This textbook is designed to introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as possible. The principal topics treated are 2-dimensional manifolds, the fundamental group, and covering spaces, plus the group theory needed in these topics. The only prerequisites are some group theory, such as that normally contained in an undergraduate algebra course on the junior-senior level, and a one-semester undergraduate course in general topology.

CHAPTER ONE Two-Dimensional Manifolds

 1 Introduction

 2 Definition and examples of n-manifolds

 3 Orientable vs. nonorientable manifolds

 4 Examples of compact, connected 2-manifolds

 5 Statement of the classification theorem for compact surfaces

 6 Triangulations of compact surfaces

 7 Proof of Theorem 5.1

 8 The Euler characteristic of a surface

 9 Manifolds with boundary

 10 The classification of compact, connected 2-manifolds with boundary

 11 The Euler characteristic of a bordered surface

 12 Models of compact bordered surfaces in Euclidean 3-space

 13 Remarks on noncompact surfaces

CHAPTER TWO The Fundamental Group

 1 Introduction

 2 Basic notation and terminology

 3 Definition of the fundamental group of a space

 4 The effect of a continuous mapping on the fundamental group

 5 The fundamental group of a circle is infinite cyelic

 6 Application: The Brouwer fixed-point theorem in dimension 2

 7 The fundamental group of a product space

 8 Homotopy type and homotopy equivalence of spaces

CHAPTER THREE Free Groups and Free Products of Groups

 1 Introduction

 2 The weak product of abelian groups

 3 Free abelian groups

 4 Free products of groups

 5 Free groups

 6 The presentation of groups by generators and relations

 7 Universal mapping problems

CHAPTER FOUR Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces.Applica tions

 1 Introduction

 2 Statement and proof of the theorem of Seifert and Van Kampen

 3 First application of Theorem 2.1

 4 Second application of Theorem 2.1

 5 Structure of the fundamental group of a compact surface

 6 Application to knot theory

CHAPTER FIVE Covering Spaces

 1 Introduction

 2 Definition and some examples of covering spaces

 3 Lifting of paths to a covering space

 4 The fundamental group of a covering space

 5 Lifting of arbitrary maps to a covering space

 6 Homomorphisms and automorphisms of covering spaces

 7 The action of the group π(X, x) on the set p-τ(x)

 8 Regular covering spaces and quotient spaces

 9 Application: The Borsuk-Ulam theorem for the 2-sphere

 10 The existence theorem for covering spaces

 ll The induced covering space over a subspace

 12 Point set topology of covering spaces

CHAPTER SIX The Fundamental Group and Covering Spaces of a Graph.

 Applications to Group Theory

 1 Introduction

 2 Definition and examples

 3 Basic properties of graphs

 4 Trees

 5 The fundamental group of a graph

 6 The Euler characteristic of a finite graph

 7 Covering spaces of a graph

 8 Generators for a subgroup of free group

CHAPTER SEVEN The Fundamental Group of Higher Dimensional Spaces

 1 Introduction

 2 Adjunction of 2-cells to a space

 3 Adjunction of higher dimensional cells to a space

 4 CW-complexes

 5 The Kurosh subgroup theorem

 6 Grushko's Theorem

CHAPTER EIGHT

Epilogue

APPENDIX A

The Quotient Space or |dentification Space Topology

 1 Definitions and basic properties

 2 A generalization of the quotient space topology

 3 Quotient spaces and product spaces

 4 Subspaee of a quotient space vs. quotient space of a subspace

 5 Conditions for a quotient space to be a Hausdorff space

APPENDIX B

Permutation Groups or Transformation Groups

 1 Basic definitions

 2 Homogeneous G-spaces

Index