Discrete homotopy theory and critical values of metric spaces

Jim Conant¹ ; Victoria Curnutte² ; Corey Jones³ ; Conrad Plaut¹ ; Kristen Pueschel⁴ ; Maria Lusby⁵ ; Jay Wilkins⁶

¹ Department of Mathematics University of Tennessee Knoxville, TN 37996, U.S.A.
² 7078 W. Rainbow Rd. Sedalia, CO 80135, U.S.A.
³ Department of Mathematics Vanderbilt University Nashville, TN 37240, U.S.A.
⁴ Department of Mathematics Cornell University Ithaca, NY 14853-4201, U.S.A.
⁵ 17 Sutherland Rd. Hicksville, NY 11801, U.S.A.
⁶ Department of Mathematics and Computer Science University of North Carolina at Pembroke Pembroke, NC 28372, U.S.A.

Fundamenta Mathematicae, Tome 227 (2014) no. 2, pp. 97-128.

Résumé

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii–Plaut, we define the critical spectrum ${\rm Cr}(X)$ 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, ${\rm Cr}(X)$ 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 $\varepsilon $-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.

Détail

DOI : 10.4064/fm227-2-1

Keywords: utilizing discrete homotopy methods developed uniform spaces berestovskii plaut define critical spectrum metric space generalizing non geodesic covering spectrum defined sormani wei homotopy critical spectrum defined plaut wilkins geodesic the homotopy critical spectrum which differs covering spectrum factor latter spectra known discrete compact geodesic spaces correspond values which certain special covering maps called delta covers sormani wei varepsilon covers plaut wilkins change equivalence type paper initiate study these ideas non geodesic spaces motivated understand extent which accompanying covering maps topological invariants discreteness critical spectrum general metric spaces fail several ways which classify newcomer critical values compact non geodesic spaces completely determined homotopy critical values refinement critical values latter which many cases removed changing metric bi lipschitz