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Some Aspects of Generalized Covering Space Theory

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Some Aspects of Generalized Covering Space Theory SOME ASPECTS OF GENERALIZED COVERING SPACE THEORY A THESIS SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE MASTER OF SCIENCE BY JACOB D. GARCIA DR. HANSPETER FISCHER - ADVISOR BALL STATE UNIVERSITY MUNCIE, INDIANA MAY 2018 ABSTRACT THESIS: Some Aspects of Generalized Covering Space Theory STUDENT: Jacob Garcia DEGREE: Master of Science COLLEGE: Sciences and Humanities DATE: May 2018 PAGES: 21 Covering space theory is a classical tool used to characterize the geometry and topology of real or abstract spaces. It seeks to separate the main geometric features from certain algebraic properties. For each conjugacy class of a subgroup of the fundamental group, it supplies a corresponding covering of the underlying space and encodes the interplay between algebra and geometry via group actions. The full applicability of this theory is limited to spaces that are, in some sense, locally simple. However, many modern areas of mathematics, such as fractal geometry, deal with spaces of high local complexity. This has stimulated much recent research into generalizing covering space theory by weakening the covering requirement while maintaining most of the classical utility. This project focuses on the relationships between generalized covering projections, fibrations with unique path lifting, separation properties of the fibers, and continuity of the monodromy. iii Acknowledgements I would first like to thank my thesis advisor Professor Hanspeter Fischer of the Department of Mathematical Sciences at Ball State University. The door to Professor Fischer's office was always open whenever I ran into difficulty or had a question about my research or writing. He consistently steered me in the right direction and provided incredible advice and support during this thesis. I would also like to acknowledge Professor Dan Rutherford and Professor Richard Stankewitz of the Department of Mathematical Sciences at Ball State University as members of the committee for this thesis, and I am grateful to them for their time and helpful comments on this thesis. Finally, I must express my gratitude to my family and to my fianc´eRebecca Kurtz for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them. iv Contents Abstract iii Acknowledgements iv 1 Results from Classical Theory1 2 Generalized Covering Space Theory5 3 Small Loop Transfer and Related Extensions 11 4 Extensions and Generalizations 15 Bibliography 23 v Chapter 1 Results from Classical Theory We assume familiarity with the basic notions of point set topology and the fundamental group, as presented in [12]. The classical way of defining a covering projection is through the use of a local homeomorphism condition. We begin by providing the definition of a classical covering projection, as defined in [14], and some well known examples of covering spaces. Then we introduce the notion of unique path lifting and restate the lifting criterion for covering spaces found in [14]. Definition 1.1 (Evenly Covered). Let p : E ! X be a continuous surjective map between topo- logical spaces. An open set U of X is said to be evenly covered by p if the inverse image p−1(U) can be written as the union [i2I Vi of disjoint open sets Vi in E such that for each i, the restriction of p to Vi is a homeomorphism of Vi onto U. Definition 1.2 (Covering Space and Covering Projection). Let p : E ! X be a continuous and surjective map between topological spaces. If every point x 2 X has a neighborhood U that is evenly covered by p, then p is called a covering projection, and E is said to be a covering space of X. 1 The real line R is a covering space for the circle S with covering projection p(t) = (cos t; sin t). 