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Book DetailsPure and Applied Undergraduate TextsVolume: 10; 2005; 236 ppMSC: Primary 55; 57
Beginning Topology is designed to give undergraduate students a broad notion of the scope of topology in areas of pointset, geometric, combinatorial, differential, and algebraic topology, including an introduction to knot theory. A primary goal is to expose students to some recent research and to get them actively involved in learning. Exercises and openended projects are placed throughout the text, making it adaptable to seminarstyle classes.
The book starts with a chapter introducing the basic concepts of pointset topology, with examples chosen to captivate students' imaginations while illustrating the need for rigor. Most of the material in this and the next two chapters is essential for the remainder of the book. One can then choose from chapters on map coloring, vector fields on surfaces, the fundamental group, and knot theory.
A solid foundation in calculus is necessary, with some differential equations and basic group theory helpful in a couple of chapters. Topics are chosen to appeal to a wide variety of students: primarily upperlevel math majors, but also a few freshmen and sophomores as well as graduate students from physics, economics, and computer science. All students will benefit from seeing the interaction of topology with other fields of mathematics and science; some will be motivated to continue with a more indepth, rigorous study of topology.
ReadershipUndergraduate students interested in topology.

Table of Contents

Front Cover

CONTENTS

PREFACE

AN OVERVIEW

CHAPTER1: INTRODUCTION TO POINT SET TOPOLOGY

1.1 OPEN AND CLOSED SETS

1.2 CONTINUOUS FUNCTIONS

1.3 SOME TOPOLOGICAL PROPERTIES

1.4 A BRIEF INTRODUCTION TO DIMENSION (OPTIONAL)

CHAPTER 2 SURFACES

2.1 DEFINITION OF A SURFACE

2.2 CONNECTED SUM CONSTRUCTION

2.3 PLANE MODELS OF SURFACES

2.4 ORIENTABILITY

2.5 PLANE MODELS OF NONORIENTABLE SURFACES

2.6 CLASSIFICATION OF SURFACES

2.7 PROOF OF THE CLASSIFICATION THEOREM FOR SURFACES (OPTIONAL)

CHAPTER 3: THE EULER CHARACTERISTIC

3.1 CELL COMPLEXES AND THE EULER CHARACTERISTIC

3.2 TRIANGULATIONS

3.3 GENUS

3.4 REGULAR COMPLEXES

3.5 bVALENT COMPLEXES

CHAPTER: 4 MAPS AND GRAPHS

4.1 MAPS AND MAP COLORING

4.2 THE FIVECOLOR THEOREM FOR S^2

4.3 INTRODUCTION TO GRAPHS

4.4 GRAPHS IN SURFACES

4.5 EMBEDDING THE COMPLETE GRAPHS AND GRAPH COLORING

CHAPTER 5: VECTOR FIELDS ON SURFACES

5.1 VECTOR FIELDS IN THE PLANE

5.2 INDEX OF A CRITICAL POINT

5.3 LIMIT SETS IN THE PLANE

5.4 A LOCAL DESCRIPTION OF A CRITICAL POINT

5.5 VECTOR FIELDS ON SURF ACES

CHAPTER 6: THE FUNDAMENTAL GROUP

6.1 PATH HOMOTOPY AND THE FUNDAMENTAL GROUP

6.2 THE FUNDAMENTAL GROUP OF THE CIRCLE

6.3 DEFORMATION RETRACTS

6.4 FURTHER CALCULATIONS

6.5 PRESENTATIONS OF GROUPS

6.6 THE SEIFERTVAN KAMPEN THEOREM AND THE FUNDAMENTAL GROUPS OF SURFACES

6.7 PROOF OF THE SEIFERTVAN KAMPEN THEOREM

CHAPTER 7: INTRODUCTION TO KNOTS

7.1 KNOTS: WHAT THEY ARE AND HOW TO DRAW THEM

7 .2 PRIME KNOTS

7.3 ALTERNATING KNOTS

7.4 REIDEMEISTER MOVES

7.5 SOME SIMPLE KNOT INVARIANTS

7.6 SURFACES WITH BOUNDARY

7.7 KNOTS AND SURFACES

7.8 KNOT POLYNOMIALS

BIBLIOGRAPHY AND READING LIST

INDEX

Back Cover


Additional Material

Reviews

This text is an interesting introduction to some of the various aspects of topology . . . [A] very attractive way to learn more and discover new things in topology.
Corina Mohorianu, Zentralblatt MATH


