DOCSLIB.ORG

15 Universal Covering Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
15 Universal Covering Spaces 15 Universal Covering Spaces Uniformization theorem Let us assert the following theorem: Theorem 15.1 (Uniformization theorem) Every connected Riemann surface is biholomor- phic to one of the following: ∗ CP1, C , C , or C/L , ∆(1)/Γ where Γ ⊂ P SU(1, 1) is a discrete group of automorphisms acting freely and properly dis- continuously. The uniformization theorem breaks up into two parts. Theorem 15.2 (hard) Every simply connected Riemann surface is biholomorphic to CP1, C or ∆(1). Theorem 15.3 (fairly easy) Every connected Riemann surface X biholomorphic to a quo- e e e tient X/Γ of a simply connected Riemann surface X by a group Γ ⊂ Aut(X) of automor- e phisms acting freely and properly discontinuously. Moreover, X and Γ are unique up to biholomorphism. To prove the hard theorem, the difficulty is to construct a biholomorphic map from the simply connected Riemann surface to CP1, C or ∆(1). You may recall that the proof of Riemann Mapping Theorem is quite non-trivial. To prove the easy theorem, the proof turns out to be purely topological, which involves “covering space” and “fundamental group” and are discussed in this section. Covering space and covering map We first introduce a definition from Topology. Let X,Y be topological spaces and f : Y → X is called a covering map if and only if every −1 x ∈ X has a neighborhood U in X so that f (U)= ∪jVj is a disjoint union with so that for each Vj, the restriction map f : Vj → U is a homeomorphism. Such Y is called a covering space of X. Similarly, we can define holomorphic covering space and holomorphic covering map (for one dimension one or higher dimension). Here is the definition for one dimension. Let X,Y be Riemann surfaces and a holomorphic map f : Y → X is called a holomorphic covering 93 −1 map if and only if every x ∈ X has a neighborhood U in X so that f (U) = ∪jVj is a disjoint union with so that for each Vj, the restriction map f : Vj → U is a biholomorphic map. Here are some examples: 1. Let f(z)= ez : C → C∗ := C −{0}. It is a holomorphic covering map. 2. Let g(z)= zn : C∗ → C∗. It is also a holomorphic covering map. However, g(z)= zn : C → C is not a covering map when n> 1. 3. In general, a quotient space M/G where M is a complex manifold and G is a subgroup of Aut(M) may not be a complex manifold. Some conditions are required. For a covering map π : X → Y , in addition, if X is simply connected, such covering map π : X → Y is called a universal covering map, and the X is called a universal covering space. Similar definitions for holomorphic case. Lifting properties of covering spaces in topology Let us assume existence of covering space and discuss its some basic properties. Our basic model is a complex torus X = C/Γ e where the natural quotient map π : X = C → X, z 7→ [z], is a universal covering map. e e Proposition 15.4 Let f : X → X be a covering spaces where X and X are topological spaces. e 1. (path lifting property) For any p ∈ X, pe ∈ X with f(pe)= p, and any continuous curve e e φ : [0, 1] → X with φ(0) = p, there is a unique continuous curve φ : [0, 1] → X such e that f ◦ φ = φ. e 2. (lifting of simply connected spaces) For any w0 ∈ W , p ∈ X and pe ∈ X with f(pe)= p, and a continuous map g : W → X from a path connected, simply connected space W to X, with g(w0) = p where w0 ∈ W and p ∈ X. Then there is a unique lifting e ge : W → X satisfying f ◦ ge = g with ge(x0)= pe. e 3. (Uniqueness of simply connected covering spaces) Assume that X is connected and X is simply connected, then f is a homeomorphism. Proof: 1. By the local homeomorphism property in the definition of covering spaces. 2. By the part 1. and the same argument on monodromy theorem. 