Discrete homotopy theory and critical values of metric spaces

Jim Conant, et al. "Discrete homotopy theory and critical values of metric spaces." Fundamenta Mathematicae 227.2 (2014): 97-128. <http://eudml.org/doc/283023>.

@article{JimConant2014,
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 δ-covers (Sormani-Wei) or ε-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 the refinement critical values, the latter of which can, in many cases, be removed by changing the metric in a bi-Lipschitz way.},
author = {Jim Conant, Victoria Curnutte, Corey Jones, Conrad Plaut, Kristen Pueschel, Maria Lusby, Jay Wilkins},
journal = {Fundamenta Mathematicae},
keywords = {metric space; discrete homotopy; critical spectrum; homotopy critical value},
language = {eng},
number = {2},
pages = {97-128},
title = {Discrete homotopy theory and critical values of metric spaces},
url = {http://eudml.org/doc/283023},
volume = {227},
year = {2014},
}

TY - JOUR
AU - Jim Conant
AU - Victoria Curnutte
AU - Corey Jones
AU - Conrad Plaut
AU - Kristen Pueschel
AU - Maria Lusby
AU - Jay Wilkins
TI - Discrete homotopy theory and critical values of metric spaces
JO - Fundamenta Mathematicae
PY - 2014
VL - 227
IS - 2
SP - 97
EP - 128
AB - 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 δ-covers (Sormani-Wei) or ε-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 the refinement critical values, the latter of which can, in many cases, be removed by changing the metric in a bi-Lipschitz way.
LA - eng
KW - metric space; discrete homotopy; critical spectrum; homotopy critical value
UR - http://eudml.org/doc/283023
ER -