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Pro-Covering Fibrations of the Hawaiian Earring
Brigham Young University BYU ScholarsArchive Theses and Dissertations 2014-12-01 Pro-Covering Fibrations of the Hawaiian Earring Nickolas Brenten Callor Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mathematics Commons BYU ScholarsArchive Citation Callor, Nickolas Brenten, "Pro-Covering Fibrations of the Hawaiian Earring" (2014). Theses and Dissertations. 4324. https://scholarsarchive.byu.edu/etd/4324 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. Pro-Covering Fibrations of the Hawaiian Earring Nickolas B. Callor A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Greg Conner, Chair Jessica Purcell Eric Swenson Department of Mathematics Brigham Young University December 2014 Copyright c 2014 Nickolas B. Callor All Rights Reserved abstract Pro-Covering Fibrations of the Hawaiian Earring Nickolas B. Callor Department of Mathematics, BYU Master of Science Let H be the Hawaiian Earring, and let H denote its fundamental group. Assume (Bi) is an inverse system of bouquets of circles whose inverse limit is H. We give an explicit bijection between finite normal covering spaces of H and finite normal covering spaces of Bi. This bijection induces a correspondence between a certain family of inverse sequences of these covering spaces. The correspondence preserves the inverse limit of these sequences, thus offering two methods of constructing the same limit. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Introduction to Algebraic Topology MAST31023 Instructor: Marja Kankaanrinta Lectures: Monday 14:15 - 16:00, Wednesday 14:15 - 16:00 Exercises: Tuesday 14:15 - 16:00
Introduction to Algebraic Topology MAST31023 Instructor: Marja Kankaanrinta Lectures: Monday 14:15 - 16:00, Wednesday 14:15 - 16:00 Exercises: Tuesday 14:15 - 16:00 August 12, 2019 1 2 Contents 0. Introduction 3 1. Categories and Functors 3 2. Homotopy 7 3. Convexity, contractibility and cones 9 4. Paths and path components 14 5. Simplexes and affine spaces 16 6. On retracts, deformation retracts and strong deformation retracts 23 7. The fundamental groupoid 25 8. The functor π1 29 9. The fundamental group of a circle 33 10. Seifert - van Kampen theorem 38 11. Topological groups and H-spaces 41 12. Eilenberg - Steenrod axioms 43 13. Singular homology theory 44 14. Dimension axiom and examples 49 15. Chain complexes 52 16. Chain homotopy 59 17. Relative homology groups 61 18. Homotopy invariance of homology 67 19. Reduced homology 74 20. Excision and Mayer-Vietoris sequences 79 21. Applications of excision and Mayer - Vietoris sequences 83 22. The proof of excision 86 23. Homology of a wedge sum 97 24. Jordan separation theorem and invariance of domain 98 25. Appendix: Free abelian groups 105 26. English-Finnish dictionary 108 References 110 3 0. Introduction These notes cover a one-semester basic course in algebraic topology. The course begins by introducing some fundamental notions as categories, functors, homotopy, contractibility, paths, path components and simplexes. After that we will study the fundamental group; the Fundamental Theorem of Algebra will be proved as an application. This will take roughly the first half of the semester. During the second half of the semester we will study singular homology. -
A COUNTABLE SPACE with an UNCOUNTABLE FUNDAMENTAL GROUP 1. Introduction Since Fundamental Groups Are Defined in Terms of Maps Fr
A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP JEREMY BRAZAS AND LUIS MATOS Abstract. Traditional examples of spaces that have uncountable fundamental group (such as the Hawaiian earring space) are path-connected compact metric spaces with uncountably many points. In this paper, we construct a T0 compact, path-connected, locally path-connected topological space H with countably many points but with an uncountable fundamental group. The construction of H, which we call the “coarse Hawaiian earring” is based on the construction of the usual S Hawaiian earring space H = n 1 Cn where each circle Cn is replace with a copy of the four-point “finite circle.” ≥ 1. Introduction Since fundamental groups are defined in terms of maps from the unit interval [0; 1], students are often surprised to learn that spaces with finitely many points can be path-connected and have non-trivial fundamental groups. In fact, it has been known