Elementary Topology

Elementary Topology

A Combinatorial and Algebraic Approach

1st Edition - January 28, 1982

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  • Author: Donald W. Blackett
  • eBook ISBN: 9781483262536

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Description

Elementary Topology: A Combinatorial and Algebraic Approach focuses on the application of algebraic methods to topological concepts and theorems. The publication first elaborates on some examples of surfaces and their classifications. Discussions focus on combinatorial invariants of a surface, combinatorial equivalence, surfaces and their equations, topological surfaces, coordinates on a sphere and torus, and properties of the sphere and torus. The text then examines complex conics and covering surfaces and mappings into the sphere, including applications of the winding number in complex analysis, mappings into the plane, winding number of a plane curve, covering surfaces, and complex conies. The book examines vector fields, network topology, and three-dimensional topology. Topics include topological products and fiber bundles, manifolds of configurations, paths, circuits, and trees, vector fields and hydrodynamics, vector fields on a sphere, and vector fields and differential equations. The publication is highly recommended for sophomores, juniors, and seniors who have completed a year of calculus.

Table of Contents


  • Preface

    1. Some Examples of Surfaces

    1.1 Coordinates On A Sphere And Torus

    1.2 The Topological Sphere And Torus

    1.3 Properties Of The Sphere And Torus

    1.4 The Cylinder And Möbius Band

    1.5 Additional Representations Of The Möbius Band

    1.6 The Projective Plane

    Exercises

    2. The Classification of Surfaces

    2.1 Surfaces And Their Equations

    2.2 Combinatorial Equivalence

    2.3 The Canonical Equation

    2.4 Combinatorial Invariants Of A Surface

    2.5 Topological Surfaces

    Exercises

    3. Complex Conies and Covering Surfaces

    3.1 Complex Conics

    3.2 Covering Surfaces

    3.3 Some Additional Examples Of Riemann Surfaces

    Exercises

    4. Mappings into the Sphere

    4.1 Winding Number Of A Plane Curve

    4.2 Mappings Into The Plane

    4.3 The Brouwer Degree

    4.4 Applications Of The Winding Number In Complex Analysis

    Exercises

    5. Vector Fields

    5.1 Vector Fields On The Plane

    5.2 A Geographical Application

    5.3 Vector Fields And Hydrodynamics

    5.4 Vector Fields And Differential Equations

    5.5 Vector Fields On A Sphere

    5.6 Mappings Of A Sphere Into Itself

    Exercises

    6. Network Topology

    6.1 Introduction

    6.2 Boundary And Coboundary

    6.3 Paths, Circuits, And Trees

    6.4 Basic Circuits

    6.5 The Kirchhoff-Maxwell Laws

    6.6 A Transportation Problem

    Exercises

    7. Some Three-Dimensional Topology

    7.1 Three-Dimensional Manifolds

    7.2 Orientability

    7.3 Manifolds Of Configurations

    7.4 Topological Products And Fiber Bundles

    Exercises

    Bibliography

    Subject Index

Product details

  • No. of pages: 236
  • Language: English
  • Copyright: © Academic Press 1982
  • Published: January 28, 1982
  • Imprint: Academic Press
  • eBook ISBN: 9781483262536

About the Author

Donald W. Blackett

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