Discrete Homotopy Theory and Critical Values of Metric Spaces

TL;DR

This paper extends discrete homotopy theory to non-geodesic metric spaces, defining the critical spectrum Cr(X) and exploring its properties, including cases where discreteness fails and the impact of metric changes.

Contribution

It generalizes the concept of critical spectra to non-geodesic spaces and analyzes their topological invariance and behavior under metric modifications.

Findings

Discreteness of the critical spectrum can fail in various ways.

New critical values are determined by homotopy and refinement critical values.

Refinement critical values can often be removed via bi-Lipschitz metric changes.

Abstract

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3/2. The latter two spectra are known to be discrete for compact geodesic spaces, and correspond to the values at which certain special covering maps, called delta-covers (Sormani-Wei) or epsilon-covers (Plaut-Wilkins), change equivalence type. In this paper we initiate the study of these ideas for non-geodesic spaces, motivated by the need to understand the extent to which the accompanying covering maps are topological invariants. We show that discreteness of the critical spectrum for general metric spaces can fail in several ways, which we classify. The "newcomer" critical values for compact, non-geodesic spaces are completely determined by the homotopy critical values and refinement critical values, the latter of which can, in many cases, be removed by changing the metric in a bi-Lipschitz way.