University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Spring 2011 Homotopy mapping spaces Jeremy Brazas University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Brazas, Jeremy, "Homotopy mapping spaces" (2011). Doctoral Dissertations. 568. https://scholars.unh.edu/dissertation/568 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. HOMOTOPY MAPPING SPACES BY JEREMY BRAZAS B.S., Harding University, 2005 M.S.E., Harding University, 2006 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics May, 2011 UMI Number: 3467362 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMT Dissertation Publishing UMI 3467362 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. uest ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 This dissertation has been examined and approved. ^•f\TAi_ Thesis Director, Maria Basterra Associate Professor of Mathematics (rOannJ^ ($ZJJ-XV> L*/»V_^ David Feldman Associate Professor of Mathematics Eric Grinberg Professor of Mathematics Dmitri Nikshyeh Professor of Mathematics Sam Shore Professor of Mathematics l/mzoW Date DEDICATION To Clarice iii ACKNOWLEDGMENTS I would like to give special thanks to my advisor Dr. Maria Basterra for her wisdom, encouragement, and technical help throughout the research and writing process. I am indebted to the members of my thesis committee: Dr. D. Feld- man, Dr. E. Grinberg, Dr. D. Nikshych, and Dr. S. Shore who provided helpful suggestions and valuable comments. I would also like to express my gratitude to Dr. R. Hibschweiler for her advice and encouragement throughout my time at the University of New Hampshire and to Jan Jankowsky, April Flore, and Ellen O'Keefe for their patience and various forms of support. Many thanks are also due to my fellow graduate students: Todd Abel, Justin Greenough and Maryna Yeroshkina, for their friendship and helpful conversations and to Dr. Paul Fabel for many helpful conversations. I am so grateful for my best friend and wife Clarice who has supported me emotionally and financially throughout my graduate studies. This work would not have been possible without her. Thanks to all of my family and friends, especially my parents, for encouraging me to pursue what I love. Thanks are due to the UNH Department of Mathematics and Statistics for financial support in the form of teaching assistantships and travel funds. Thanks are also due to the UNH Graduate School for funding in the form of a Summer TA Fellowship, Dissertation Year Fellowship, and travel funds which all aided in the completion of this research. iv TABLE OF CONTENTS DEDICATION iii ACKNOWLEDGMENTS iv ABSTRACT vii INTRODUCTION 1 0.1 Notation 4 0.2 Outline 6 PRELIMINARIES: TOPOLOGY 9 1.1 Function spaces 9 1.1.1 Restricted paths and neighborhoods 14 1.1.2 A convenient basis for free path spaces 16 1.2 Quotient spaces 21 HOMOTOPY MAPPING SPACES 26 2.1 Homotopy mapping sets as quotient spaces 26 2.2 Multiplication in homotopy mapping spaces 30 2.2.1 H-spaces 30 2.2.2 co-H-spaces 40 2.3 Homotopy sequences 46 2.4 Discreteness of homotopy mapping spaces 51 2.5 Alternative topologies for homotopy mapping sets 57 2.5.1 The functor T and [X, Y]J 58 2.5.2 k-spaces, the functor k, and [X, Y]k, 60 v 2.5.3 The topological shape groups and n^h(X) 63 PATH COMPONENT SPACES 70 THE TOPOLOGICAL FUNDAMENTAL GROUP 85 4.1 The topological properties of n°p 85 4.1.1 Attaching cells 87 4.1.2 Discreteness and separation 92 4.1.3 Covering spaces and n°p 104 4.2 A computation of n^(L(X+)) 108 4.2.1 The spaces E(X+) 108 4.2.2 The fundamental group 7i1(Z(X+)) 112 p n x p 4.2.3 n° (L(X+)) = F R (n° (X)) 123 4.2.4 The weak suspension spaces wL(X+) and n^p(w£(X+)) .... 133 4.3 The topological properties of n°p(L(X+)) 135 T 4.4 The topological properties of n x 142 CONCLUSIONS AND FUTURE DIRECTIONS 161 APPENDIX 163 A.l Monoids with topology 164 A.2 Groups with topology 171 A.3 Topologies on free groups . .