View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Tennessee, Knoxville: Trace University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2011 Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces Leonard Duane Wilkins [email protected] Recommended Citation Wilkins, Leonard Duane, "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces. " PhD diss., University of Tennessee, 2011. https://trace.tennessee.edu/utk_graddiss/1039 This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Leonard Duane Wilkins entitled "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Conrad P. Plaut, Major Professor We have read this dissertation and recommend its acceptance: James Conant, Fernando Schwartz, Michael Guidry Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) To the Graduate Council: I am submitting herewith a dissertation written by Leonard Duane Wilkins entitled \Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Conrad Plaut, Major Professor We have read this dissertation and recommend its acceptance: James Conant Fernando Schwartz Michael Guidry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with offical student records.) Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces A Dissertation Presented for the Doctor of Philosophy Mathematics The University of Tennessee, Knoxville Leonard Duane Wilkins, Jr. May 2011 Acknowledgments I am very grateful to the members of my doctoral committee - Dr. Jim Conant, Dr. Fernando Schwarz, and Dr. Michael Guidry - for their advice and assistance. I am also deeply indebted to my family and my close friends - Anne and Duane Wilkins, Jill Wilkins, Christine Kraft, Shannon Mathis, Nicholas Underwood, and many others - for their patience and support during this process. Finally, none of this would have been possible without the guidance and support of my doctoral advisor, Dr. Conrad Plaut. He has been an invaluable source of information, advice, and knowledge. I will forever be thankful for having had the opportunity to work with him. ii Abstract Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines and develops discrete homotopy theory for metric spaces, which can be thought of as a discrete analog of classical path-homotopy and covering space theory. Given a metric space, X, this leads to the construction of a collection of covering spaces of X - and corresponding covering groups - parameterized by the positive real numbers, which we call the -covers [epsilon-covers] and the -groups [epsilon-groups]. These covers and groups evolve dynamically as the parameter decreases, changing topological type at specific parameter values which depend on the topology and local geometry of X. This leads to the definition of a critical spectrum for metric spaces, which is the set of all values at which the topological type of the covers change. Several results are proved regarding the critical spectrum and its connections to topology and local geometry, particularly in the context of geodesic spaces, refinable spaces, and Gromov-Hausdorff limits of compact metric spaces. We investigate the relationship between the critical spectrum and covering spectrum in the case when X is geodesic, connections between the geometry of the -groups [epsilon-groups] and the metric and topological structure of the -covers [epsilon-covers], as well as the behavior of the -covers [epsilon-covers] and critical values under Gromov-Hausdorff convergence. iii Contents 1 Introduction and Background 1 1.1 Introduction . 1 1.2 Metric Geometry Basics . 2 1.3 Uniform Spaces and Entourage Covers . 6 1.4 Spanier Covers and the Covering Spectrum . 10 2 Discrete Homotopy Theory and the Critical Spectrum 15 2.1 Basic Definitions . 15 2.2 The -covers . 17 2.3 Critical Values of Metric Spaces . 25 2.4 Metric, Topological, and Lifting Properties of X ....................... 30 2.5 The -intrinsic Property . 34 2.6 Group Action of π(X) on X .................................. 38 2.7 Convergence Results in X .................................... 43 3 The Critical and Covering Spectra 46 3.1 Path Lifting . 46 3.2 Covering Comparison for Geodesic Spaces . 55 3.3 Spectral Comparison . 57 4 Some Examples 59 4.1 Essential Gaps . 59 4.2 Variations on a Theme . 62 4.3 A Space with a Dense Critical Spectrum . 68 5 Refinability and -group Geometry 79 5.1 Refinability and Generators of π(X).............................. 79 5.2 Refinability and Proper -covers . 81 6 Convergence of -covers and Critical Values 93 6.1 More on Gromov-Hausdorff Convergence . 93 6.2 Structure of the Limit Cover . 95 6.3 Precompactness Results . 102 6.4 Convergence of Critical Values . 110 Bibliography 118 Vita 120 iv List of Figures 1.1 Loops generating π1(X; ∗; U ).................................. 11 2.1 Geodesic Circle with a Gap (the -covers) . 22 2.2 The Tuning Fork . 36 4.1 An essential gap. 61 4.2 Rapunzel's Comb . 63 4.3 Rapunzel's Comb - Variation 1 . 66 4.4 Unit Interval Dyadic Rationals . 71 4.5 The Basic Piece . 72 4.6 The Imbedded Basic Piece . 73 4.7 Organization and Attachment Order of the Basic Pieces . 74 5.1 An -ladder . 90 v Chapter 1 Introduction and Background 1.1 Introduction Spectral analysis has long been a very useful tool for studying objects in mathematics, from the eigenvalue spectrum students encounter in a basic linear algebra course to the operator spectra studied in functional analysis. Several of these spectra have proved useful in the study of geometry, including the eigenvalue spectrum of the Laplacian operator on a Riemannian manifold and the length spectrum of a geodesic space. In recent years, Christina Sormani and Guofang Wei developed a covering spectrum for geodesic spaces ([13],[14]), which is somewhat unique among other spectra in that it can be considered both a geometric and topological spectrum. More specifically, the covering spectrum detects fundamental group generators of geodesic spaces, and the values in this spectrum are directly related to the diameters of the loops that generate the fundamental group. The work of Sormani and Wei grew out of their efforts to show that Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature uniformly bounded below have categorical universal covers ([12],[13],[14],[15]). This effort is part of another thread of research that has gained momentum in recent years, namely the geometric and topological study of metric spaces that are not necessarily smooth and may have singular structure. Classical differential geometry focuses on the study of smooth manifolds with various metric structures (e.g. Riemannian, Finsler, simplectic, etc.), and, in this context, one has a very rich structure of analytic and algebraic tools at his or her disposal. Recent uses of non-smooth spaces in various engineering and technological applications, however, have prompted the need for a more systematic study of singular geometry and topology. For instance, certain fractals have been used recently as design models for antennae, since such a structure allows a greater length to be enclosed in a more compact space. As another example, network and optimal transport theorists who study the flow of various materials and information between points in a space often use \simple looking" spaces as their underlying structures, but the metrics they employ are not always standard or intuitive and can lead to strange phenomena, such as the space having no rectifiable curves. The Gromov-Hausdorff limits studied by Sormani and Wei make up just a small part of the class of metric spaces that one might consider non-standard, since passing to a limit space may induce singular topology and/or geometry. Their goal of finding universal covers for these spaces is really a special case of the general effort to understand the topology of singular spaces, since the existence of such a cover implies certain topological properties of the space in question. 1 Following a similar goal, and around the same time, Conrad Plaut and Valera Berestovskii began an effort to generalize the classical universal cover of a topological space in the context of uniform spaces, a general class of spaces that includes, for example, metric spaces and topo- logical groups ([1],[8]). Their efforts actually began several years earlier when they undertook a study