Discrete Homotopy Theory and Critical Values of Metric Spaces

@article{Conant2012DiscreteHT,
  title={Discrete Homotopy Theory and Critical Values of Metric Spaces},
  author={James F. Conant and Victoria Curnutte and Corey Jones and Conrad Plaut and Kristen Pueschel and Maria Walpole and Jay Wilkins},
  journal={arXiv: Metric Geometry},
  year={2012}
}
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… 

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