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Beginning Topology is designed to give undergraduate students a broad notion of the scope of topology in areas of pointset, geometric, combinatorial, differential, and algebraic topology, including an introduction to knot theory. A primary goal is to expose students to some recent research and to get them actively involved in learning. Exercises and openended projects are placed throughout the text, making it adaptable to seminarstyle classes.
The book starts with a chapter introducing the basic concepts of pointset topology, with examples chosen to captivate students' imaginations while illustrating the need for rigor. Most of the material in this and the next two chapters is essential for the remainder of the book. One can then choose from chapters on map coloring, vector fields on surfaces, the fundamental group, and knot theory.
A solid foundation in calculus is necessary, with some differential equations and basic group theory helpful in a couple of chapters. Topics are chosen to appeal to a wide variety of students: primarily upperlevel math majors, but also a few freshmen and sophomores as well as graduate students from physics, economics, and computer science. All students will benefit from seeing the interaction of topology with other fields of mathematics and science; some will be motivated to continue with a more indepth, rigorous study of topology.
Undergraduate students interested in topology.

Front Cover

CONTENTS

PREFACE

AN OVERVIEW

CHAPTER1: INTRODUCTION TO POINT SET TOPOLOGY

1.1 OPEN AND CLOSED SETS

1.2 CONTINUOUS FUNCTIONS

1.3 SOME TOPOLOGICAL PROPERTIES

1.4 A BRIEF INTRODUCTION TO DIMENSION (OPTIONAL)

CHAPTER 2 SURFACES

2.1 DEFINITION OF A SURFACE

2.2 CONNECTED SUM CONSTRUCTION

2.3 PLANE MODELS OF SURFACES

2.4 ORIENTABILITY

2.5 PLANE MODELS OF NONORIENTABLE SURFACES

2.6 CLASSIFICATION OF SURFACES

2.7 PROOF OF THE CLASSIFICATION THEOREM FOR SURFACES (OPTIONAL)

CHAPTER 3: THE EULER CHARACTERISTIC

3.1 CELL COMPLEXES AND THE EULER CHARACTERISTIC

3.2 TRIANGULATIONS

3.3 GENUS

3.4 REGULAR COMPLEXES

3.5 bVALENT COMPLEXES

CHAPTER: 4 MAPS AND GRAPHS

4.1 MAPS AND MAP COLORING

4.2 THE FIVECOLOR THEOREM FOR S^2

4.3 INTRODUCTION TO GRAPHS

4.4 GRAPHS IN SURFACES

4.5 EMBEDDING THE COMPLETE GRAPHS AND GRAPH COLORING

CHAPTER 5: VECTOR FIELDS ON SURFACES

5.1 VECTOR FIELDS IN THE PLANE

5.2 INDEX OF A CRITICAL POINT

5.3 LIMIT SETS IN THE PLANE

5.4 A LOCAL DESCRIPTION OF A CRITICAL POINT

5.5 VECTOR FIELDS ON SURF ACES

CHAPTER 6: THE FUNDAMENTAL GROUP

6.1 PATH HOMOTOPY AND THE FUNDAMENTAL GROUP

6.2 THE FUNDAMENTAL GROUP OF THE CIRCLE

6.3 DEFORMATION RETRACTS

6.4 FURTHER CALCULATIONS

6.5 PRESENTATIONS OF GROUPS

6.6 THE SEIFERTVAN KAMPEN THEOREM AND THE FUNDAMENTAL GROUPS OF SURFACES

6.7 PROOF OF THE SEIFERTVAN KAMPEN THEOREM

CHAPTER 7: INTRODUCTION TO KNOTS

7.1 KNOTS: WHAT THEY ARE AND HOW TO DRAW THEM

7 .2 PRIME KNOTS

7.3 ALTERNATING KNOTS

7.4 REIDEMEISTER MOVES

7.5 SOME SIMPLE KNOT INVARIANTS

7.6 SURFACES WITH BOUNDARY

7.7 KNOTS AND SURFACES

7.8 KNOT POLYNOMIALS

BIBLIOGRAPHY AND READING LIST

INDEX

Back Cover

This text is an interesting introduction to some of the various aspects of topology . . . [A] very attractive way to learn more and discover new things in topology.
Corina Mohorianu, Zentralblatt MATH