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Regular Fibrations Over the Hawaiian Earring Stewart Mason Mcginnis Brigham Young University

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Regular Fibrations Over the Hawaiian Earring Stewart Mason Mcginnis Brigham Young University Brigham Young University Masthead Logo BYU ScholarsArchive All Theses and Dissertations 2019-04-01 Regular Fibrations over the Hawaiian Earring Stewart Mason McGinnis Brigham Young University Follow this and additional works at: https://scholarsarchive.byu.edu/etd BYU ScholarsArchive Citation McGinnis, Stewart Mason, "Regular Fibrations over the Hawaiian Earring" (2019). All Theses and Dissertations. 7366. https://scholarsarchive.byu.edu/etd/7366 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. Regular Fibrations over the Hawaiian Earring Stewart Mason Cecil McGinnis A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Curtis Kent, Chair Greg Conner Eric Swenson Mark Hughes Department of Mathematics Brigham Young University Copyright c 2019 Stewart Mason Cecil McGinnis All Rights Reserved abstract Regular Fibrations over the Hawaiian Earring Stewart Mason Cecil McGinnis Department of Mathematics, BYU Master of Science We present a family of fibrations over the Hawaiian earring that are inverse limits of regular covering spaces over the Hawaiian earring. These fibrations satisfy unique path lifting, and as such serve as a good extension of covering space theory in the case of non- semi-locally simply connected spaces. We give a condition for when these fibrations are path-connected. Keywords: Hawaiian earring, inverse limit, fibration, path-connected Acknowledgments I would like to thank my advisor, Curt Kent, for his guidance and encouragement in preparing this thesis. Also, I would like to express my gratitude for the support of instructors and family as I have pursued mathematics. Contents Contents iv 1 Introduction 1 2 Preliminaries 1 2.1 Covering Spaces . .1 2.2 Fibrations from towers of covering Spaces . .6 3 Path-connectedness and lim1 7 − 3.1 Towers of Groups and lim1 ............................8 − 3.2 Path-connectedness . 12 4 Product structures on towers of regular covering spaces 14 5 Product towers that yield to path-connected regular fibrations 18 6 Conclusion 21 Bibliography 22 iv Chapter 1. Introduction The fundamental group is an important algebraic invariants in topology. Classically, covering space theory is the primary approach to studying and computing the fundamental group. There is a correspondence between automorphisms of covering spaces over a base space X and subgroups of the fundamental group of X. However, this correspondence is incomplete if X is not a path-connected, locally path-connected, semilocally simply connected space. One of the most simple nonsemilocally simply connected spaces is the Hawaiian earring. Every neighborhood of the basepoint contains an essential loop. Consequently, the Hawaiian earring does not admit a universal covering space. We probe deeper into the structure of the Hawaiian earring by generalizing covering spaces to inverse limits of towers of covering spaces. Chapter 2. Preliminaries We first recall the definitions of covering spaces and fibrations, as well as their connection to the fundamental group of a space. 2.1 Covering Spaces Definition 2.1. [1] A covering space is a triple (E; p; X) where p : E ! X is a continuous surjective map which evenly covers X. That is, for each x 2 X there is an open neighborhood −1 Ux of x such that p (Ux) is homeomorphic to a disjoint union of copies of Ux, called slices, and that p restricted to one of these slices is a homeomorphism to Ux. The map p is called a covering map. Definition 2.2. A deck transformation of a covering space p : E ! X is a homeomorphism f : E ! E such that f = p ◦ f. 1 f E E p p X The set of deck transformations of a covering space form a group under composition called the deck transformation group, denoted Aut(E ! X) or Aut(E) when X is understood. Definition 2.3. [2] A triple (E; p; X) where p : E ! X is a continuous map is said to have the homotopy lifting property with respect to a space Y if for all homotopies H : I × Y ! X and maps f0 : 