An Introduction to Topological Groups
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Arxiv:1006.1489V2 [Math.GT] 8 Aug 2010 Ril.Ias Rfie Rmraigtesre Rils[14 Articles Survey the Reading from Profited Also I Article
Pure and Applied Mathematics Quarterly Volume 8, Number 1 (Special Issue: In honor of F. Thomas Farrell and Lowell E. Jones, Part 1 of 2 ) 1—14, 2012 The Work of Tom Farrell and Lowell Jones in Topology and Geometry James F. Davis∗ Tom Farrell and Lowell Jones caused a paradigm shift in high-dimensional topology, away from the view that high-dimensional topology was, at its core, an algebraic subject, to the current view that geometry, dynamics, and analysis, as well as algebra, are key for classifying manifolds whose fundamental group is infinite. Their collaboration produced about fifty papers over a twenty-five year period. In this tribute for the special issue of Pure and Applied Mathematics Quarterly in their honor, I will survey some of the impact of their joint work and mention briefly their individual contributions – they have written about one hundred non-joint papers. 1 Setting the stage arXiv:1006.1489v2 [math.GT] 8 Aug 2010 In order to indicate the Farrell–Jones shift, it is necessary to describe the situation before the onset of their collaboration. This is intimidating – during the period of twenty-five years starting in the early fifties, manifold theory was perhaps the most active and dynamic area of mathematics. Any narrative will have omissions and be non-linear. Manifold theory deals with the classification of ∗I thank Shmuel Weinberger and Tom Farrell for their helpful comments on a draft of this article. I also profited from reading the survey articles [14] and [4]. 2 James F. Davis manifolds. There is an existence question – when is there a closed manifold within a particular homotopy type, and a uniqueness question, what is the classification of manifolds within a homotopy type? The fifties were the foundational decade of manifold theory. -
On S-Topological Groups
Mathematica Moravica Vol. 18-2 (2014), 35–44 On s-Topological Groups Muhammad Siddique Bosan, Moiz ud Din Khan and Ljubiša D.R. Kočinac∗) Abstract. In this paper we study the class of s-topological groups and a wider class of S-topological groups which are defined by us- ing semi-open sets and semi-continuity introduced by N. Levine. It is shown that these groups form a generalization of topological groups, and that they are different from several distinct notions of semitopolog- ical groups which appear in the literature. Counterexamples are given to strengthen these concepts. Some basic results and applications of s- and S-topological groups are presented. Similarities with and differ- ences from topological groups are investigated. 1. Introduction If a set is endowed with algebraic and topological structures, then it is natural to consider and investigate interplay between these two structures. The most natural way for such a study is to require algebraic operations to be continuous. It is the case in investigation of topological groups: the multi- plication mapping and the inverse mapping are continuous. Similar situation is with topological rings, topological vector spaces and so on. However, it is also natural to see what will happen if some of algebraic operations sat- isfy certain weaker forms of continuity. Such a study in connection with topological groups started in the 1930s and 1950s and led to investigation of semitopological groups (the multiplication mapping is separately con- tinuous), paratopological groups (the multiplication is jointly continuous), quasi-topological groups (which are semitopological groups with continu- ous inverse mapping). -
Functorial Constructions in Paratopological Groups Reflecting Separation Axioms