94 − e 3. As the part 2, we construct the inverse map f 1 from X o X. e Quotient space and covering spaces Let f : X → X be a covering map. Let a group e e Γ act on X by homeomorphisms. The action is properly discontinuous if every xe ∈ X has e e some neighbourhood U whose translates γU, as γ ranges over Γ, are mutually disjoint. The connection between quotient space and covering space can be seen in the following result, which is immediate from the definition. e e Proposition 15.5 If Γ acts properly discontinuously on X, then the quotient map X → e X/Γ is a covering space. Moreover we have e e Lemma 15.6 Let f : X → X be a covering map where X is simply connected. Let Γ be e the group of self-homeomorphisms of X which commutes with f (i.e., f ◦ φ = f for any e e e e φ : X → X in Γ). Then Γ acts X properly discontinuously and X ≃ X/Γ. Proof: For any p ∈ X and any a, b ∈ f −1(p), by the lifting property of covering maps, e e ′ e e there exists a unique γ : X → X such that γ(a)= b, and another unique γ : X → X such that γ′(b) = a. Then γ′ ◦ γ fixes a and γ ◦ γ′ fixes b, so that they are the identity maps e of X, by the uniqueness of lifting. So γ is a homeomorphism commuting with f. We just established that Γ acts simply transitively on f −1(p). As p is arbitrary, this implies that e − X = X/Γ. Choosing a small connected neighbourhood U of p for which f 1(U) is a disjoint union of copies of U, it follows that Γ acts simply transitively on these copies, hence it acts e properly discontinuously on X. What is application for “covering space?” or “lifting ?” As an application, we’ll prove Picard theorem by assuming the exitence of holomorphic covering space for every Riemann surface. Theorem 15.7 (Picard Little Theorem) If an entire function f ∈ Hol(C) misses two values, 36 then f = constant. 36An entire function f : C → C missing one point is not necessary to be a constant. For example, f(z)= ez misses only one value 0. 95 To prove this theorem, we consider f : C → C −{0, 1} and want to show: f = constant. We claim that the universal covering space of C = {0, 1, } = ∆(1), which will be proved in the next section. Then we apply Proposition 22.4 to left f to fˆ: ∆(1) fˆ ր ↓ π f C −→ C −{0, 1} and shall prove that fˆ is holomorphic. By Louville’s theorem, from fˆ : C → ∆(1), it implies that f = constant. It remains to prove existence of the universal covering space for C −{0, 1}. To prove this, we need Schwarz Reflection principle and the following result. Theorem 15.8 Let D ⊂ Cˆ be a simply connected domain such that D =6 Cˆ. Assume that the boundary ∂D of D is given by a closed Jordan curve. Then every biholomorphic map f : ∆(1) → D extends to a homeomorphism f : ∆(1) → D. In particular, the restriction f|∂∆(1) : ∂∆(1) → ∂D is homeomorphic. 96.
Load more

Recommended publications
  • 1.6 Covering Spaces 1300Y Geometry and Topology
    1.6 Covering spaces 1300Y Geometry and Topology 1.6 Covering spaces Consider the fundamental group π1(X; x0) of a pointed space. It is natural to expect that the group theory of π1(X; x0) might be understood geometrically. For example, subgroups may correspond to images of induced maps ι∗π1(Y; y0) −! π1(X; x0) from continuous maps of pointed spaces (Y; y0) −! (X; x0). For this induced map to be an injection we would need to be able to lift homotopies in X to homotopies in Y . Rather than consider a huge category of possible spaces mapping to X, we restrict ourselves to a category of covering spaces, and we show that under some mild conditions on X, this category completely encodes the group theory of the fundamental group. Definition 9. A covering map of topological spaces p : X~ −! X is a continuous map such that there S −1 exists an open cover X = α Uα such that p (Uα) is a disjoint union of open sets (called sheets), each homeomorphic via p with Uα. We then refer to (X;~ p) (or simply X~, abusing notation) as a covering space of X. Let (X~i; pi); i = 1; 2 be covering spaces of X. A morphism of covering spaces is a covering map φ : X~1 −! X~2 such that the diagram commutes: φ X~1 / X~2 AA } AA }} p1 AA }}p2 A ~}} X We will be considering covering maps of pointed spaces p :(X;~ x~0) −! (X; x0), and pointed morphisms between them, which are defined in the obvious fashion.