since the 1960s that the homotopy theory of finite spaces is quite rich [10, 14]. The algebraic topology has finite topological spaces has gained significant interest since Peter May’s sequence of REU Notes [7, 8, 9]. See [2, 3, 5] and the references therein for more recent theory and applications. While it is reasonable to expect that all finite connected spaces have finitely gen- erated fundamental groups, it is rather remarkable that for every finitely generated group G one can construct a finite space X so that π1(X; x0) G. In fact, every finite simplicial complex is weakly homotopy equivalent to a finite space [10]. In the same spirit, we consider the consider fundamental groups of spaces with coarse topologies. -
HOMOTOPIES and DEFORMATION RETRACTS THESIS Presented To
A181 HOMOTOPIES AND DEFORMATION RETRACTS THESIS Presented to the Graduate Council of the University of North Texas in Partial Fulfillment of the Requirements For the Degree of MASTER OF ARTS By Villiam D. Stark, B.G.S. Denton, Texas December, 1990 Stark, William D., Homotopies and Deformation Retracts. Master of Arts (Mathematics), December, 1990, 65 pp., 9 illustrations, references, 5 titles. This paper introduces the background concepts necessary to develop a detailed proof of a theorem by Ralph H. Fox which states that two topological spaces are the same homotopy type if and only if both are deformation retracts of a third space, the mapping cylinder. The concepts of homotopy and deformation are introduced in chapter 2, and retraction and deformation retract are defined in chapter 3. Chapter 4 develops the idea of the mapping cylinder, and the proof is completed. Three special cases are examined in chapter 5. TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS . .0 . a. 0. .a . .a iv CHAPTER 1. Introduction . 1 2. Homotopy and Deformation . 3. Retraction and Deformation Retract . 10 4. The Mapping Cylinder .. 25 5. Applications . 45 REFERENCES . .a .. ..a0. ... 0 ... 65 iii LIST OF ILLUSTRATIONS Page FIGURE 1. Illustration of a Homotopy ... 4 2. Homotopy for Theorem 2.6 . 8 3. The Comb Space ...... 18 4. Transitivity of Homotopy . .22 5. The Mapping Cylinder . 26 6. Function Diagram for Theorem 4.5 . .31 7. The Homotopy "r" . .37 8. Special Inverse . .50 9. Homotopy for Theorem 5.11 . .61 iv CHAPTER I INTRODUCTION The purpose of this paper is to introduce some concepts which can be used to describe relationships between topological spaces, specifically homotopy, deformation, and retraction. -
Degree in Mathematics
Degree in Mathematics Title: The fundamental group and some applications in robotics. Author: Homs Moreno, Josep. Advisor: Pascual Gainza, Pere. Department: Department I of Applied Mathematics. Academic year: 2014-2015. Universitat Polit`ecnicade Catalunya Facultat de Matem`atiquesi Estad´ıstica Bachelor degree thesis The fundamental group and some applications in robotics. Josep Homs Moreno Advisor: Pere Pascual Gainza Abstract The aim of this project is to study important techniques to determine if two topolog- ical spaces are homeomorphic (or homotopy equivalent) or not. Such techniques are the ones related with the fundamental group of a topological space. This project is aimed to cover widely the study of this area of knowledge, always in a very intuitive approach. In chapters one and two there is a motivation of the fundamental group concept. Chapter three shows some important computations throughout some ex- amples. Chapter four introduces the Seifert-Van Kampen theorem and shows its power throughout more examples. This theorem will give some characterisations of really important topological spaces such as topological graphs, connected com- pact surfaces and torus knots among others. The proof of the Seifert-Van Kampen theorem is given in chapter five. Finally, chapter six contains a brief analysis of a re- search paper. In this paper there appear lots of the concepts covered in the project in an applied framework, different from the strictly theoric one. This project uses concepts seen in introductory courses of topology and group theory. Contents Chapter 1. Review of homotopy of maps 1 1. Definitions 1 2. Basic homotopy properties 5 Chapter 2. -
Arc-Reduced Forms for Peano Continua ✩ ∗ G