Functorial constructions in paratopological groups reflecting separation axioms Mikhail Tkachenko Universidad Aut´onomaMetropolitana, Mexico City [email protected] Brazilian Conference on General Topology and Set Theory S~aoSebasti~ao,Brazil, 2013 In honor of Ofelia T. Alas Contents: 1. Three known functorial constructions 2. Each axiom of separation has its functorial reflection 3. `Internal' description of the groups Tk (G) 4. Properties of the functors Tk 's 5. Products and functors 6. Some applications A paratopological group is a group G with topology such that multiplication in G is jointly continuous. `topological' =) `paratopological' =) `semitopological' Let (G; τ) be a paratopological group and τ −1 = fU−1 : U 2 τg be the conjugate topology of G. Then G 0 = (G; τ −1) is also a paratopological group and the inversion in G is a homeomorphism of (G; τ) onto (G; τ −1). Let τ ∗ = τ _ τ −1 be the least upper bound of τ and τ −1. Then G ∗ = (G; τ ∗) is a topological group associated to G. ∗ For the Sorgenfrey line S, the topological group S is discrete. Paratopological and semitopological groups A semitopological group is an abstract group G with topology τ such that the left and right translations in G are continuous or, equivalently, multiplication in G is separately continuous. `topological' =) `paratopological' =) `semitopological' Let (G; τ) be a paratopological group and τ −1 = fU−1 : U 2 τg be the conjugate topology of G. Then G 0 = (G; τ −1) is also a paratopological group and the inversion in G is a homeomorphism of (G; τ) onto (G; τ −1). -
Syllabus for M.Phil
Structure of Syllabus for M.Phil. in Mathematics Semester-I Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) Research Methodology: MPHILMA-101 Theory Research Foundation 20 5 25 2 Research Methodology: Computer Application in MPHILMA-102 Practical 20 5 25 2 Research Elective: Any two papers to be chosen from Table-I, 40 10 50 4 based on research interest of MPHILMA-103 Theory + + + + the students and availability of suitable teachers/ 40 10 50 4 Supervisors. Total 120 30 150 12 Semester-II Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) Elective: Any three papers to 40 10 50 4 be chosen from Table-II, + + + + MPHILMA-201 Theory based on research interest of 40 10 50 4 the students and availability of + + + + suitable teachers/Supervisors. 40 10 50 4 Total 120 30 150 12 Semester-III Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) Preliminary MPHILMA-301 Theory 75 75 6 Dissertation MPHILMA-302 Theory Viva-voce 25 25 2 Total 100 100 8 Semester-IV Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) MPHILMA-401 Theory Final Dissertation 75 75 6 MPHILMA-402 Theory Viva-voce 25 25 2 Total 100 100 8 Table: I Elective Papers for MPHILMA-103 (M1) to MPHILMA-103 (M18) Elective Paper Title of the Paper Sub-Code M1 Theory of Convergence -
An Overview of Topological Groups: Yesterday, Today, Tomorrow
axioms Editorial An Overview of Topological Groups: Yesterday, Today, Tomorrow Sidney A. Morris 1,2 1 Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia; [email protected]; Tel.: +61-41-7771178 2 Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia Academic Editor: Humberto Bustince Received: 18 April 2016; Accepted: 20 April 2016; Published: 5 May 2016 It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups. Walter Rudin’s book “Fourier Analysis on Groups” [2] includes an elegant proof of the Pontryagin-van Kampen Duality Theorem. Much gentler than these is “Introduction to Topological Groups” [3] by Taqdir Husain which has an introduction to topological group theory, Haar measure, the Peter-Weyl Theorem and Duality Theory. Of course the book “Topological Groups” [4] by Lev Semyonovich Pontryagin himself was a tour de force for its time. P. S. Aleksandrov, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko described this book in glowing terms: “This book belongs to that rare category of mathematical works that can truly be called classical - books which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians”. The final book I mention from my graduate studies days is “Topological Transformation Groups” [5] by Deane Montgomery and Leo Zippin which contains a solution of Hilbert’s fifth problem as well as a structure theory for locally compact non-abelian groups. -
Lie Group and Geometry on the Lie Group SL2(R)
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear -
Extensions of Topological and Semitopological Groups and the Product Operation
