COVERING SPACES Contents 1. Introduction 1 2. Fundamental
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Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19Th, 2020
Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19th, 2020 Group Isomorphism Theorems + Group Actions The Isomorphism Theorems for Groups Group Actions Polynomial Invariants and An This material represents x3.2.3-3.3.2 from the course notes. Quotients and Homomorphisms, I Like with rings, we also have various natural connections between normal subgroups and group homomorphisms. To begin, observe that if ' : G ! H is a group homomorphism, then ker ' is a normal subgroup of G. In fact, I proved this fact earlier when I introduced the kernel, but let me remark again: if g 2 ker ', then for any a 2 G, then '(aga−1) = '(a)'(g)'(a−1) = '(a)'(a−1) = e. Thus, aga−1 2 ker ' as well, and so by our equivalent properties of normality, this means ker ' is a normal subgroup. Thus, we can use homomorphisms to construct new normal subgroups. Quotients and Homomorphisms, II Equally importantly, we can also do the reverse: we can use normal subgroups to construct homomorphisms. The key observation in this direction is that the map ' : G ! G=N associating a group element to its residue class / left coset (i.e., with '(a) = a) is a ring homomorphism. Indeed, the homomorphism property is precisely what we arranged for the left cosets of N to satisfy: '(a · b) = a · b = a · b = '(a) · '(b). Furthermore, the kernel of this map ' is, by definition, the set of elements in G with '(g) = e, which is to say, the set of elements g 2 N. Thus, kernels of homomorphisms and normal subgroups are precisely the same things. -
Categories of Sets with a Group Action
Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given. -
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. -
GROUP ACTIONS 1. Introduction the Groups Sn, An, and (For N ≥ 3)
GROUP ACTIONS KEITH CONRAD 1. Introduction The groups Sn, An, and (for n ≥ 3) Dn behave, by their definitions, as permutations on certain sets. The groups Sn and An both permute the set f1; 2; : : : ; ng and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If we label the vertices of the n-gon in a definite manner by the numbers from 1 to n then we can view Dn as a subgroup of Sn. For instance, the labeling of the square below lets us regard the 90 degree counterclockwise rotation r in D4 as (1234) and the reflection s across the horizontal line bisecting the square as (24). The rest of the elements of D4, as permutations of the vertices, are in the table below the square. 2 3 1 4 1 r r2 r3 s rs r2s r3s (1) (1234) (13)(24) (1432) (24) (12)(34) (13) (14)(23) If we label the vertices in a different way (e.g., swap the labels 1 and 2), we turn the elements of D4 into a different subgroup of S4. More abstractly, if we are given a set X (not necessarily the set of vertices of a square), then the set Sym(X) of all permutations of X is a group under composition, and the subgroup Alt(X) of even permutations of X is a group under composition. If we list the elements of X in a definite order, say as X = fx1; : : : ; xng, then we can think about Sym(X) as Sn and Alt(X) as An, but a listing in a different order leads to different identifications 1 of Sym(X) with Sn and Alt(X) with An. -
Right Product Quasigroups and Loops
RIGHT PRODUCT QUASIGROUPS AND LOOPS MICHAEL K. KINYON, ALEKSANDAR KRAPEZˇ∗, AND J. D. PHILLIPS Abstract. Right groups are direct products of right zero semigroups and groups and they play a significant role in the semilattice decomposition theory of semigroups. Right groups can be characterized as associative right quasigroups (magmas in which left translations are bijective). If we do not assume associativity we get right quasigroups which are not necessarily representable as direct products of right zero semigroups and quasigroups. To obtain such a representation, we need stronger assumptions which lead us to the notion of right product quasigroup. If the quasigroup component is a (one-sided) loop, then we have a right product (left, right) loop. We find a system of identities which axiomatizes right product quasigroups, and use this to find axiom systems for right product (left, right) loops; in fact, we can obtain each of the latter by adjoining just one appropriate axiom to the right product quasigroup axiom system. We derive other properties of right product quasigroups and loops, and conclude by show- ing that the axioms for right product quasigroups are independent. 1. Introduction In the semigroup literature (e.g., [1]), the most commonly used definition of right group is a semigroup (S; ·) which is right simple (i.e., has no proper right ideals) and left cancellative (i.e., xy = xz =) y = z). The structure of right groups is clarified by the following well-known representation theorem (see [1]): Theorem 1.1. A semigroup (S; ·) is a right group if and only if it is isomorphic to a direct product of a group and a right zero semigroup. -