0 × Y ! E such that H0 = f0 ◦ p, there exists a lift He : I × Y ! E of H with He0 = f0. f 0 × Y 0 E 9H e p I × Y H X Definition 2.4. A (Hurewicz) fibration is a triple (E; p; X) where p : E ! X has the homotopy lifting with respect to all spaces. A unique path lifting (UPL) fibration is a fibration for which lifts of paths are unique. We now show that covering spaces are fibrations with unique path lifting. It is this property that is preserved when we pass to the inverse limit of a tower. Throughout the following proofs we denote the backwards parametrization of a path by overlining. Proposition 2.5. [2] Let p :(E; e0) ! (X; x0) be a covering map, and f :(Y; y0) ! (X; x0) where Y is path-connected and locally path-connected. A lift fe : Y ! E of f exists if and only if f∗(π1(Y; y0)) is in a conjugacy class of p∗(π1(E; e0)). Proof. Suppose we have a lift fe, and α :(I; 0; 1) ! (E; e0; fe0(y0)). So we have a change of basepoint homomorphism hα : π1(E; fe(y0)) ! π1(E; e0). Then since p ◦ fe = f, hp◦α ◦ p∗ ◦ hα ◦ fe∗ = f∗, so we must have im f∗ ⊆ [p ◦ α](im p∗)[p ◦ α]. 2 0 Now suppose im f∗ ⊆ [p◦α](im p∗)[p◦α]. Let γ; γ :(I; 0; 1) ! (Y; y0; y) and define fe(y) = 0 p ◦ ^α ∗ f ◦ γe0 . Since γ ∗ γ is a loop in Y , and im f∗ ⊆ [p ◦ α](im p∗)[p ◦ α], p ◦ ^α ∗ f ◦ γe0 ∗ 0 0 p ◦ α^∗ f ◦ γ e0 is a loop in E. So p ◦ ^α ∗ f ◦ γe0 (1) = p ◦ α^∗ f ◦ γ e0 (1). Thus fe is well defined. Also p ◦ fe = p(fe(y)) = p(p ◦ ^α ∗ f ◦ γe0 (1)) = f ◦ (p ◦ α ∗ γ(1)) = f(y). Now let e 2 fe(y) and U be an evenly covered neighborhood about e. Then take V ⊆ f −1(p(U)) a path connected neighborhood of y. Then let δ be a path based at y contained in V . Then f(δ(1)) = f ◦^(γ ∗ δ) (1). By construction f ◦ (γ ∗ δ)(1) 2 p(U), so f ◦^(γ ∗ δ) (1) e e0 e0 is in U. Thus fe(V ) ⊆ U. So fe is continuous. If we have two such lifts that agree on a point, then they are the same lift. 0 0 Proof. Suppose fe1(y) = fe2(y). Then given y 2 Y , let γ :(I; 0; 1) ! (Y; y; y ). Then f (y0) = f(γ(1)) = f]◦ γ (1) = f]◦ γ (1) = f (γ(1)) = f (y0). e1 e fe1(y) fe2(y) e2 e2 Corollary 2.6. Covering spaces are fibrations. Proof. Let p : E ! X be a covering space. Given a homotopy H : Y × I ! X and a lift of He0 of H0 : Y ! X, we have H∗(π1(Y × I) = H0∗(π1(Y )) because H, and H0 are homotopic maps. Since H0 admits a lift, by Proposition 2.5, H0∗(π1(Y )) is contained in a conjugacy class of p∗(π1(E)). Hence H admits a lift as well. We now show unique path lifting. Proposition 2.7. Given a covering space p : E ! X, a path α : [0; 1] ! X and e 2 p−1α(0), there exists a unique lift αe of α with p ◦ αe = α and αe(0) = e. E 9!αe p I α X Proof. Since I is compact, its image in X is compact, so we can cover the image of I with finitely many evenly covered neighborhoods. Call this cover U. Take one such neighborhood −1 U1 2 U containing α(0), and define α = pj ◦ αjI where U1;e is the slice of U1 containing e U1;e 1 −1 e and I1 is the open interval of α (U1) that contains 0. We then choose Ui+1 2 U such 3 −1 that α (Ui+1) has as a component an open interval Ii+1 which intersects and extends Ii, −1 and extend α = pj ◦ αjI where Ui+1;e is the slice of Ui+1 that intersects the slice of e Ui+1;e i+1 Ui;e containing αe(Ii \ Ii+1). This process terminate because U is finite, and completes the definition of αe since U covers the image of α. Given β any other lift beginning at e, we can find that β = α on U1 by homeomorphically pushing β down and back up along p, and then continuing to Ui as before. Corollary 2.8. Covering spaces are UPL fibrations with unique lifts of homotopies. Proof. Suppose we have a map f :(I × Y; (0; y0)) ! (X; x0). Then there exists a map fe0 :(f0g × Y; (0; y0)) ! (X; x0) if and only if fjf0g × Y ∗(π1(f0g × Y; (0; y0)) ⊆ p∗(π1(E; e0)). If this is the case then f∗(π1(I × Y; (0; y0))) ⊆ p∗(π1(E; e0)) since I × Y is homotopic to Y . So the homotopy f lifts to fe with fe(0; y) = fe0(y). Furthermore, this lift is uniquely determined by fe0(y0). We now discuss the relationship between covering spaces and the fundamental group of the base space. The correspondence is more general than we present here, but we restrict our attention to what is relevant for this thesis.
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