Thorough, modern treatment, essentially from a homotopy theoretic viewpoint. Topics include homotopy and simplicial complexes, the fundamental group, homology theory, homotopy theory, homotopy groups and CW-Complexes, and other topics. Each chapter contains exercises and suggestions for further reading. 1980 corrected edition.

Table of Contents for Algebraic Topology

CHAPTER 1 ALGEBRAIC AND TOPOLOGICAL PRELIMINARIES
1.1 Introduction
1.2 Set theory
1.3 Algebra
1.4 Analytic Topology
CHAPTER 2 HOMOTOPY AND SIMPLICIAL COMPLEXES
2.1 Introduction
2.2 The classification problem; homotopy
2.3 Simplicial complexes
2.4 Homotopy and homeomorphism of polyhedra
2.5 Subdivision and the Simplicial Approximation Theorem
Exercises
Notes on Chapter 2
CHAPTER 3 THE FUNDAMENTAL GROUP
3.1 Introduction
3.2 Definition and elementary properties of the fundamental group
3.3 Methods of calculation
3.4 Classification of triangulable 2-manifolds
Exercises
Notes on Chapter 3
CHAPTER 4 HOMOLOGY THEORY
4.1 Introduction
4.2 Homology groups
4.3 Methods of calculation: simplicial homology
4.4 Methods of calculation: exact sequences
4.5 "Homology groups with arbitrary coefficients, and the Lefschetz Fixed-Point Theorem"
Exercises
Notes on Chapter 4
CHAPTER 5 COHOMOLOGY AND DUALITY THEOREMS
5.1 Introduction
5.2 Definitions and calculation theorems
5.3 The Alexander-Poincaré Duality Theorem
5.4 Manifolds with boundary and the Lefschetz Duality Theorem
Exercises
Notes on Chapter 5
CHAPTER 6 GENERAL HOMOTOPY THEORY
6.1 Introduction
6.2 Some geometric constructions
6.3 Homotopy classes of maps
6.4 Exact sequences
6.5 Fibre and cofibre maps
Exercises
Notes on Chapter 6
CHAPTER 7 HOMOTOPY GROUPS AND CW-COMPLEXES
7.1 Introduction
7.2 Homotopy groups
7.3 CW-complexes
7.4 Homotopy groups of CW-complexes
7.5 The theorem of J. H. C. Whitehead and the Cellular Approximation Theorem
Exercises
Notes on Chapter 7
CHAPTER 8 HOMOLOGY AND COHOMOLOGY OF CW-COMPLEXES
8.1 Introduction
8.2 The Excision Theorem and cellular homology
8.3 The Hurewicz theorem
8.4 Cohomology and Eilenberg-MacLane spaces
8.5 Products
Exercises
Notes on Chapter 8
References
Index

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