UNIVERSIDAD SAN FRANCISCO DE QUITO Colegio de Ciencias e Ingenier´ıa THE FUNDAMENTAL GROUP OF A GROUP ACTING ON A TOPOLOGICAL SPACE Bryan Patricio Maldonado Puente Director de Tesis: John R. Skukalek Ph.D Tesis de grado presentada como requisito para la obtenci´ondel t´ıtulode: Licenciado en Matem´aticas Quito, Diciembre 2013 UNIVERSIDAD SAN FRANCISCO DE QUITO Colegio de Ciencias e Ingenier´ıa HOJA DE APROBACION´ DE TESIS The Fundamental Group of a Group Acting on a Topological Space Bryan Patricio Maldonado Puente John R. Skukalek Ph.D Director de Tesis Andrea Moreira Ph.D Miembro del Comit´ede Tesis Carlos Jimenez Ph.D Miembro del Comit´ede Tesis Eduardo Alba Ph.D Director Departamento de Matem´atica Cesar Zambrano Ph.D Decano Escuela de Ciencias Colegio de Ciencias e Ingenier´ıa Quito, Diciembre 2013 c DERECHOS DE AUTOR Por medio del presente documento certifico que he le´ıdola Pol´ıticade Propiedad Intelectual de la Universidad San Francisco de Quito y estoy de acuerdo con su contenido, por lo que los derechos de propiedad intelectual del presente trabajo de investigacin quedan sujetos a lo dispuesto en la Pol´ıtica. Asimismo, autorizo a la USFQ para que realice la digitalizaci´ony publicaci´onde este trabajo de investigaci´onen el repositorio virtual, de conformidad a lo dispuesto en el Art. 144 de la Ley Org´anicade Educaci´onSuperior. Firma: Nombre: Bryan Patricio Maldonado Puente C.I.: 1720579075 Fecha: 13 de diciembre de 2013 Exotic or ordinary, glamorous or plain, exciting or boring - it's all in our point-of-view. - Jonathan Lockwood Huie 5 Acknowledgements First of all, I want to thank my thesis advisor John R. Skukalek Ph.D in Mathematics for giving me the support and guidance I needed to continue with my work during this learning process of writing an academic article; since I started with this project I have discovered a new field in mathematics thanks to him. I also would like to mention James Montaldi Ph.D in Mathematics from University of Manchester, without whose scholarly correspondence I would not have been able to analyze so well the topics explained in Chapters 4 and 5 of this paper. Despite the distance, Professor Montaldi helped me selflessly with my doubts in algebraic topology. Finally, to Emily Sabo who reviewed this paper for grammatical correctness. 6 Abstract In 1966, F. Rhodes introduced the idea of the fundamental group of a group G of homeomor- phisms of a topological space X. His article contains summarized proofs of important results and has been studied since then primarily because the category of transformation group is more general than the category of topological spaces. In this thesis, a thorough study for Rhodes's work is presented providing examples to enrich the theory. Dr. James Montaldi from University of Manchester has recently contributed to this theory with a more general and applicable way of Rhodes's main theorem. His results are also analyzed here. 7 Resumen En 1966, F. Rhodes introdujo la idea del grupo fundamental de un grupo G de homeomorfis- mos de un espacio topol´ogico X. Su art´ıculocontiene demostraciones que resumen los resulta- dos importantes y ha sido estudiado desde entonces, principalmente debido a que la categor´ıa de grupo de transformacin es m´asgeneral que la categor´ıade los espacios topol´ogicos.En esta tesis, un estudio a fondo del trabajo de Rhodes se presenta con ejemplos para enriquecer la teora. El Dr. James Montaldi de la Universidad de Manchester ha contribuido recientemente a esta teor´ıacon una forma m´asgeneral y aplicable del teorema principal de Rhodes. Sus resultados tambi´ense analizan aqu´ı. Contents Declaration of Authorship 3 Acknowledgements 5 Abstract 6 List of Figures 10 Symbols 11 1 Introduction 13 2 Background 16 2.1 Topological Spaces and Continuous Functions . 16 2.2 The Fundamental Group . 21 2.2.1 The Euclidean Space . 22 2.2.2 The Sphere . 23 2.3 Isometries of Topological Spaces . 26 2.3.1 Dihedral Group . 27 2.3.2 The Orthogonal Group . 28 3 Groups acting on Topological Spaces 30 3.1 The Transformation Group . 30 3.2 The Fundamental Group of a Transformation Group . 31 8 9 3.3 Trivial Groups and Simple Connected Spaces . 39 4 Relationship between σ(X; x0;G) and π1(X; x0) 41 4.1 Exact Sequences . 41 4.2 Connection between π1(X; x0) and the group G ................. 42 4.3 Group of Automorphisms of π1(X; x0) induced by k .............. 44 4.4 Connection between σ(X; x0;G) and the group G ................ 47 4.5 Subgroups of σ(X; x0;G).............................. 51 4.6 Rhodes's Theorem . 52 4.7 Finite Groups . 55 5 Morphisms between Transformation Groups 58 5.1 The Category . 