  We show that various classes of products of manifolds do not support
transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology
class, we prove that the product of a negatively curved manifold with a
rational homology sphere does not support transitive Anosov diffeomorphisms. We
extend this result to products of finitely many negatively curved manifolds of
dimensions at least three with a rational homology sphere that has vanishing
simplicial volume. As an application of this study, we obtain new examples of
manifolds that do not support transitive Anosov diffeomorphisms, including
certain manifolds with non-trivial higher homotopy groups and certain products
of aspherical manifolds.