• 185 A.3.1 Free topological groups FM(X) 185 A.3.2 Reduction topologies PR(Y) 190 REFERNCES 211 VI ABSTRACT HOMOTOPY MAPPING SPACES by Jeremy Brazas University of New Hampshire, May 2011 Advisor: Dr. Maria Basterra In algebraic topology, one studies the group structure of sets of homotopy classes of maps (such as the homotopy groups nn(X)) to obtain information about the spaces in question. It is also possible to place natural topologies on these groups that remember local properties ignored by the algebraic structure. Upon choosing a topology, one is left to wonder how well the added topological structure interacts with the group structure and which results in homotopy theory admit topological analogues. A natural place to begin is to view the n-th homotopy group n„ (X) as the quotient space of the iterated loop space Q"(X) with the compact-open topology. This dissertation contains a systematic study of these quotient topologies, giving special attention to the fundamental group. The quotient topology is shown to be a complicated and somewhat naive ap­ proach to topologizing sets of homotopy classes of maps. The resulting groups with topology capture a great deal of information about the space in question but unfortunately fail to be a topological group quite often. Examples of this failure oc- vii curs in the context of a computation, namely, the topological fundamental group of a generalized wedge of circles. This computation introduces a surprising connec­ tion to the well-studied free Markov topological groups and indicates that similar failures are likely to appear in higher dimensions. The complications arising with the quotient topology motivate the introduction of well-behaved, alternative topologies on the homotopy groups. Some alterna­ tives are presented, in particular, free topological groups are used to construct the finest group topology on nn(X) such that the map Q"(X) —» nn(X) identifying ho­ motopy classes is continuous. This new topology agrees with the quotient topology precisely when the quotient topology does result in a topological group and admits a much nicer theory. vui INTRODUCTION The fact that classical homotopy theory is insufficient for studying spaces with homotopy type other than that of a CW-complex has motivated the introduction of a number of invariants useful for studying spaces with complicated local structure. For instance, in Cech theory, one typically approximates complicated spaces with "nice" spaces and takes the limit or colimit of an algebraic invariant evaluated on the approximating spaces. The approach taken in this dissertation is to directly transfer topological data to algebraic invariants by endowing them with natural topologies that behave nicely with respect to the algebraic structure. While this second approach does not yield purely algebraic objects, it does have the advantage of allowing direct application of the rich theory of topological algebra. The notion of "topologized" homotopy invariant seems to have been introduced by Hurewicz in [Hur35] and studied subsequently by Dugundji in [Dug50]. Whereas these early methods focused on "finite step homotopies" through open covers of spaces, we are primarily interested in the properties of spaces of homotopy classes of maps pen. The topological fundamental group 7i*op(X) of a based space (X, x), as first spec­ ified by Biss [Bis02], is the fundamental group n^Xx) endowed with the natural topology that arises from viewing it as a quotient space of the space of loops based at x. This choice of topological structure makes n^ particularly useful for studying 1 the homotopy of spaces that lack 1-connected covers. The brevity of this construc­ tion is rather deceiving since the topology of TC^P(X, X) is typically very complicated. In fact, for over 10 years [May90,Bis02], it was thought that this construction always results in a topological group. The initial intention of this research was to determine the validity of this assertion. We actually produce a plethora of counterexamples and shed light on a number of "defects" of the functor n°p. Recently, Fabel [Fab09] has shown that the Hawaiian earring group n^HE) fails to be a topological group independently of this work. In a first algebraic topology course, one learns early on that the fundamental group of a wedge of circles is the free group on the set indexing the wedge. One might similarly expect a generalized or "non-discrete" wedge of circles (constructed here as a suspension space £(X+)) to have topological fundamental group with some similar notion of "freeness." This computation is one of the main contributions of this dissertation to the theory of topological fundamental groups. A surprising consequence is that n^(L{X+)) either fails to be a topological group or is one of the well-studied but notoriously complicated free (Markov) topological groups.