Manifolds which do not Admit Anosov Diffeomorphisms

Abstract

In [3], M. W. Hirsch obtained some necessary conditions for the existence of an Anosov diffeomorphism on a differentiable manifold. As an application, he constructed many manifolds which do not admit Anosov diffeomorphisms.

Aspherical Products Which do not Support Anosov Diffeomorphisms

“…• Shiraiwa [Sh73] noted that an Anosov diffeomorphism with orientable stable (or unstable) distribution cannot induce the identity map on homology in all dimensions. It follows, for example, that spheres, lens spaces and projective spaces do not admit Anosov diffeomorphisms.…”

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

“…We deduce that y k−n M = ±λ · x k−n M , and so (17) f * (x k−n M × 1) = ±λ · (x k−n M × 1). As before, the last conclusion is impossible because M is negatively curved and m > 2.…”

“…In particular, H k (M; R) = 0. (c) (Shiraiwa [17]). If f : M −→ M is an Anosov diffeomorphism, then there is some k such that the induced homomorphism f * : H k (M; Q) −→ H k (M; Q) is not the identity.…”

Hofer growth of $C^1$-generic Hamiltonian flows

“…If deg(α) = n, we get that rank(E s ) = rank(E u ) = n. Hence rank(E c ) = 0. On the other hand f is isotopic to ½, so rank(E c ) > 0 (see [13]) and this is a contradiction. We get that 0 < deg(α) < n. This completes the proof.…”