An Introduction to Algebraic Topology

Cover
Title Page
Preface
To the Reader
0 Introduction
Notation
Brouwer Fixed Point Theorem
Categories and Functors
1 Some Basic Topological Notions
Homotopy
Convexity, Contractibility, and Cones
Paths and Path Connectedness
2 Simplexes
Affine Spaces
Affine Maps
3 The Fundamental Group
The Fundamental Groupoid
The Functor pi1
Pi1(S1)
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
5 Long Exact Sequences
The Category Comp
Exact Homology Sequences
Reduced Homology
6 Excision and Applications
Excision and Mayer-Vietoris
Homology of Spheres and Some Applications
Barycentric Subdivision and the Proof of Excision
More Applications to Euclidean Space
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
8 CW Complexes
Hausdorff Quotient Spaces
Attaching Cells
Homology and Attaching Cells
CW Complexes
Cellular Homology
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 K�nneth Formula
10 Covering Spaces
Basic Properties
Covering Transformations
Existence
Orbit Spaces
11 Homotopy Groups
Function Spaces
Group Objects and Cogroup Objects
Loop Space and Suspension
Homotopy Groups
Exact Sequences
Fibrations
A Glimpse Ahead
12 Cohomology
Differential Forms
Cohomology Groups
Universal Coefficients Theorems for Cohomology
Cohomology Rings
Computations and Applications
Bibliography
Notation
Index