COVERING SPACES, GRAPHS, and GROUPS Contents 1. Covering
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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. -
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. -
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. -
Chapter 9 Partitions of Unity, Covering Maps ~
Chapter 9 Partitions of Unity, Covering Maps ~ 9.1 Partitions of Unity To study manifolds, it is often necessary to construct var- ious objects such as functions, vector fields, Riemannian metrics, volume forms, etc., by gluing together items con- structed on the domains of charts. Partitions of unity are a crucial technical tool in this glu- ing process. 505 506 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ The first step is to define “bump functions”(alsocalled plateau functions). For any r>0, we denote by B(r) the open ball n 2 2 B(r)= (x1,...,xn) R x + + x <r , { 2 | 1 ··· n } n 2 2 and by B(r)= (x1,...,xn) R x1 + + xn r , its closure. { 2 | ··· } Given a topological space, X,foranyfunction, f : X R,thesupport of f,denotedsuppf,isthe closed set! supp f = x X f(x) =0 . { 2 | 6 } 9.1. PARTITIONS OF UNITY 507 Proposition 9.1. There is a smooth function, b: Rn R, so that ! 1 if x B(1) b(x)= 2 0 if x Rn B(2). ⇢ 2 − See Figures 9.1 and 9.2. 1 0.8 0.6 0.4 0.2 K3 K2 K1 0 1 2 3 Figure 9.1: The graph of b: R R used in Proposition 9.1. ! 508 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ > Figure 9.2: The graph of b: R2 R used in Proposition 9.1. ! Proposition 9.1 yields the following useful technical result: 9.1. PARTITIONS OF UNITY 509 Proposition 9.2. Let M be a smooth manifold. -
Theory of Covering Spaces
THEORY OF COVERING SPACES BY SAUL LUBKIN Introduction. In this paper I study covering spaces in which the base space is an arbitrary topological space. No use is made of arcs and no assumptions of a global or local nature are required. In consequence, the fundamental group ir(X, Xo) which we obtain fre- quently reflects local properties which are missed by the usual arcwise group iri(X, Xo). We obtain a perfect Galois theory of covering spaces over an arbitrary topological space. One runs into difficulties in the non-locally connected case unless one introduces the concept of space [§l], a common generalization of topological and uniform spaces. The theory of covering spaces (as well as other branches of algebraic topology) is better done in the more general do- main of spaces than in that of topological spaces. The Poincaré (or deck translation) group plays the role in the theory of covering spaces analogous to the role of the Galois group in Galois theory. However, in order to obtain a perfect Galois theory in the non-locally con- nected case one must define the Poincaré filtered group [§§6, 7, 8] of a cover- ing space. In §10 we prove that every filtered group is isomorphic to the Poincaré filtered group of some regular covering space of a connected topo- logical space [Theorem 2]. We use this result to translate topological ques- tions on covering spaces into purely algebraic questions on filtered groups. In consequence we construct examples and counterexamples answering various questions about covering spaces [§11 ]. In §11 we also -
Topology and Geometry of 2 and 3 Dimensional Manifolds
Topology and Geometry of 2 and 3 dimensional manifolds Chris John May 3, 2016 Supervised by Dr. Tejas Kalelkar 1 Introduction In this project I started with studying the classification of Surface and then I started studying some preliminary topics in 3 dimensional manifolds. 2 Surfaces Definition 2.1. A simple closed curve c in a connected surface S is separating if S n c has two components. It is non-separating otherwise. If c is separating and a component of S n c is an annulus or a disk, then it is called inessential. A curve is essential otherwise. An observation that we can make here is that a curve is non-separating if and only if its complement is connected. For manifolds, connectedness implies path connectedness and hence c is non-separating if and only if there is some other simple closed curve intersecting c exactly once. 2.1 Decomposition of Surfaces Lemma 2.2. A closed surface S is prime if and only if S contains no essential separating simple closed curve. Proof. Consider a closed surface S. Let us assume that S = S1#S2 is a non trivial con- nected sum. The curve along which S1 n(disc) and S2 n(disc) where identified is an essential separating curve in S. Hence, if S contains no essential separating curve, then S is prime. Now assume that there exists a essential separating curve c in S. This means neither of 2 2 the components of S n c are disks i.e. S1 6= S and S2 6= S . -