    [Show full text]
  • [Math.AT] 4 Sep 2003 and H Uhri Ebro DE Eerhtann Ewr HPRN-C Network Programme
    ON RATIONAL HOMOTOPY OF FOUR-MANIFOLDS S. TERZIC´ Abstract. We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply con- nected four-manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature. 1. Introduction In this paper we consider the problem of computation of the rational homotopy groups and the problem of rational homotopy classification of simply connected closed four-manifolds. Our main results could be collected as follows. Theorem 1. Let M be a closed oriented simply connected four-manifold and b2 its second Betti number. Then: (1) If b2 =0 then rk π4(M)=rk π7(M)=1 and πp(M) is finite for p =46 , 7 , (2) If b2 =1 then rk π2(M)=rk π5(M)=1 and πp(M) is finite for p =26 , 5 , (3) If b2 =2 then rk π2(M)=rk π3(M)=2 and πp(M) is finite for p =26 , 3 , (4) If b2 > 2 then dim π∗(M) ⊗ Q = ∞ and b (b + 1) b (b2 − 4) rk π (M)= b , rk π (M)= 2 2 − 1, rk π (M)= 2 2 . 2 2 3 2 4 3 arXiv:math/0309076v1 [math.AT] 4 Sep 2003 When the second Betti number is 3, we can prove a little more. Proposition 2. If b2 =3 then rk π5(M)=10. Regarding rational homotopy type classification of simply connected closed four-manifolds, we obtain the following. Date: November 21, 2018; MSC 53C25, 57R57, 58A14, 57R17.
    [Show full text]
  • Monodromy Representations Associated to a Continuous Map
    Monodromy representations associated to a continuous map Salvador Borrós Cullell Supervisor: Dr. Pietro Polesello ALGANT Master Program Universität Duisburg-Essen; Università degli studi di Padova This dissertation is submitted for the degree of Master in Science July 2017 Acknowledgements First and foremost, I would like to thank my advisor Dr. Polesello for his infinite patience and his guidance while writing the thesis. His emphasis on writing mathematics in a precise and concise manner will forever remain a valuable lesson. I would also like to thank my parents for their constant support, specially during these two years I have been out of home. Despite the distance, they have kept me going when things weren’t going my way. A very special mention must be made of Prof. Marc Levine who introduced me to many of the notions that now appear in this thesis and showed unparalleled patience and kindness while helping me understand them in depth. Finally, I would like to thank my friends Shehzad Hathi and Jan Willem Frederik van Beck for taking up the role of a supportive family while living abroad. Abstract The aim of this thesis is to give a geometrical meaning to the induced monodromy rep- resentation. More precisely, given f : X ! Y a continuous map, the associated functor ind f f : P1(X) ! P1(Y) induces a functor Repk(P1(X)) ! Repk(P1(Y)) of the corresponding LCSH categories of representations. We will define a functor f∗ : LCSH(kX ) ! LCSH(kY ) from the category of locally constant sheaves on X to that of locally constant sheaves on Y in a way that the monodromy representation m LCSH is given by ind (m ), where m denotes f∗ F f F F the monodromy representation of a locally constant sheaf F on X.
