Discreteness of Flux Groups

Let （M, ω） be a closed symplectic 2n-dimensional manifold. Donaldson in his paper showed that there exist 2m-dimensional symplectie submanifolds （V^2m,ω） of （M,ω）, 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^2m, 2 ≤ m ≤ n - 1. Then, using this relation, we show that the flux group of M is discrete if the action of π1 （M） on π2（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.

Discreteness of Flux Groups

Let (M, ω) be a closed symplectic 2n–dimensional manifold. Donaldson in his paper showed that there exist 2m–dimensional symplectic submanifolds (V ^{2 m} , ω) of (M,ω), 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 π₁(M) on π₂(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. This work is supported by Korea Research Foundation, Grant (KRF 2002-070-C0010)