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

Part of: Classical Topics in Algebraic Topology Homology and cohomology theories Differential topology Dynamical systems with hyperbolic behavior Fiber spaces and bundles

Published online by Cambridge University Press:  18 October 2019

CHRISTOFOROS NEOFYTIDIS
CHRISTOFOROS NEOFYTIDIS*
Affiliation:
Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211Genève 4, Switzerland email Christoforos.Neofytidis@unige.ch
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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 dimension 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.

Keywords

Anosov diffeomorphismdirect productnegative curvaturerational homology sphere

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 55R10: Fiber bundles 57R19: Algebraic topology on manifolds 55M05: Duality
Secondary: 55N45: Products and intersections 55R37: Maps between classifying spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 553 - 569
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Copyright
© Cambridge University Press, 2019

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