Manifolds with higher homotopy which do not support Anosov diffeomorphisms

Abstract

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.

18 pages; We thank the referee for suggestions that resulted in a better exposition in the second version

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