Preface

Summary

Since the introduction of homotopy groups by Hurewicz in 1935, homotopy theory has been occupying an increasingly prominent place in the field of algebraic topology. Important new advances are continually being made in the subject by various workers; and the recent developments emanating from the French school of topologists underline the desirability of having available a basic introduction to homotopy theory suitable for those who wish to undertake research in the subject and for those who wish to be in a position to understand the modern techniques and results.

At the moment, no text-book of homotopy theory exists at any level, with the result that the newcomer to this branch of mathematics is obliged to plunge straight into the study of original papers, often of very considerable complexity. This monograph is designed to fill the gap. It does not claim to be a comprehensive treatment of its subject (the recent work of the French school, for example, is not included, except for a brief introduction to it in the last section of Chapter V); but it is hoped that a reader familiar with Lefschetz's Introduction to Topology will obtain an understanding of the fundamental ideas of homotopy theory from the first six chapters of this book.

The final two chapters are somewhat different in kind from the first six, being an account of the homotopy theory of complexes. J. H. C. Whitehead's generalization of the simplicial complex has been very fruitful, and it seems proper that a book of this kind should contain a brief treatment of it.