    [Show full text]
  • Arxiv:Math/0507171V1 [Math.AG] 8 Jul 2005 Monodromy
    Monodromy Wolfgang Ebeling Dedicated to Gert-Martin Greuel on the occasion of his 60th birthday. Abstract Let (X,x) be an isolated complete intersection singularity and let f : (X,x) → (C, 0) be the germ of an analytic function with an isolated singularity at x. An important topological invariant in this situation is the Picard-Lefschetz monodromy operator associated to f. We give a survey on what is known about this operator. In particular, we re- view methods of computation of the monodromy and its eigenvalues (zeta function), results on the Jordan normal form of it, definition and properties of the spectrum, and the relation between the monodromy and the topology of the singularity. Introduction The word ’monodromy’ comes from the greek word µoνo − δρoµψ and means something like ’uniformly running’ or ’uniquely running’. According to [99, 3.4.4], it was first used by B. Riemann [135]. It arose in keeping track of the solutions of the hypergeometric differential equation going once around arXiv:math/0507171v1 [math.AG] 8 Jul 2005 a singular point on a closed path (cf. [30]). The group of linear substitutions which the solutions are subject to after this process is called the monodromy group. Since then, monodromy groups have played a substantial rˆole in many areas of mathematics. As is indicated on the webside ’www.monodromy.com’ of N. M. Katz, there are several incarnations, classical and l-adic, local and global, arithmetic and geometric. Here we concentrate on the classical lo- cal geometric monodromy in singularity theory. More precisely we focus on the monodromy operator of an isolated hypersurface or complete intersection singularity.
    [Show full text]
  • Arxiv:2010.15657V1 [Math.NT] 29 Oct 2020
    LOW-DEGREE PERMUTATION RATIONAL FUNCTIONS OVER FINITE FIELDS ZHIGUO DING AND MICHAEL E. ZIEVE Abstract. We determine all degree-4 rational functions f(X) 2 Fq(X) 1 which permute P (Fq), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational func- tions. We also determine all degree-8 rational functions f(X) 2 Fq(X) 1 which permute P (Fq) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for several other positive integers n < 4096, for each sufficiently large q we determine all degree-n rational functions f(X) 2 Fq(X) which permute 1 P (Fq) but which are not compositions of lower-degree rational func- tions in Fq(X). Some of these results are proved by using a new Galois- theoretic characterization of additive (linearized) polynomials among all rational functions, which is of independent interest. 1. Introduction Let q be a power of a prime p.A permutation polynomial is a polyno- mial f(X) 2 Fq[X] for which the map α 7! f(α) is a permutation of Fq. Such polynomials have been studied both for their own sake and for use in various applications. Much less work has been done on permutation ra- tional functions, namely rational functions f(X) 2 Fq(X) which permute 1 P (Fq) := Fq [ f1g. However, the topic of permutation rational functions seems worthy of study, both because permutation rational functions have the same applications as permutation polynomials, and because of the con- struction in [24] which shows how to use permutation rational functions over Fq to produce permutation polynomials over Fq2 .
    [Show full text]
  • Riemann Mapping Theorem
    Riemann surfaces, lecture 8 M. Verbitsky Riemann surfaces lecture 7: Riemann mapping theorem Misha Verbitsky Universit´eLibre de Bruxelles December 8, 2015 1 Riemann surfaces, lecture 8 M. Verbitsky Riemannian manifolds (reminder) DEFINITION: Let h 2 Sym2 T ∗M be a symmetric 2-form on a manifold which satisfies h(x; x) > 0 for any non-zero tangent vector x. Then h is called Riemannian metric, of Riemannian structure, and (M; h) Riemannian manifold. DEFINITION: For any x:y 2 M, and any path γ :[a; b] −! M connecting R dγ dγ x and y, consider the length of γ defined as L(γ) = γ j dt jdt, where j dt j = dγ dγ 1=2 h( dt ; dt ) . Define the geodesic distance as d(x; y) = infγ L(γ), where infimum is taken for all paths connecting x and y. EXERCISE: Prove that the geodesic distance satisfies triangle inequality and defines metric on M. EXERCISE: Prove that this metric induces the standard topology on M. n P 2 EXAMPLE: Let M = R , h = i dxi . Prove that the geodesic distance coincides with d(x; y) = jx − yj. EXERCISE: Using partition of unity, prove that any manifold admits a Riemannian structure. 2 Riemann surfaces, lecture 8 M. Verbitsky Conformal structures and almost complex structures (reminder) REMARK: The following theorem implies that almost complex structures on a 2-dimensional oriented manifold are equivalent to conformal structures. THEOREM: Let M be a 2-dimensional oriented manifold. Given a complex structure I, let ν be the conformal class of its Hermitian metric. Then ν is determined by I, and it determines I uniquely.
