University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2006 Coarse Structures and Higson Compactification Christian Stuart Hoffland University of Tennessee, Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Mathematics Commons Recommended Citation Hoffland, Christian Stuart, "Coarse Structures and Higson Compactification. " Master's Thesis, University of Tennessee, 2006. https://trace.tennessee.edu/utk_gradthes/4485 This Thesis 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 Masters Theses 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 thesis written by Christian Stuart Hoffland entitled "Coarse Structures and Higson Compactification." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. Jurek Dydak, Major Professor We have read this thesis and recommend its acceptance: Nikolay Brodskiy, James Conant Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Christian Stuart Hoffland entitled "Coarse Structures and Higson Compactification." I have examined the finalpaper copy of this the­ sis for form and content and recommend that it be accepted in partial fulfillment of the requirements forthe degree of Master of Science, with a major in Mathematics . (� We have read this thesis and recommend its acceptance: Accepted forthe Council: Dean of Graduate Studies COARSE STRUCTURES AND HIGSON COMPACTIFICATION A thesis presented for the Master of Science degree University of Tennessee, Knoxville CHRISTIAN STUART HOFFLAND August2006 Copyright © 2006 Christian Stuart Hoffland Allrights reserved. ii To John, for everything, and to my parents, for always believing in me iii Acknowledgments I AM indebted to my supervisor, Professor Jurek Dydak, for assigning this thesis topic, and forhis guidance, advice, suggestions, corrections, and encouragement throughout. Conver­ sations about topology and geometry-as well as what is good forstudents to study-made a significant difference to this book, and without his involvement it would not have been possible to complete the text. I must thank him also for the opportunity to collaborate on work beyond the context of this thesis: many students are not as fortunateas I to have had such a supervisor. Thanks are extended to the larger thesis committee, comprising also Professors Nikolay Brodskiy and James Conant, whose comments and suggestions were invaluable: the com­ mitment to reading any thesis or dissertation deserves much appreciation. I must also thank them fortheir involvement in the topology seminar, which provided me with some very use­ fulideas for this text. Proofreadersfor any thesis are hard to come by, and I must express my gratitude to John Rumple forhelping with some of the text sections of this work. I can only hope that when proofreading his work that he will receive from me the same level of skill and patience he lavished on mine. Ziga Virk pointed out some errors in an early version, and his various suggestions proved extremely useful. Any remaining errors, however, are completely my responsibility. Conversations with Brendon LaBuz and Atish Mitra were helpfulin discussing possible counterexamples and eliminating potential errors: my thanks. Making �'IF,X do what one wants can sometimes be a hard task. Fortunately for me, Mike Saum has both the technical ability and saint-like patience to deal with the many, many queries and problems that came fromme on what seemed like a daily basis. The look of the text is very much the better for his presence in the department. Any graduate student in mathematics knows that the graduate secretary is the first per­ son to go for assistance. Pam Armentrout has provided support and advice in that capacity before I even arrived at the department, and has always been available since. To her goes the warmest appreciation forher dedication to the graduate students she serves. V Finally, my special thanks go again to my partner, John Rumple, for, among countless other things, his advice and thoughts on the process of writing. This,as one discovers when one begins any serious writing, is a lot harder than it looks, and his generosity in sharing his experience-and the experience of others-in the written word has benefitedthis small book in more ways than I can mention here. � vi Abstract F oLL ow ING John Roe in his Lectureson Coarse Geometry, we begin by describing the large­ scale structure of metric spaces by means of coarse maps between them, those being maps which preserve distances at large scales. Using these techniques, we demonstrate that the real numbers and the integers have the same large scale structure-or are coarsely equivalent­ but that the real line is coarsely equivalent to neither the Euclidean plane nor the set of positive real numbers. Following a generalization of these concepts forgeneral topological spaces with the introduction of an abstract coarse structure on the space, we show, among other things, that the real line is not coarsely equivalent to the long line of the countable ordinals with the order topology. We depart fromRoe to describe a connection between locally compact Hausdorff spaces and a sub-class of Banach algebras known as C* -algebras, where we findthat every such algebra can be described as a set of continuous functions on a particular locally compact Hausdorffspace. In particular, we see that there is a very strong relationship between the two: the categoriesof C* -algebras and * -homomorphisms, and the opposite of locally com­ pact Hausdorffspaces and continuous functions, are dual. Returning to coarse structures, we examine compactifications of locally compact Haus­ dorffspaces with a view to construction of a topological coarse structure, one in which those maps which are coarse are precisely those which may be continuously extended to the bound­ ary of the space. We complete our investigation by describing the inverse process: given a coarse structure, can we finda compactificationpossessing the same properties? We provide an partial answer to this question, called a Higson compactification, and end by calculating Higson compactifi.cationsof some familiar spaces. � vii Preface THE typical study of metric or topological spaces involves examination of the small-scale structure of the space in varying ways. The quintessential property, continuity, depends on either functionalvalues taken at small distances, or on the inverse image of a neighborhood: that is to say, small-scale properties. To study the large-scale structure of a metric or topological space, one naturally becomes concerned with properties which hold at large scales, such as boundedness, degreesof free­ dom, and restriction of movement: continuity becomes correspondingly less important, since it has little impact on these qualities of a space. Coarse geometry provides a set of tools for discussion oflarge-scale structure by consideration of maps which preserve these properties. This thesis examines the foundations of coarse geometry, with a particular emphasis in describing a certain compactificationof a locally compact Hausdorffspace , by following John Roe in [16). Although it is assumed that the reader has a basic knowledge of geometry and topology, many of the terms used are defined either in the textitself, or in the Appendix. Care has been taken to ensure that any topological space or structure which might be unfamiliar is given at least a brief definition. Particular attention is given to descriptions of the long line and the sphere at infinitysince these spaces are familiar probably only to those who have taken a course in topology. Examples have been provided throughout to illustrate the ideas presented. Any back­ ground material which was thought necessary-but not central to the main arguments-has been included in an extensive appendix to the text, including a summary of some ideas from category theory. The firsttwo chapters outline the basic principles of coarse geometry, firstby definingthe required concepts formetric spaces, and then generalizing them for, primarily, topological spaces. Some knowledge of metric and topological spaces is assumed: [5) provides an excel­ lent introduction to the former,while the firstfew chapters of [ 14] would provide the rest of what is required. Familiarity with C* -algebras is not assumed, and since this material was deemed impor­ tant enough for inclusion, an entire chapter on it is justifiable. There were two reasons for ix providing the material. Thefirst is that C*-algebras are generally studied in the context of functional analysis, and as a result, geometers or topologists may have had no cause to be­ come acquainted with the subject. Since this text is primarily of interest to them, an account has been given of the basic facts connected with C* -algebras. The second reason is the very strong connection between the compactifications of locally compact Hausdorffspaces and C*-algebras which Roe assumes in [16), and is vital for an understanding of most of the fourthchapter. The third chapter can be skipped if the reader is conversant with the main concepts involved. In the fourthchapter some familiaritywith locally compact Hausdorffspaces is assumed, although introductory material on these is again included in the Appendix. All proofs and figures are mine unless otherwise stated. When a proof is not original, or if it is an adaptation of another author's proof, the reference is given in the text. � August 1, 2006 Christian Stuart Hoffland Knoxville, Tennessee X Contents 1 Elementary Coarse Geometry 1 1.1 Choice of metric .