Elementary Topology

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