Topology and its Applications 159 (2012) 3538–3543 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Arc-reduced forms for Peano continua ✩ ∗ G. Conner a, M. Meilstrup b, a Math Department, Brigham Young University, Provo, UT 84602, USA b Math Department, Southern Utah University, Cedar City, UT 84720, USA article info abstract Article history: Define a point in a topological space to be homotopically fixed if it is fixed by every Received 18 May 2011 self-homotopy of the space, i.e. every self-map of the space which is homotopic to Received in revised form 14 May 2012 the identity, and define a point to be one-dimensional if it has a neighborhood whose Accepted 20 August 2012 covering dimension is one. In this paper, we show that every Peano continuum is homotopy equivalent to a reduced form in which the one-dimensional points which are MSC: 55P10 not homotopically fixed form a disjoint union of open arcs. In the case of one-dimensional 57M99 Peano continua, this presents the space as a compactification of a null sequence of open arcs by the homotopically fixed subspace. Keywords: © 2012 Elsevier B.V. All rights reserved. Peano continua Homotopy equivalence Reduced forms One-dimensional 1. Introduction The main result of this paper is Theorem 3.1, which describes a reduced form for homotopy types of one-dimensional Peano continua. In the theorem, I(X) represents the points in X with one-dimensional neighborhoods, and B(X) is the ho- motopically fixed subset of I(X). -
Topology: Notes and Problems
TOPOLOGY: NOTES AND PROBLEMS Abstract. These are the notes prepared for the course MTH 304 to be offered to undergraduate students at IIT Kanpur. Contents 1. Topology of Metric Spaces 1 2. Topological Spaces 3 3. Basis for a Topology 4 4. Topology Generated by a Basis 4 4.1. Infinitude of Prime Numbers 6 5. Product Topology 6 6. Subspace Topology 7 7. Closed Sets, Hausdorff Spaces, and Closure of a Set 9 8. Continuous Functions 12 8.1. A Theorem of Volterra Vito 15 9. Homeomorphisms 16 10. Product, Box, and Uniform Topologies 18 11. Compact Spaces 21 12. Quotient Topology 23 13. Connected and Path-connected Spaces 27 14. Compactness Revisited 30 15. Countability Axioms 31 16. Separation Axioms 33 17. Tychonoff's Theorem 36 References 37 1. Topology of Metric Spaces A function d : X × X ! R+ is a metric if for any x; y; z 2 X; (1) d(x; y) = 0 iff x = y. (2) d(x; y) = d(y; x). (3) d(x; y) ≤ d(x; z) + d(z; y): We refer to (X; d) as a metric space. 2 Exercise 1.1 : Give five of your favourite metrics on R : Exercise 1.2 : Show that C[0; 1] is a metric space with metric d1(f; g) := kf − gk1: 1 2 TOPOLOGY: NOTES AND PROBLEMS An open ball in a metric space (X; d) is given by Bd(x; R) := fy 2 X : d(y; x) < Rg: Exercise 1.3 : Let (X; d) be your favourite metric (X; d). How does open ball in (X; d) look like ? Exercise 1.4 : Visualize the open ball B(f; R) in (C[0; 1]; d1); where f is the identity function. -
5. Connectedness We Begin Our Introduction to Topology with the Study of Connectedness—Traditionally the Only Topic Studied in Both Analytic and Algebraic Topology
5. Connectedness We begin our introduction to topology with the study of connectedness—traditionally the only topic studied in both analytic and algebraic topology. C. T. C. Wall, 1972 The property at the heart of certain key results in analysis is connectedness. The definition, however, applies to any topological space. Definition 5.1. A space X is disconnected by a separation U, V if U and V are open, non-empty, and disjoint (U V = ) subsets of X with X = {U V}. If no separation of the space X exists, then X is connected∩ ∅ . ∪ Notice that V = X U is closed and likewise U is closed. A subset that is both open and closed is sometimes− called clopen. Closure leads to an equivalent condition. Theorem 5.2. A space X is connected if and only if whenever X = A B with A, B, non-empty, then A (cls B) = or (cls A) B = . ∪ ∩ ∅ ∩ ∅ Proof: If A (cls B)= and (cls A) B = , then, since A B = X, it will follow that X cls A, X∩ cls B is a∅ separation of∩X. To∅ see this, consider∪x (X cls A) (X cls B); then{ −x/cls A−and x/} cls B. But then x/cls A cls B = X, a∈ contradiction.− ∩ Therefore− (X cls∈ A) (X cls∈ B)= . Thus we have∈ a separation.∪ − ∩ − ∅ Conversely, if U, V is a separation of X, let A = X V = U and B = X U = V . Since U and V are{ open,} A and B are closed. Then X −= U V = A B.− However, A cls B = A B = U V = . -
A Core-Free Semicovering of the Hawaiian Earring
