Manifolds with higher homotopy which do not support Anosov diffeomorphisms

TL;DR

This paper demonstrates that certain closed manifolds with complex higher homotopy structures, including products of spheres with even dimensions, cannot support transitive Anosov diffeomorphisms, highlighting topological constraints.

Contribution

It establishes new topological obstructions to the existence of transitive Anosov diffeomorphisms on manifolds with non-trivial higher homotopy groups.

Findings

Finite products of spheres with at least one even-dimensional sphere do not support transitive Anosov diffeomorphisms.

Manifolds with non-trivial higher homotopy groups generally cannot support such dynamical systems.

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.