Commentationes Mathematicae Universitatis Carolinae Aleksander V. Arhangel'skii; Miroslav Hušek Extensions of topological and semitopological groups and the product operation Commentationes Mathematicae Universitatis Carolinae, Vol. 42 (2001), No. 1, 173--186 Persistent URL: http://dml.cz/dmlcz/119232 Terms of use: © Charles University in Prague, Faculty of Mathematics and Physics, 2001 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Comment.Math.Univ.Carolin. 42,1 (2001)173–186 173 Extensions of topological and semitopological groups and the product operation A.V. Arhangel’skii, M. Huˇsek Abstract. The main results concern commutativity of Hewitt-Nachbin realcompactifica- tion or Dieudonn´ecompletion with products of topological groups. It is shown that for every topological group G that is not Dieudonn´ecomplete one can find a Dieudonn´e complete group H such that the Dieudonn´ecompletion of G × H is not a topological group containing G×H as a subgroup. Using Korovin’s construction of Gδ-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups. Keywords: topological group, Dieudonn´ecompletion, PT-group, realcompactness, Moscow space, C-embedding, product Classification: 22A05, 54H11, 54D35, 54D60 §0. Introduction Although many of our general results are valid for more general spaces, we shall assume that all the spaces under consideration are Tychonoff. -
Topological Groups
Topological Groups Siyan Daniel Li-Huerta September 8, 2020 Definition A topological group is a group G with a topological space structure such that The multiplication map µ : G × G ! G is continuous, The inverse map ι : G ! G is continuous. Example Any group G with the discrete topology, The group R under addition with the Euclidean topology, × The group C under multiplication with the Euclidean topology, 1 × × The subgroup S = fz 2 C j jzj = 1g ⊂ C , The group GLn(R) = fA 2 Matn×n(R) j det A 6= 0g under multiplication with the Euclidean topology. Remark Any finite Hausdorff topological group G must be discrete. 2 / 9 Let G be a topological group, and let g be in G. Then left translation (i.e. multiplication) by g equals the composite (g;id) µ G / fgg × G / G; so it's continuous. Its inverse is left translation by g −1, so it's even a homeomorphism. The same holds for right translation. Definition Let X be a topological space. We say X is homogeneous if, for every x and y in X , there exists a homeomorphism f : X ! X such that f (x) = y. Example Any topological group G is homogeneous (using left translation), n n+1 The n-sphere S = f~v 2 R j k~vk= 1g is homogeneous (using rotations). Thus topological groups are very special topological spaces: they look \the same" around every point. So it often suffices to study neighborhoods of one point. Let's pick the point 1! 3 / 9 Proposition Let G be a topological group, and let U be a neighborhood of 1. -
Introducing Group Theory with Its Raison D'etre for Students
Introducing group theory with its raison d’etre for students Hiroaki Hamanaka, Koji Otaki, Ryoto Hakamata To cite this version: Hiroaki Hamanaka, Koji Otaki, Ryoto Hakamata. Introducing group theory with its raison d’etre for students. INDRUM 2020, Université de Carthage, Université de Montpellier, Sep 2020, Cyberspace (virtually from Bizerte), Tunisia. hal-03113982 HAL Id: hal-03113982 https://hal.archives-ouvertes.fr/hal-03113982 Submitted on 18 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Introducing group theory with its raison d’être for students 1 2 3 Hiroaki Hamanaka , Koji Otaki and Ryoto Hakamata 1Hyogo University of Teacher Education, Japan, [email protected], 2Hokkaido University of Education, Japan, 3Kochi University, Japan This paper reports results of our sequence of didactic situations for teaching fundamental concepts in group theory—e.g., symmetric group, generator, subgroup, and coset decomposition. In the situations, students in a preservice teacher training course dealt with such concepts, together with card-puzzle problems of a type. And there, we aimed to accompany these concepts with their raisons d’être. Such raisons d’être are substantiated by the dialectic between tasks and techniques in the praxeological perspective of the anthropological theory of the didactic. -
Group Theory
