Basic Topology 2: Topological Groups, Topology of Manifolds and Lie Groups

Preface
A Note on Basic Topology—Volumes 1–3
Basic Topology—Volume 1: Metric Spaces and General Topology
Basic Topology—Volume 2: Topological Groups, Topology of Manifolds and Lie Groups
Basic Topology—Volume 3: Algebraic Topology and Topology of Fiber Bundles
Contents
About the Authors
1 Background on Algebra, Topology and Analysis
1.1 Some Basic Concepts on Algebraic Structures and Their Homomorphisms
1.1.1 Groups and Fundamental Homomorphism Theorem
1.1.2 Concept of Group Actions
1.1.3 Ring of Real-Valued Continuous Functions
1.1.4 Fundamental Theorem of Algebra
1.1.5 Vector Space and Linear Transformations
1.1.6 Basis for a Vector Space
1.1.7 Algebra over a Field
1.2 Some Basic Results of Set Topology
1.2.1 Continuity of Functions
1.2.2 Identification Topology
1.2.3 Open Base for a Topology
1.2.4 Product Topology
1.2.5 Connectedness and Path Connectedness
1.2.6 Compactness
1.2.7 Axioms of Countability and Regularity of a Topological Space
1.2.8 Urysohn Lemma
1.2.9 Completely Regular Space
1.3 Some Basic Results on Analysis
1.4 Category and Functor
1.4.1 Introductory Concept of Category
1.4.2 Introductory Concept of Functors
1.5 Euclidean Spaces and Some Standard Notations
References
2 Topological Groups and Topological Vector Spaces
2.1 Topological Groups: Introductory Concepts
2.1.1 Basic Definitions and Examples
2.1.2 Homomorphism, Isomorphism and Automorphism for Topological Groups
2.2 Local Isomorphism and Local Characterization of Topological Groups
2.2.1 Neighborhood System of the Identity Element of Topological Group
2.2.2 Local Properties and Local Isomorphism
2.2.3 Local Characterization of Topological Groups
2.3 Topological Subgroups and Normal Subgroups
2.3.1 Discrete Normal Subgroups
2.3.2 Some Topological Properties of Groups, Subgroups and Normal Subgroups
2.4 Topological Group Action and Transformation Group
2.5 Local Compactness and its Characterization
2.5.1 Locally Compact Topological Groups
2.5.2 Characterization of Locally Compact Topological Groups
2.6 Quotient Structure of a Topological Group
2.7 Product and Direct Product of Topological Groups
2.7.1 Product of Topological Groups
2.7.2 Intersection of Subgroups of a Topological Group
2.7.3 Product of Subgroups of a Topological Group
2.7.4 Direct Product of Topological Groups
2.7.5 Isomorphism Theorems
2.8 Classical Topological Groups of Matrices
2.8.1 Topological Groups of Matrices
2.8.2 Rotation Group SO(2,R)
2.9 Topological Rings and Topological Semirings
2.10 Topological Vector Space and Topological Algebra
2.10.1 Basic Concepts of Topological Vector Spaces
2.10.2 Topological Algebra
2.10.3 Representation of a Topological Group
2.10.4 Topological Quotient Vector Space
2.10.5 Topological Modules
2.11 Linear Transformations and Linear Functionals
2.11.1 Continuity of Linear Transformations
2.11.2 Continuity of Linear Functionals
2.12 Applications
2.12.1 Linear Isomorphism Theorem
2.12.2 Finite Dimensional Topological Vector Spaces
2.12.3 Completeness of Topological Vector Spaces
2.12.4 Torus, Lens Space, Real and Complex Projective Spaces as Orbit Spaces
2.12.5 Subgroups of the Topological Group ( R, +)
