Discrete homotopy theory and critical values of metric spaces

• Autorzy: Jim Conant , Victoria Curnutte , Corey Jones , Conrad Plaut , Kristen Pueschel , Maria Lusby , Jay Wilkins
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54E45: Compact (locally compact) metric spaces
• 54G20: Counterexamples

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 227
• Numer: 2
• Strony: 97-128

Daty

wydano

2014

Twórcy

autor

Jim Conant

• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.

autor

Victoria Curnutte

• 7078 W. Rainbow Rd., Sedalia, CO 80135, U.S.A.

autor

Corey Jones

• Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A.

autor

Conrad Plaut

• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.

autor

Kristen Pueschel

• Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, U.S.A.

autor

Maria Lusby

• 17 Sutherland Rd., Hicksville, NY 11801, U.S.A.

autor

Jay Wilkins

• Department of Mathematics and, Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, U.S.A.

Identyfikatory

DOI

10.4064/fm227-2-1