Configuration Spaces of Graphs
Configuration Spaces of Graphs Daniel Lütgehetmann Master’s Thesis Berlin, October 2014 Supervisor: Prof. Dr. Holger Reich Selbstständigkeitserklärung Hiermit versichere ich, dass ich die vorliegende Masterarbeit selbstständig und nur unter Zuhilfenahme der angegebenen Quellen erstellt habe. Daniel Lütgehetmann Erstkorrektor: Prof. Dr. Holger Reich Zweitkorrektor: Prof. Dr. Elmar Vogt Contents Introduction iii 1 Preliminaries1 1.1 Configuration Spaces . .1 1.2 Reduction to Connected Spaces . .3 1.3 The Rational Cohomology as Σn-Representation . .3 1.4 Some Representation Theory of the Symmetric Group . .4 2 A Deformation Retraction7 2.1 The Poset PnΓ ................................8 2.2 Cube Complexes and Cube Posets . 11 2.3 The Geometric Realization of a Cube Poset . 14 2.4 The Cube Complex KnΓ .......................... 16 2.5 The Embedding into the Configuration Space . 18 2.6 The Deformation Retraction . 21 2.7 The Path Metric on KnΓ .......................... 25 2.8 The Curvature of KnΓ ........................... 26 3 The Homology of Confn(Γ) 29 3.1 Representation Stability . 32 3.2 The Generalized Euler Characteristic of Confn(Γ) ........... 34 3.3 Graphs With At Most One Branched Vertex . 35 3.4 The General Case of Finite Graphs . 38 3.5 Applications . 39 3.6 Next Steps . 41 4 A Sheaf-Theoretic Approach 43 4.1 Sheaf Theory . 43 4.2 The Spectral Sequence . 46 4.3 The Stalks . 50 i Contents 4.4 Decomposition of Rq(inc )(Q) into Easier Sheaves . 52 ∗ 4.5 Sketch of the Proof for Manifolds . 53 BibliographyII Appendix III ii Introduction Let Γ be a graph and n a natural number. We want to understand the ordered configuration space Confn(Γ) consisting of all n-tuples x = (x1; : : : ; xn) of elements in Γ such that x = x for i = j, endowed with the subspace topology induced by the i 6 j 6 inclusion Conf (Γ) Γ n. -
Brauer Groups of Linear Algebraic Groups
Volume 71, Number 2, September 1978 BRAUERGROUPS OF LINEARALGEBRAIC GROUPS WITH CHARACTERS ANDY R. MAGID Abstract. Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d - l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, characterless, linear algebraic group over an algebraically closed field of characteristic zero: the Brauer group is finite-in fact, it is the Schur multiplier of the fundamental group of the algebraic group [6, Theorem 4.1, p. 299]. In this note, we extend these calculations to an arbitrary connected linear group in characteristic zero. The main result is the determination of the Brauer group of a ¿/-dimen- sional affine algebraic torus, which is shown to be (Q/Z)(n) where n = d(d — l)/2. (This result is noted in [6, 4.8, p. 301] when d = 2.) We then show that if G is a connected linear algebraic group whose character group has Z-rank d, then the Brauer group of G is C X (Q/Z)<n) where n is as above and C is finite. We adopt the following notational conventions: 7ms an algebraically closed field of characteristic zero, and T = (F*)id) is a ^-dimensional affine torus over F. -
Introduction
COVERING GROUPS OF NON-CONNECTED TOPOLOGICAL GROUPS REVISITED Ronald Brown and Osman Mucuk School of Mathematics University of Wales Bangor, Gwynedd LL57 1UT, U.K. Received 11 March 1993 , Revision 31 May 1993 Published in Math. Proc. Camb. Phil. Soc, 115 (1994) 97-110. This version: 18 Dec. 2006, with some new references. Introduction All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a con- nected topological group with identity e, and let p : X~ ! X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any pointe ~ in X~ with pe~ = e there is a unique structure of topological group on X~ such thate ~ is the identity and p: X~ ! X is a morphism of groups. We say that the structure of topological group on X lifts to X~. It is less generally appreciated that this result fails for the non-connected case. The set ¼0X of path components of X forms a non-trivial group which acts on the abelian group ¼1(X; e) via conjugation in X. R.L. Taylor [20] showed that the topological group X determines an obstruction class kX in 3 H (¼0X; ¼1(X; e)), and that the vanishing of kX is a necessary and su±cient condition for the lifting of the topological group structure on X to a universal covering so that the projection is a morphism. Further, recent work, for example Huebschmann [15], shows there are geometric applications of the non-connected case. -
Arxiv:2003.07426V2 [Math.CT] 22 Jul 2021 Modifying Some Constructions of Adams [1] for Connected CW Complexes with Base Point