Group theory March 7, 2016 Nearly all of the central symmetries of modern physics are group symmetries, for simple a reason. If we imagine a transformation of our fields or coordinates, we can look at linear versions of those transformations. Such linear transformations may be represented by matrices, and therefore (as we shall see) even finite transformations may be given a matrix representation. But matrix multiplication has an important property: associativity. We get a group if we couple this property with three further simple observations: (1) we expect two transformations to combine in such a way as to give another allowed transformation, (2) the identity may always be regarded as a null transformation, and (3) any transformation that we can do we can also undo. These four properties (associativity, closure, identity, and inverses) are the defining properties of a group. 1 Finite groups Define: A group is a pair G = fS; ◦} where S is a set and ◦ is an operation mapping pairs of elements in S to elements in S (i.e., ◦ : S × S ! S: This implies closure) and satisfying the following conditions: 1. Existence of an identity: 9 e 2 S such that e ◦ a = a ◦ e = a; 8a 2 S: 2. Existence of inverses: 8 a 2 S; 9 a−1 2 S such that a ◦ a−1 = a−1 ◦ a = e: 3. Associativity: 8 a; b; c 2 S; a ◦ (b ◦ c) = (a ◦ b) ◦ c = a ◦ b ◦ c We consider several examples of groups. 1. The simplest group is the familiar boolean one with two elements S = f0; 1g where the operation ◦ is addition modulo two. -
A Linear Algebraic Group? Skip Garibaldi
WHATIS... ? a Linear Algebraic Group? Skip Garibaldi From a marketing perspective, algebraic groups group over F, a Lie group, or a topological group is are poorly named.They are not the groups you met a “group object” in the category of affine varieties as a student in abstract algebra, which I will call over F, smooth manifolds, or topological spaces, concrete groups for clarity. Rather, an algebraic respectively. For algebraic groups, this implies group is the analogue in algebra of a topological that the set G(K) is a concrete group for each field group (fromtopology) or a Liegroup (from analysis K containing F. and geometry). Algebraic groups provide a unifying language Examples for apparently different results in algebra and The basic example is the general linear group number theory. This unification can not only sim- GLn for which GLn(K) is the concrete group of plify proofs, it can also suggest generalizations invertible n-by-n matrices with entries in K. It is the and bring new tools to bear, such as Galois co- collection of solutions (t, X)—with t ∈ K and X an homology, Steenrod operations in Chow theory, n-by-n matrix over K—to the polynomial equation etc. t ·det X = 1. Similar reasoning shows that familiar matrix groups such as SLn, orthogonal groups, Definitions and symplectic groups can be viewed as linear A linear algebraic group over a field F is a smooth algebraic groups. The main difference here is that affine variety over F that is also a group, much instead of viewing them as collections of matrices like a topological group is a topological space over F, we view them as collections of matrices that is also a group and a Lie group is a smooth over every extension K of F. -
An Introduction to Computational Group Theory Ákos Seress
seress.qxp 5/21/97 4:13 PM Page 671 An Introduction to Computational Group Theory Ákos Seress an one rotate only one corner piece in groups to a certain extent, there are two systems Rubik’s cube? What are the energy which are particularly well suited for computa- levels of the buckyball molecule? Are tions with groups: GAP and Magma. Also, nu- the graphs on Figure 1 isomorphic? merous stand-alone programs and smaller sys- What is the Galois group of the poly- tems are available. Cnomial x8 +2x7 +28x6 + 1728x + 3456? What GAP can be obtained by anonymous ftp from are the possible symmetry groups of crystals? servers on three continents; the addresses can These are all questions which, directly or in a not be found on the World Wide Web page so obvious way, lead to problems in computa- http://www-groups.dcs.st-and.ac.uk/. tional group theory. For the availability of Magma, please see the Algebraic structures are well suited for ma- World Wide Web page http://www.maths. chine computations. One reason for that is that usyd.edu.au:8000/comp/magma/. we can describe large objects very concisely by The important subareas of CGT correspond a set of generators: for example, 50 bits are to the most frequently used representations of enough to define GL5(2), a group of order groups: permutation groups, matrix groups, and 9999360, by two 0-1 matrices of size 5 5. groups defined by generators and relators, as Even more importantly, often we can find a gen-× well as to perhaps the most powerful tool for the erating set which reflects the structure of the investigation of groups, representation theory.