Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces

Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines and develops discrete homotopy theory for metric spaces, which can be thought of as a discrete analog of classical path-homotopy and covering space theory. Given a metric space, X, this leads to the construction of a collection of covering spaces of X - and corresponding covering groups - parameterized by the positive real numbers, which we call the [epsilon]-covers and the [epsilon]-groups. These covers and groups evolve dynamically as the parameter decreases, changing topological type at specific parameter values which depend on the topology and local geometry of X. This leads to the definition of a critical spectrum for metric spaces, which is the set of all values at which the topological type of the covers change. Several results are proved regarding the critical spectrum and its connections to topology and local geometry, particularly in the context of geodesic spaces, refinable spaces, and Gromov-Hausdorff limits of compact metric spaces. We investigate the relationship between the critical spectrum and covering spectrum in the case when X is geodesic, connections between the geometry of the [epsilon]-groups and the metric and topological structure of the [epsilon]-covers, as well as the behavior of the [epsilon]-covers and critical values under Gromov-Hausdorff convergence.