    [Show full text]
  • The Monodromy Groups of Schwarzian Equations on Closed
    Annals of Mathematics The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces Author(s): Daniel Gallo, Michael Kapovich and Albert Marden Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 151, No. 2 (Mar., 2000), pp. 625-704 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/121044 . Accessed: 15/02/2013 18:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded on Fri, 15 Feb 2013 18:57:11 PM All use subject to JSTOR Terms and Conditions Annals of Mathematics, 151 (2000), 625-704 The monodromy groups of Schwarzian equations on closed Riemann surfaces By DANIEL GALLO, MICHAEL KAPOVICH, and ALBERT MARDEN To the memory of Lars V. Ahlfors Abstract Let 0: 7 (R) -* PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for 0 to be the monodromy represen- tation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that 0(7ri(R)) be nonelementary.
    [Show full text]
  • Galois Theory Towards Dessins D'enfants
    Galois Theory towards Dessins d'Enfants Marco Ant´onioDelgado Robalo Disserta¸c~aopara obten¸c~aodo Grau de Mestre em Matem´aticae Aplica¸c~oes J´uri Presidente: Prof. Doutor Rui Ant´onioLoja Fernandes Orientador: Prof. Doutor Jos´eManuel Vergueiro Monteiro Cidade Mour~ao Vogal: Prof. Doutor Carlos Armindo Arango Florentino Outubro de 2009 Agradecimentos As` Obsess~oes Em primeiro lugar e acima de tudo, agrade¸co`aminha M~aeFernanda. Por todos os anos de amor e de dedica¸c~ao.Obrigado. Agrade¸coao meu Irm~aoRui. Obrigado. As` minhas Av´os,Diamantina e Maria, e ao meu Av^o,Jo~ao.Obrigado. Este trabalho foi-me bastante dif´ıcilde executar. Em primeiro lugar pela vastid~aodo tema. Gosto de odisseias, da proposi¸c~aoaos objectivos inalcan¸c´aveis e de vis~oespanor^amicas.Seria dif´ıcilter-me dedicado a uma odisseia matem´aticamaior que esta, no tempo dispon´ıvel. Agrade¸coprofundamente ao meu orientador, Professor Jos´eMour~ao: pela sua enorme paci^enciaperante a minha constante mudan¸cade direc¸c~aono assunto deste trabalho, permitindo que eu pr´oprio encontrasse o meu caminho nunca deixando de me apoiar; por me ajudar a manter os p´esnos ch~aoe a n~aome perder na enorme vastid~aodo tema; pelos in´umeros encontros regulares ao longo de mais de um ano. A finaliza¸c~aodeste trabalho em tempo ´utils´ofoi poss´ıvel com a ajuda da sua sensatez e contagiante capacidade de concretiza¸c~ao. Agrade¸cotamb´emao Professor Paulo Almeida, que talvez sem o saber, me ajudou a recuperar o entu- siasmo pela procura da ess^enciadas coisas.
    [Show full text]
  • Rational Homotopy Theory: a Brief Introduction
    Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article.
