Title:

Remarks on symplectically aspherical manifolds

Symplectically aspherical manifolds rst appeared in the work of Floer where he proved a version of the Arnold Conjecture. Since then their topological properties have been studied. One of the results of the present thesis are new characterisations of symplectically aspherical manifolds. Namely, we prove in Chapter 4 that a closed symplectic manifold is symplectically aspherical if and only if one of the following conditions hold: Its universal cover can be symplectically embedded into the standard sym plectic Euclidean space. Its fundamental group is large (see De nition 4.7 for the de nition of large ness). The latter condition has a well known counterpart in algebraic geometry. It has been conjectured by Shafarevich (see [32]) that a closed algebraic variety has a Stein universal cover if and only if its fundamental group is large (in the algebraic sense). A manifold is Stein if it is holomorphically embedded into the standard complex vector space. Thus the characterisations of symplectically aspherical manifolds mentioned above prove a symplectic analog of the Shafarevich conjecture. One may ask whether the universal cover of a symplectically aspherical manifold is Stein. In Chapter 5, we construct an example of a closed symplectically aspherical manifold whose universal cover is not Stein. This is the second main result of the thesis. In Chapter 6, we focus on the properties of the fundamental group of symplectically aspherical manifolds. The main result here is to relate the Flux group of a symplecti cally aspherical manifold with its geometric properties. More precisely, we prove that if the Flux group is nontrivial then the manifold is not symplectically hyperbolic. It is not known that if a nontrivial free product of groups can be realised as the fundamental group of a closed symplectically aspherical manifold. In Chapter 5, we construct an example of a closed symplectically aspherical manifold whose universal cover is not Stein. This is the second main result of the thesis.In Chapter 6, we focus on the properties of the fundamental group of symplectically aspherical manifolds. The main result here is to relate the Flux group of a symplecti cally aspherical manifold with its geometric properties. More precisely, we prove that if the Flux group is nontrivial then the manifold is not symplectically hyperbolic. It is not known that if a nontrivial free product of groups can be realised as the fundamental group of a closed symplectically aspherical manifold. In Chapter 5, we investigate a more general problem to obtain some partial results. The remaining parts of the thesis introduce symplectic and symplectically aspherical manifolds and review what is known in the subject.