2.12.6 Generators of Connected Topological Groups
2.12.7 More Topological Applications to Matrix Algebra
2.12.8 Special Properties Linear Functionals and Linear Transformations
2.12.9 More Applications
2.12.10 Special Properties of O(n, R), U(n, C) and Sp ( n, H)
2.13 Exercises
2.13.1 Multiple Choice Exercises on Topological Groups
2.13.2 Multiple Choice Exercises on Topological Groups of Matrices
References
3 Topology and Manifolds
3.1 Different Types of Manifolds with Motivation
3.1.1 Motivation of Manifolds
3.1.2 Approach to Topological Manifold
3.1.3 Approach to Differentiable Manifolds
3.1.4 The Nature of Differential Topology
3.1.5 Approach to Piecewise Linear Manifold
3.1.6 The Stiefel and Grassmann Manifolds
3.1.7 Surfaces
3.2 Calculus of Several Variables Related to Differentiable Manifolds
3.2.1 Derivative of a Map
3.2.2 Smooth Maps in Rn
3.2.3 Jacobian Matrix and Jacobian Determinant
3.2.4 Rank of a Differentiable Map
3.2.5 The Inverse Function and Implicit Function Theorems on R n
3.3 Manifolds in Rn
3.3.1 Smooth Manifolds in Rn
3.3.2 Tangent Spaces of Manifolds in Rn and Derivative Operations
3.3.3 Basic Properties of the Derivative Operation
3.3.4 Regular and Critical Values of Smooth Maps Between Manifolds of Same Dimension
3.4 Differentiable Manifolds
3.4.1 Differentiable Manifolds: Introductory Concepts
3.4.2 Cinfty-Structures on Manifolds and Cinfty-Diffeomorphism
3.5 Tangent Spaces and Differentials of Smooth Maps
3.6 The Topology on a Smooth Manifold Induced by Its C infty-Structure
3.6.1 The Topology Induced by C infty -Structure on a Manifold
3.6.2 A Necessary and Sufficient Condition for τM= τ
3.6.3 Properties of the Induced Topology on a Differentiable Manifold
3.6.4 Topological Restriction on a Differentiable Manifold
3.7 The Rank of a Smooth Map and Constant Rank Theorem
3.8 Manifolds With and Without Boundary
3.8.1 Topological Manifold with Boundary
3.8.2 Topological Manifold Without Boundary
3.8.3 Differential Manifold with Boundary
3.8.4 Surfaces with Boundaries
3.9 Immersion, Submersion and Transversality Theorem
3.10 The Inverse Function, Local Immersion and Local …
3.10.1 Canonical Embedding and Canonical Submersion
3.10.2 Local Immersion and Local Submersion Theorems
3.11 Submanifolds of Smooth Manifolds
3.12 Germs of Differentiable Functions
3.13 Product and Quotient Manifolds
3.13.1 Product Manifold and Its Induced Topology
3.13.2 Quotient Manifold and Its Induced Topology
3.14 Embedding of Compact Manifold in Euclidean Spaces and Whitney's Theorem
3.14.1 Smooth Partition of Unity
3.14.2 Embedding of Compact Manifold in Euclidean Spaces
3.15 Vector Bundles Over Smooth Manifolds and Their Homotopy Properties
3.15.1 Vector Bundles on Smooth Manifolds
3.15.2 The Whitney Sum of Vector Bundles Over a Smooth Manifold
3.15.3 Equivalence of Vector Bundles
3.15.4 Homotopy of Smooth Maps and Homotopy Property of Vector Bundles
3.16 Tangent, Normal and Cotangent Bundle of a Smooth Manifold
3.16.1 Tangent Bundle of a Smooth Manifold
3.16.2 Normal Bundle Over a Manifold
3.16.3 Cotangent Space and Cotangent Bundle of a Smooth Manifold
3.16.4 Vector Fields and Flows on Smooth Manifolds
3.17 Regular Values, Sard's, Brown's and Morse–Sard Theorems
3.18 Fundamental Theorem of Algebra
