Geometric structures in Topology, Geometry, Global Analysis and Dynamics

Abstract

Following Thurston's geometrisation picture in dimension three, we study geometric manifolds in a more general setting in arbitrary dimensions, with respect to the following problems: (i) The existence of maps of non-zero degree (domination relation or Gromov's order); (ii) The Gromov-Thurston monotonicity problem for numerical homotopy invariants with respect to the domination relation; (iii) The existence of Anosov diffeomorphisms (Anosov-Smale conjecture).

arXiv comment36 pages, survey chapter; v2: final version, to appear in the book "In the tradition of Thurston III", edited by K. Ohshika and A. Papadopoulos

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