Doing exercises is an essential part of learning mathematics, and the serious reader of this book should attempt to solve all the exercises as they arise. An asterisk indicates only that an exercise is cited elsewhere in the text, sometimes in a proof (those exercises used in proofs, however, are always routine).

Preface

To the Reader

CHAPTER 0 Introduction

 Notation

 Brouwer Fixed Point Theorem

 Categories and Functors

CHAPTER 1 Some Basic Topological Notions

 Homotopy

 Convexity, Contractibility, and Cones

 Paths and Path Connectedness

CHAPTER 2 Simplexes

 Affine Spaces

 Aftine Maps

CHAPTER 3 The Fundamental Group

 The Fundamental Groupoid

 The Functor π

 π1(S1)

CHAPTER 4 Singular Homology

 Holes and Green's Theorem

 Free Abelian Groups

 The Singular Complex and Homology Functors

 Dimension Axiom and Compact Supports

 The Homotopy Axiom

 The Hurewicz Theorem

CHAPTER 5 Long Exact Sequences

 The Category Comp

 Exact Homology Sequences

 Reduced Homology

CHAPTER 6 Excision and Applications

 Excision and Mayer-Vietoris

 Homology of Spheres and Some Applications

 Barycentric Subdivision and the Proof of Excision

 Moxe Applications to Euclidean Space

CHAPTER 7 Simplicial Complexes

 Definitions

 Simplicial Approximation

 Abstract Simplicial Complexes

 Simplicial Homology

 Comparison with Singular Homology

 Calculations

 Fundamental Groups of Polyhedra

 The Seifert-van Kampen Theorem

CHAPTER 8 CW Complexes

 Hausdorff Quotient Spaces

 Attaching Calls

 Homology and Attaching Cells

 CW Complexes

 Cellular Homology

CHAPTER 9 Natural Transformations

 Definitions and Examples

 Eilenberg-Steenrod Axioms

 Chain Equivalences

 Acyclic Models

 Lefschetz Fixed Point Theorem

 Tensor Products

 Universal Coefficients

 Eilenberg-Zilber Theorem and the Kiinneth Formula

CHAPTER 10 Covering Spaces

 Basic Properties

 Covering Transformations

 Existence

 Orbit Spaces

CHAPTER 11 Homotopy Groups

 Function Spaces

 Group Objects and Cogroup Objects

 Loop Space and Suspension

 Homotopy Groups

 Exact Sequences

 Fibrations

 A Glimpse Ahead

CHAPTER 12 Cohomology

 Differential Forms

 Cohomoiogy Groups

 Universal Coefficients Theorems for Cohomology

 Cohomology Rings

 Computations and Applications

Bibliography

Notation

Index