BROWN REPRESENTABILITY FOR DIRECTED GRAPHS ZACHARY MCGUIRK † AND BYUNGDO PARK ‡ ABSTRACT. We prove that any contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying the additivity axiom and the Mayer-Vietoris axiom is repre- sentable. 1. INTRODUCTION The homotopy theory of directed graphs is a discrete analogue of homotopy theory in algebraic topology. In topology, a homotopy between two continuous maps is defined by an interpolating family of continuous maps parametrized by a closed interval [0; 1]. Its discrete analogue we study uses a directed line graph keeping track of a discrete change of directed graph maps. See for example Grigor’yan, Lin, Muranov, and Yau [8]. Efforts towards a homotopy theory for graphs extend back to the 1970’s and 1980’s with some results by Gianella [6] and Malle [16]. However, the most recent variant of graph homotopy theory appears to have taken off with a paper by Chen, Yau, and Yeh from 2001 [2], before culminating in the 2014 work of Grigor’yan, Lin, Muranov, and Yau [8]. Recently, there is increased interest in this new notion of homotopy for directed graphs because it was shown by Grigor’yan, Jimenez, Muranov, and Yau in [7] that the path-space homology theory of directed graphs is invariant under this version of graph homotopy. The path-space homology and cohomology theories for directed graphs were studied by Grigor’yan, Lin, Muranov, and Yau [9] and by Grigor’yan, Muranov, and Yau in [10] and [11]. The homology theory developed in the references above is quite natural, computable, and can be non-trivial for degrees greater than one, depending on the lengths of admissible, @-invariant paths in a directed graph. -
On Distinct Finite Covers of 3-Manifolds
ON DISTINCT FINITE COVERS OF 3-MANIFOLDS STEFAN FRIEDL, JUNGHWAN PARK, BRAM PETRI, JEAN RAIMBAULT, AND ARUNIMA RAY Abstract. Every closed orientable surface S has the following property: any two connected covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic questions. 1. Introduction Finite covers are an important tool in manifold topology. For instance, they provide us with many examples of manifolds: in hyperbolic geometry, covers have been used to construct many pairwise non-isometric manifolds of the same volume [BGLM02, Zim94] and pairs of hyperbolic surfaces that are isospectral, yet not isometric [Sun85]. Finite cyclic covers of knot complements yield the torsion invariants of knots. In general, covering spaces provide a geometric means of studying the subgroup structure of the fundamental group, which is particularly powerful in the case of 3-manifolds, which in almost all cases are determined by their fundamental group. Given a manifold, it is natural to ask which other manifolds appear as its covering spaces. Famous examples of questions of this form are Thurston's conjectures on covers of hyperbolic 3-manifolds: the virtual positive Betti number, virtual Haken, and virtual fibering conjectures, that were resolved by Agol [Ago13] based on the work of Wise and many others. In this paper, we study the wealth (or poverty) of finite covers of a given 3-manifold. Observe that every compact 3-manifold with infinite fundamental group has infinitely many finite covers. -
Algebraic Topology I Fall 2016
31 Local coefficients and orientations The fact that a manifold is locally Euclidean puts surprising constraints on its cohomology, captured in the statement of Poincaré duality. To understand how this comes about, we have to find ways to promote local information – like the existence of Euclidean neighborhoods – to global information – like restrictions on the structure of the cohomology. Today we’ll study the notion of an orientation, which is the first link between local and global. The local-to-global device relevant to this is the notion of a “local coefficient system,” which is based on the more primitive notion of a covering space. We merely summarize that theory, since it is a prerequisite of this course. Definition 31.1. A continuous map p : E ! B is a covering space if (1) every point pre-image is a discrete subspace of E, and (2) every b 2 B has a neighborhood V admitting a map p−1(V ) ! p−1(b) such that the induced map =∼ p−1(V ) / V × p−1(b) p pr1 # B y is a homeomorphism. The space B is the “base,” E the “total space.” Example 31.2. A first example is given by the projection map pr1 : B × F ! B where F is discrete. A covering space of this form is said to be trivial, so the covering space condition can be rephrased as “local triviality.” The first interesting example is the projection map Sn ! RPn obtained by identifying antipodal maps on the sphere. This example generalizes in the following way. Definition 31.3.