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The revised and uniform fundamental groups and universal covers of geodesic spaces
 Topology Appl
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Wilkins Discrete Homotopy Theory and Critical Values of Metric Spaces
, 2012
"... Abstract. Utilizing the discrete homotopy methods developed for uniform spaces by BerestovskiiPlaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the nongeodesic case the covering spectrum defined by SormaniWei and the homotopy critical spectrum defined by PlautWilkin ..."
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Abstract. Utilizing the discrete homotopy methods developed for uniform spaces by BerestovskiiPlaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the nongeodesic case the covering spectrum defined by SormaniWei and the homotopy critical spectrum defined by PlautWilkins. 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 δcovers (SormaniWei) or εcovers (PlautWilkins), change equivalence type. In this paper we initiate the study of these ideas for nongeodesic 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, nongeodesic 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 biLipschitz way. Key words and phrases: metric space, discrete homotopy, critical spectrum, homotopy critical value,