    [Show full text]
  • Lecture 34: Hadamaard's Theorem and Covering Spaces
    Lecture 34: Hadamaard’s Theorem and Covering spaces In the next few lectures, we’ll prove the following statement: Theorem 34.1. Let (M,g) be a complete Riemannian manifold. Assume that for every p M and for every x, y T M, we have ∈ ∈ p K(x, y) 0. ≤ Then for any p M,theexponentialmap ∈ exp : T M M p p → is a local diffeomorphism, and a covering map. 1. Complete Riemannian manifolds Recall that a connected Riemannian manifold (M,g)iscomplete if the geodesic equation has a solution for all time t.Thisprecludesexamplessuch as R2 0 with the standard Euclidean metric. −{ } 2. Covering spaces This, not everybody is familiar with, so let me give a quick introduction to covering spaces. Definition 34.2. Let p : M˜ M be a continuous map between topological spaces. p is said to be a covering→ map if (1) p is a local homeomorphism, and (2) p is evenly covered—that is, for every x M,thereexistsaneighbor- 1 ∈ hood U M so that p− (U)isadisjointunionofspacesU˜ for which ⊂ α p ˜ is a homeomorphism onto U. |Uα Example 34.3. Here are some standard examples: (1) The identity map M M is a covering map. (2) The obvious map M →... M M is a covering map. → (3) The exponential map t exp(it)fromR to S1 is a covering map. First, it is clearly a local → homeomorphism—a small enough open in- terval in R is mapped homeomorphically onto an open interval in S1. 113 As for the evenly covered property—if you choose a proper open in- terval inside S1, then its preimage under the exponential map is a disjoint union of open intervals in R,allhomeomorphictotheopen interval in S1 via the exponential map.
    [Show full text]
  • COVERING SPACES, GRAPHS, and GROUPS Contents 1. Covering
    COVERING SPACES, GRAPHS, AND GROUPS CARSON COLLINS Abstract. We introduce the theory of covering spaces, with emphasis on explaining the Galois correspondence of covering spaces and the deck trans- formation group. We focus especially on the topological properties of Cayley graphs and the information these can give us about their corresponding groups. At the end of the paper, we apply our results in topology to prove a difficult theorem on free groups. Contents 1. Covering Spaces 1 2. The Fundamental Group 5 3. Lifts 6 4. The Universal Covering Space 9 5. The Deck Transformation Group 12 6. The Fundamental Group of a Cayley Graph 16 7. The Nielsen-Schreier Theorem 19 Acknowledgments 20 References 20 1. Covering Spaces The aim of this paper is to introduce the theory of covering spaces in algebraic topology and demonstrate a few of its applications to group theory using graphs. The exposition assumes that the reader is already familiar with basic topological terms, and roughly follows Chapter 1 of Allen Hatcher's Algebraic Topology. We begin by introducing the covering space, which will be the main focus of this paper. Definition 1.1. A covering space of X is a space Xe (also called the covering space) equipped with a continuous, surjective map p : Xe ! X (called the covering map) which is a local homeomorphism. Specifically, for every point x 2 X there is some open neighborhood U of x such that p−1(U) is the union of disjoint open subsets Vλ of Xe, such that the restriction pjVλ for each Vλ is a homeomorphism onto U.
    [Show full text]
  • Generalized Covering Spaces and the Galois Fundamental Group
    GENERALIZED COVERING SPACES AND THE GALOIS FUNDAMENTAL GROUP CHRISTIAN KLEVDAL Contents 1. Introduction 1 2. A Category of Covers 2 3. Uniform Spaces and Topological Groups 6 4. Galois Theory of Semicovers 9 5. Functoriality of the Galois Fundamental Group 15 6. Universal Covers 16 7. The Topologized Fundamental Group 17 8. Covers of the Earring 21 9. The Galois Fundamental Group of the Harmonic Archipelago 22 10. The Galois Fundamental Group of the Earring 23 References 24 Gal Abstract. We introduce a functor π1 from the category of based, connected, locally path connected spaces to the category of complete topological groups. We then compare this groups to the fundamental group. In particular, we show that there is a topological σ Gal group π1 (X; x) whose underlying group is π1(X; x) so that π1 (X; x) is the completion σ of π1 (X; x). 1. Introduction Gal In this paper, we give define a topological group π1 (X; x) called the Galois funda- mental group, which is associated to any based, connected, locally path connected space X. As the notation suggests, this group is intimately related to the fundamental group π1(X; x). The motivation of the Galois fundamental group is to look at automorphisms of certain covers of a space instead of homotopy classes of loops in the space. It is well known that if X has a universal cover, then π1(X; x) is isomorphic to the group of deck transformations of the universal cover. However, this approach breaks down if X is not semi locally simply connected.
    [Show full text]