreteness of Flux Groups

Let (M, .) be a closed symplectic 2n.dimensional manifold. Donaldson in his paper showed that there exist 2m.dimensional symplectic submanifolds (V 2 m, .) of (M,.), 1 . m . n . 1, with (m . 1).equivalent inclusions. On the basis of this fact we obtain isomorphic relations between kernel of Lefschetz map of M and kernels of Lefschetz maps of Donaldson submanifolds V 2 m, 2 . m . n.1. Then, using this relation, we show that the flux group of M is discrete if the action of .1(M) on .2(M) is trivial and there exists a retraction r : M . V , where V is a 4.dimensional Donaldson submanifold. And, in the symplectically aspherical case, we investigate the flux groups of the manifolds