Manifolds with higher homotopy which do not support Anosov   diffeomorphisms
  We show that various classes of closed manifolds with non-trivial higher homotopy groups do not support (transitive) Anosov diffeomorphisms. In particular we show that a finite product of spheres at least one of which is even-dimensional does not support transitive Anosov diffeomorphisms. 
Andrey Gogolev and Federico Rodriguez Hertz
math.DS