Geometric structures in Topology, Geometry, Global Analysis and Dynamics

TL;DR

This paper explores the existence and properties of geometric manifolds across various dimensions, focusing on domination relations, monotonicity of invariants, and Anosov diffeomorphisms, extending Thurston's geometrisation principles.

Contribution

It generalizes Thurston's geometrisation framework to higher dimensions, addressing key problems in topology, geometry, and dynamical systems related to domination, invariants, and Anosov diffeomorphisms.

Findings

Conditions for existence of non-zero degree maps between manifolds

Monotonicity properties of homotopy invariants under domination

Results related to the existence of Anosov diffeomorphisms in higher dimensions

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).