IN THE MORE THAN TWENTY YEARS SINCE THE FIRST APPEARANCE OF Algebraic Topology the book has met with favorable response both in its use as a text and as a reference. It was the first comprehensive treatment of the fundamentals of the subject. Its continuing acceptance attests to the fact that its content and organization are still as timely as when it first appeared. Accord-ingly it has not been revised.

INTRODUCTION

 1 Set theory

 2 General topology

 3 Group theory

 4 Modules

 5 Euclidean spaces

1 ROMOTOPY AND THE FUNDAMENTAL GROUP

 1 Categories

 2 Functors

 3 Homotopy

 4 Retraction and deformation

 5 H spaces

 6 Suspension

 7 The fundamental groupoid

 8 The fundamental group

 Exercises

2 COVERING SPACES AND FIBRATIONS

 1 Covering profections

 2 The homotopy lifting property

 3 Relations with the fundamental group

 4 The lifting problem

 5 The classification of covering projections

 6 Covering transformations

 7 Fiber bundles

 8 Fibrations

 Exercises

3 POLYHEDRA

 1 Simplicial complexes

 2 Linearity in simplicial complexes

 3 Subdivision

 4 Simplicial approximation

 5 Contiguity classes

 6 The edge-path groupoid

 7 Graphs

 8 Examples and applications

 Exercises

4 HOMOLOGY

 1 Chain complexes

 2 Chain homotopy

 3 The homology of simplicial complexes

 4 singular homotogy

 5 Exactness

 6 Mayer-Vietoris sequences

 7 Some applications of homology

 8 Axiomatic characterization of homology

 Exercises

5 PRODUCTS

 1 Homology with coefficients

 2 The universal-coefficient theorem for homology

 3 The Kunneth formula

 4 Cohomology

 5 The universal-coefficient theorem for cohomology

 6 Cup and cap products

 7 Homology of fiber bundles

 8 The cohomology algebra

 9 The Steenrod squaring operations

 Exercises

6 GENERAL COHOMOLOGY THEORY AND DUALITY

 1 The slant product

 2 Duality in topological manifolds

 3 The fundamental class of a manifold

 4 The Alexander cohomology theory

 5 The homotopy axiom for the Alexander theory

 6 Tautness and continuity

 7 Presheaves

 8 Fine presheaves

 9 Applications of the cohomology of presheaves

 10 Characteristic classes

 Exercises

7 HOMOTOPY THEORY

 1 Exact sequences of sets of homotopy classes

 2 Higher homotopy groups

 3 Change of base points

 4 The Hurewics honmorphism

 5 The Hurewicz isomorphism theorem

 6 CW complexes

 7 Homotopy funetors

 8 Weak homotopy type

 Exercises

8 OBSTRUCTION THEORY

 1 Eilenberg-MacLane spaces

 2 Principal fibrations

 3 Moore-Postnikov factorizations

 4 Obstruction theory

 5 The suspension map

 Exercises

9 SPECTRAL SEQUENCES AND ROMOTOPY GROUPS OF SPHERES

 1 Spectral sequences

 2 The spectral sequence of a fibration

 3 Applications of the homology spectral sequence

 4 Multtplicative properties of spectral sequences

 5 Applications of the cohomology spectral sequence

 6 Serre classes of abeltan groups

 7 Homotopy groups of spheres

 Exercises

INDEX