1 1 1 The plane R × R, and the infinite cylinder R × S , are both covering spaces for the torus S × S . Definition 1.3 (Unique Path Lifting (UPL)). A continuous surjection p : E ! X is said to have unique path lifting if, for every f; g : [0; 1] ! E such that p ◦ f = p ◦ g and f(0) = g(0), we have f = g. 1 2 Lemma 1.4 (Lifting Criterion). Let p : E ! X be a covering projection with p(e0) = x0. Suppose f : Y ! X is a continuous map such that f(y0) = x0 and such that Y is connected and locally path-connected. Then: 0 0 0 There is a \lift" f : Y ! E such that p ◦ f = f and f (y0) = e0 () f#(π1(Y; y0)) ≤ p#(π1(E; e0)). In addition, if such a lifting exists, then it is unique. As discussed in [14], the lifting criterion shows that all paths and their homotopies lift uniquely in a covering projection. In particular, p# : π1(E; e0) ! π1(X; x0) is injective. In addition, [14] also provides a classification for when covering projections exist that preserve given subgroups of the fundamental group. The proof Spanier gives in [14, Theorem 2.5.13] is especially of note, since this is related to the construction in Definition 2.2. The following lemma combines [14, Lemma 2.5.11] and [14, Theorem 2.5.13]. Lemma 1.5. Assume X is connected and locally path-connected, x0 2 X, and H ≤ π1(X; x0). There exists a covering projection p :(E; e0) ! (X; x0) with p#(π1(E; e0)) = H if and only if there exists an open cover U of X such that π(U; x0) ≤ H, where π(U; x0) is the (normal) subgroup of − π1(X; x0) generated by homotopy classes of closed paths with a representative of the form (α∗β)∗α , − where β is a closed path in some element of U and α is a path from x0 to β(0). (Here α (t) = α(1−t) denotes the reverse path of α.) This lemma shows that the Hawaiian Earring, formed in the Euclidean plane as the union of all 1 1 circles Cn with radius n and center (0; n ), does not have a covering projection p :(E; e0) ! (X; x0 = (0; 0)) with p#(π1(E; e0)) = 1, because for any open cover U of X, there is an element of the cover U 2 U with x0 2 U, and Cn ⊆ U for some n. Since a loop around this circle Cn is not nullhomotopic in the Hawaiian Earring, U cannot be evenly covered with a simply connected covering space E. In addition to generalizing the concept of a covering projection, we are also interested in extending our results to consider fibrations: maps which have the homotopy lifting property with respect to every space. Definition 1.6 (Homotopy Lifting Property). A continuous map p : E ! X is said to have the homotopy lifting property with respect to a space Y if, given continuous maps f 0 : Y ! E and 3 F : Y × [0; 1] ! X such that F (y; 0) = p ◦ f 0(y) for all y 2 Y , there is a map F 0 : Y × [0; 1] ! E such that F 0(y; 0) = f 0(y) for all y 2 Y and p ◦ F 0 = F . The homotopy lifting property with respect to Y = fy0g is also called the path lifting property. Definition 1.7 (Fibration). A continuous map p : E ! X is called a fibration if p has the homotopy lifting property with respect to every space. We call E the total space, and X is called the base space of the fibration. For x 2 X, p−1(x) = p−1(fxg) is called the fiber over x. We now define monodromy as it relates to covering projections, which allows us to map one fiber to another in the covering space in such a way that the map between the fibers respects a given path between the images of the two fibers. Definition 1.8 (Monodromy). Let p : E ! X be a fibration with unique path lifting, y; z 2 X, and β : [0; 1] ! X a path that starts at y and ends at z. We now define the monodromy φβ : p−1(y) ! p−1(z) as follows: For y0 2 p−1(y), let β0 : [0; 1] ! E be the unique lift with p ◦ β0 = β 0 0 0 0 and β (0) = y . We define φβ(y ) = β (1). Remark 1.9. Spanier shows that every covering projection is a fibration with unique path lifting [14, Lemma 2.2.3] and shows that a fibration has unique path lifting if and only if it has totally path- disconnected fibers [14, Theorem 2.2.5]. Spanier also shows that in a fibration with unique path lifting, all monodromies are continuous [14, Theorem 2.3.7] and hence, all fibers are homeomorphic [14, Corollary 2.3.8]. Recall the following fact from [14, Corollary 2.2.13]: If p1 : E1 ! X and p2 : E2 ! X are two covering projections of a locally connected space X and f : E1 ! E2 a surjective map with p2 ◦ f = p1, then f : E1 ! E2 is a covering projection. Definition 1.10. We say that two covering projections p1 : E1 ! X and p2 : E1 ! X are equivalent if there is a homeomorphism f : E1 ! E2 such that p2 ◦ f = p1. When E1 = E2 = E and p1 = p2 = p, we call such a homeomorphism f an automorphism of p : E ! X. (Note that the p set Aut(E −! X) of automorphisms of p : E ! X forms a group under function composition.) Next, we recall the definition of a universal covering space, and state some corollaries that, when taken together, classifies every covering projection onto a base space by the projected fundamental group, and in particular, we see that if a space has a universal covering space, then any other universal covering space is equivalent.
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