Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

TL;DR

This paper demonstrates that certain products of negatively curved manifolds with rational homology spheres cannot support transitive Anosov diffeomorphisms, expanding understanding of dynamical constraints on manifold structures.

Contribution

It proves new non-existence results for transitive Anosov diffeomorphisms on specific product manifolds using cohomological methods.

Findings

Products of negatively curved manifolds with rational homology spheres lack transitive Anosov diffeomorphisms.

Extends non-existence results to products with vanishing simplicial volume.

Provides new examples of manifolds without transitive Anosov diffeomorphisms.

Abstract

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.