Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups

We present the first known nontrivial topological obstructions to the existence of partially hyperbolic diffeomorphisms. In particular, we show that there are no partially hyperbolic diffeomorphisms on the 3-sphere. More generally, we show that for a partially hyperbolic diffeomorphismof a 3-manifoldwith an Abelian fundamental group, the induced action in the first homology group is partially hyperbolic. This improves the results of [4] by dropping the assumption of dynamical coherence.