3.19 Properties of Smooth Homotopy and Isotopic Embedding
3.19.1 Properties of Smooth Homotopy of Smooth Maps
3.19.2 Isotopic Embedding of Smooth Manifolds
3.20 Riemann Surfaces
3.21 Transversability and Transversality Theorem
3.22 Tubular Nbds and Approximations
3.23 Complete Classification of Compact Surfaces
3.23.1 Construction of Projective Plane and Klein Bottle from a Square
3.23.2 Construction of Connected Sum of Tori
3.23.3 Construction of Connected Sum of Projective Planes
3.23.4 Construction of the Sphere as the Quotient Space of a Polygon
3.23.5 Methods of Constructions
3.24 Complete Classification of Connected 1-Dimensional Manifolds
3.25 Action of Topological Group on Manifolds and Transformation Group
3.25.1 Topological Transformation Group of a Topological Space
3.25.2 Topological Transformation Group of a Smooth Manifold
3.25.3 Properly Discontinuous Action of a Smooth Manifold
3.26 Applications of Morse–Sard's Theorem and the Brouwer Fixed-Point Theorem
3.26.1 More Applications
3.26.2 Configuration Space of a Hinged Pendulum
3.26.3 Solution Set of a System of Equations
3.27 Exercises
References
4 Lie Groups and Lie Algebras
4.1 Lie Group: Introductory Concepts
4.1.1 Topology of a Lie Group Induced by Its Differential Structure
4.1.2 Examples of Classical Lie Groups of Matrices and Others
4.1.3 Matrix Lie Groups
4.2 Topological Properties of Matrix Lie Groups
4.2.1 Compactness Property of Matrix Lie Groups
4.2.2 Path Connectedness and Connectedness Properties of Matrix Lie Groups
4.2.3 Simply Connectedness Property Matrix Lie Groups
4.3 Lie Group Action on Differentiable Manifolds
4.3.1 Translation and Diffeomorphism on Lie Groups
4.3.2 Actions of Lie Groups
4.3.3 Discontinuous Action of Lie Group
4.4 Lie Subgroups and Homomorphisms of Lie Groups
4.4.1 Lie Subgroups
4.4.2 Properties of Homomorphisms of Lie Groups
4.4.3 Action of O(n +1, R) on R Pn
4.5 More Properties of Lie Subgroups and Connected Lie Groups
4.6 Smooth Coverings
4.6.1 Tangent Bundles of Lie Groups
4.6.2 One-Parameter Group of Transformations
4.7 Lie Algebra
4.7.1 Lie Algebra of Vector Fields
4.7.2 Lie Algebra of a Lie Group
4.7.3 Homomorphism of Lie Algebras and Adjoint Representation
4.8 Subalgebra, Ideal, Stabilizer and Center
4.9 Some Special Lie Algebras
4.9.1 Simple Lie Algebra
4.9.2 Solvable Lie Algebra
4.9.3 Nilpotent Lie Algebra
4.10 Applications
4.11 Exercises
References
5 Brief History of Topological Groups, Manifolds and Lie Groups
5.1 A Brief History of Topological Groups
5.1.1 Motivation of the Study of Topological Groups and Topological Vector Spaces
5.1.2 Historical Development of Topological Groups and Vector Spaces
5.2 A Brief History of Manifolds
5.2.1 Motivation of the Study of Manifolds
5.2.2 Motivation of the Study of Differentiable Manifolds
5.2.3 Historical Development of Manifolds
5.3 Classification of Surfaces and 1-Dimensional Manifolds
5.4 Poincaré Conjecture on Compact Smooth n-Dimensional Manifold
5.5 Beginning of Differential Topology
5.6 A Brief History on Lie Groups
5.6.1 Motivation of the Study of Lie Groups and Lie Algebra
5.6.2 Historical Development of Lie Groups and Lie Algebra
5.6.3 Hilbert's Fifth Problem and Lie Group
References
Appendix Index
Index