A CORE-FREE SEMICOVERING OF THE HAWAIIAN EARRING HANSPETER FISCHER AND ANDREAS ZASTROW Abstract. The connected covering spaces of a connected and locally path- connected topological space X can be classified by the conjugacy classes of those subgroups of π1(X; x) which contain an open normal subgroup of π1(X; x), when endowed with the natural quotient topology of the compact-open topol- ogy on based loops. There are known examples of semicoverings (in the sense of Brazas) that correspond to open subgroups which do not contain an open normal subgroup. We present an example of a semicovering of the Hawaiian Earring H with corresponding open subgroup of π1(H) which does not contain any nontrivial normal subgroup of π1(H). 1. Introduction The fundamental group π1(X; x) of a topological space X with base point x 2 X carries a natural topology: considering the space Ω(X; x) of all continuous loops α : ([0; 1]; f0; 1g) ! (X; x) in the compact-open topology, we equip π1(X; x) with the quotient topology induced by the function [ · ] : Ω(X; x) ! π1(X; x) which assigns to each loop α its homotopy class [α]. It is known that π1(X; x) need not be a topological group, for example, when X is the Hawaiian Earring [7] (see also [3]), although left and right multiplication always constitute homeomorphisms [6]. This problem can be circumvented by removing some of the open subsets from the topology and instead giving π1(X; x) the finest group topology which makes [ · ] : Ω(X; x) ! π1(X; x) continuous [5]. -
1 the Fundamental Group of a Topological Space
Notes on Algebraic Topology 1 The fundamental group of a topological space The first part of these notes deals with the (first) homotopy group —or “fundamental group”— of a topological space. We shall see that two topological spaces which are “ho- motopic” —i.e. continuously deformable one to the other, for example homeomorphic spaces— have isomorphic fundamental groups. After defining the notion of homotopy ( 1.1), we study the subsets which are “retract” (in § various senses) of a given space ( 1.2); then we come to the definition of the fundamental § group of a topological space and investigate its invariance ( 1.3), and we study the exam- § ples of the circle and of the other quotients of topological groups by discrete subgroups ( 1.4). In computing a fundamental group, the theorem of Van Kampen allows one to § decompose the problem on the subsets of a suitable open cover ( 1.5). Deeply related § to the fundamental group is the theory of covering spaces —i.e. local homeomorphisms with uniform fibers— of a topological space ( 1.6), which enjoy the property of “lifting § homotopies” ( 1.7). The covering spaces have a simpler homotopy structure than the one § of the original topological space, at the point that the graph of subgroups of the funda- mental group of the latter describes the formers up to isomorphisms ( 1.8). In particular, § the “universal” covering space —whose characteristic subgroup is trivial— describes, by means of the covering automorphisms, the fundamental group itself ( 1.9). We end by § studying some examples, among them the fundamental group of manifolds and of real linear groups ( 1.10). -
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together with a distinguished family of subsets. This family of subsets is called a topology for the given set. Definition. Let X be a set. A collection of subsets U = {U} of X is called a topology for X, if it satisfies the following properties: 1. X ∈ U and ∅ ∈ U. 2. The intersection of finitely many members of U is again in U. 3. The union of arbitrarily many members of U is again in U. A topological space is a set together with a topology. The family U is generally called the family of open sets of the topological space. We have also observed that a topology is also determined by specifying the family of closed sets, namely, the family of sets whose complements are open. Proposition 1. Let X be a set with a topology U. Define a family of subsets C = {U c}, where U c denotes X − U, the complement of U in X. Then the family C has the following properties. 1. X ∈ C and ∅ ∈ C. 2. The union of finitely many members of C is again in C. 3. The intersection of arbitrarily many members of C is again in C. The family C is generally called the family of closed sets of the topological space X. In some respects it is a matter of convenience whether or not one chooses to work with closed sets or open sets.