58 5.2 Homotopy type . 59 5.3 Actions on S1 .................................... 61 5.4 S1 acting on Topological Spaces . 65 5.5 Product Spaces . 66 6 Conclusions and Future Work 70 List of Figures 2.1 Path described by a continuous function . 18 2.2 Homotopic loops on a simply-connected region . 20 2.3 Homotopy over S2 ................................. 25 2.4 Dihedral group of an equilateral triangle . 27 2.5 Orthogonal Group on E2 ............................. 29 3.1 Path composition for a transformation group . 32 3.2 Associativity law described over I × I ...................... 37 3.3 Equivalence classes for paths of order g ..................... 39 4.1 Automorphism induced by a preferred path at x0 ................ 43 4.2 Continuous action of a group on a topological space . 48 4.3 Homotopy between λ and its automorphism K(λ)............... 50 4.4 Figure eight . 56 4.5 Homotopies of the topological set P~n ....................... 56 1 5.1 Additive representation of the generator element of σ(S ; Zn)......... 63 1 5.2 Subtractive representation of the generator element of σ(S ; Zn)....... 64 5.3 The 3D figure eight as a cartesian product E × S1 ............... 69 5.4 The torus T2 as a cartesian product S1 × S1 .................. 69 10 Symbols N Natural Numbers Z Integers R Real Numbers C Complex Numbers ; Empty Set Ω(X; x0) Set of all loops in X with base-point x0 fλg Representative of the equivalent class of λ π1(X; x0) Fundamental Group of the Topological Space X with base-point x0 En Euclidean Space of dimension n Sn Sphere of dimension n T2 Torus Pn Regular Polygon with n sides Dn Dihedral Group of order 2n On Orthogonal Group of dimension n (X; G) Transformation Group of the group G acting on X [f; g] Equivalent class of paths of order g σ(X; x0;G) Fundamental Group of (X; G) with base-point x0 fπ1(X; x0); Kg Set of equivalent classes fλ; gg =∼ Isomorphic to ∼x0 Homotopic modulo x0 (g;x0) Paths of order g modulo x0 ≤ Subgroup of (when using groups) E Normal Subgroup of hai Group Generated by a o Semidirect Product 11 To my parents, without whose love and support I would not be able to accomplish my dreams Chapter 1 Introduction The fundamental group of a transformation group (X; G) of a group G acting on a topological space X encompasses information of the fundamental group π1(X; x0) and the action of G on X. We will prove in detail some of the results presented by F. Rhodes [10] in his article \On the Fundamental Group of a Transformation Group," providing examples involving well- know topological spaces as Euclidean space, n−sided polygons, n−spheres and the torus in which groups such as the integers, the orthogonal matrices, and cyclic groups will be acting. The objective is to take care of the detail in the proofs and to nail down this theory to a less abstract sense with visual examples of transformation groups. The last part of Chapter 4 and most of Chapter 5 talks about relatively recent studies on homology theory and planar n−body choreographies made by Dr. James Montaldi [11, 12] that provides a simpler way to state Rhodes's main results. Chapter 2 gives a small introduction to topics in topology that play an important rule in what we are trying to do. Concepts such as continuous functions, paths, and fundamental groups are introduced in order to provide a start basis for what we will build later on. Once it is clear what a topological space and its fundamental group are, we are ready to introduce 13 14 the concept of a group action. The last part of this chapter introduces to the reader some interesting groups that act as isometries of topological spaces defined previously. Chapter 3 defines what a group action is. We then define the concept of the fundamental group of a transformation group as done by Rhodes in his article. We are going to use the idea of paths to define the elements in this fundamental group. A graphic representation for the group operation is presented and it is proved that in fact this set of paths with a binary operation is a group. Finally, based on this definition, we are going to prove interesting facts about a transformation group when G is trivial, and when π1(X; x0) is trivial. Chapter 4 is the core of this undergraduate thesis. The general objective here is to generate an explicit relationship between both fundamental groups (π1(X; x0) and σ(X; x0;G)) and the group G acting over X. First we introduce a relation between these three groups (π1; σ and G) in terms of exact sequences. At this point we start thinking how these three groups should be related and how these sets can be compared.