Homology Homomorphism
|| Computations Induced (Co)homology homomorphism Daher Al Baydli Department of Mathematics National University of Ireland,Galway Feberuary, 2017 Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 1 / 11 Overview Resolution Chain and cochain complex homology cohomology Induce (Co)homology homomorphism Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 2 / 11 Resolution Definition Let G be a group and Z be the group of integers considered as a trivial ZG-module. The map : ZG −! Z from the integral group ring to Z, given by Σmg g 7−! Σmg , is called the augmentation. Definition Let G be a group. A free ZG-resolution of Z is an exact sequence of free G G @n+1 G @n G @n−1 @2 G @1 ZG-modules R∗ : · · · −! Rn+1 −! Rn −! Rn−1 −! · · · −! R1 −! G G R0 −! R−1 = Z −! 0 G with each Ri a free ZG-module for all i ≥ 0. Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 3 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules. For each n 2 Z, the nth homology module of C is defined to be the quotient module Ker@n Hn(C) = Im@n+1 G We can also construct an induced cochain complex HomZG (R∗ ; A) of abelian groups: G G δn G δn−1 HomZG (R∗ ; A): · · · − HomZG (Rn+1; A) − HomZG (Rn ; A) − G δn−2 δ1 G δ0 G HomZG (Rn−1; A) − · · · − HomZG (R1 ; A) − HomZG (R0 ; A) Chain and cochain complex G Given a ZG-resolution R∗ of Z and any ZG-module A we can use the G tensor product to construct an induced chain complex R∗ ⊗ZG A of G G @n+1 G @n abelian groups: R∗ ⊗ZG A : · · · −! Rn+1 ⊗ZG A −! Rn ⊗ZG A −! G @n−1 @2 G @1 G Rn−1 ⊗ZG A −! · · · −! R1 ⊗ZG A −! R0 ⊗ZG A Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 4 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules. -
Homology Groups of Homeomorphic Topological Spaces
An Introduction to Homology Prerna Nadathur August 16, 2007 Abstract This paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. It concludes with a proof of the equivalence of simplicial and singular homology groups. Contents 1 Simplices and Simplicial Complexes 1 2 Homology Groups 2 3 Singular Homology 8 4 Chain Complexes, Exact Sequences, and Relative Homology Groups 9 ∆ 5 The Equivalence of H n and Hn 13 1 Simplices and Simplicial Complexes Definition 1.1. The n-simplex, ∆n, is the simplest geometric figure determined by a collection of n n + 1 points in Euclidean space R . Geometrically, it can be thought of as the complete graph on (n + 1) vertices, which is solid in n dimensions. Figure 1: Some simplices Extrapolating from Figure 1, we see that the 3-simplex is a tetrahedron. Note: The n-simplex is topologically equivalent to Dn, the n-ball. Definition 1.2. An n-face of a simplex is a subset of the set of vertices of the simplex with order n + 1. The faces of an n-simplex with dimension less than n are called its proper faces. 1 Two simplices are said to be properly situated if their intersection is either empty or a face of both simplices (i.e., a simplex itself). By \gluing" (identifying) simplices along entire faces, we get what are known as simplicial complexes. More formally: Definition 1.3. A simplicial complex K is a finite set of simplices satisfying the following condi- tions: 1 For all simplices A 2 K with α a face of A, we have α 2 K. -
Algebraic Topology
Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6. -
Lecture 1.2: Group Actions
Lecture 1.2: Group actions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8510, Abstract Algebra I M. Macauley (Clemson) Lecture 1.2: Group actions Math 8510, Abstract Algebra I 1 / 29 The symmetric group Definition The group of all permutations of f1;:::; ng is the symmetric group, denoted Sn. We can concisely describe permutations in cycle notation, e.g., 1 2 3 4 as (1 2 3 4). Observation 1 Every permutation can be decomposed into a product of disjoint cycles, and disjoint cycles commute. We usually don't write 1-cycles (fixed points). For example, in S10, we can write 1 2 3 4 5 6 7 8 9 10 as (1 4 6 5) (2 3) (8 10 9). By convention, we'll read cycles from right-to-left, like function composition. [Note. Many sources read left-to-right.] M. Macauley (Clemson) Lecture 1.2: Group actions Math 8510, Abstract Algebra I 2 / 29 The symmetric group Remarks The inverse of the cycle (1 2 3 4) is (4 3 2 1) = (1 4 3 2). If σ is a k-cycle, then jσj = k. If σ = σ1 ··· σm, all disjoint, then jσj = lcm(jσ1j;:::; jσmj). A 2-cycle is called a transposition. Every cycle (and hence element of Sn) can be written as a product of transpositions: (1 2 3 ··· k) = (1 k) (1 k −1) ··· (1 3) (1 2): We say σ 2 Sn is even if it can be written as a product of an even number of transpositions, otherwise it is odd. -
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. -
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. -
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.