Geometric structures in Topology, Geometry, Global Analysis and Dynamics
arXiv:2211.06803 · doi:10.1007/978-3-031-43502-7_9
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).
36 pages, survey chapter; v2: final version, to appear in the book "In the tradition of Thurston III", edited by K. Ohshika and A. Papadopoulos