An introduction to homotopy theory / P. J. Hilton

Notes bibliogr. et bibliogr. p. 134-136. Index

The first chapter is an introduction and contains definitions of some of the basic notions of the subject. Chapter II describes various methods of definition of the absolute and relative homotopy groups, and of the operations of the fundamental groups on the higher homotopy groups. The third chapter, entitled "The classical theorems of homotopy theory'', is concerned mainly with the following three: the simplicial approximation theorem, the theorem stating that the homotopy class of a map of an n-sphere onto itself is determined completely by the Brouwer degree, and the Hurewicz theorem that in an (n−1)-connected polyhedron the nth homology and homotopy groups are isomorphic. Actually, the chapter contains only the statements and a discussion of these theorems; for the details of the proof, the reader is referred to the text by Lefschetz (op. cit.). The next chapter contains a description of the homotopy sequence of a pair consisting of a space and a subspace, a proof that this sequence is exact, and some applications of this fact. Chapter V is concerned with homotopy relations in fibre spaces; particular attention is given to the fibre maps of spheres due to Hopf. Chapter VI is about the Hopf Invariant for maps of spheres, the Freudenthal suspension theorems, and various generalizations of these concepts. Most of the more difficult proofs are omitted, the reader being referred to the original papers for full details.
The last two chapters are somewhat different from the first six, being an account of the homotopy theory of complexes. Chapter VII is devoted to a treatment of J. H. C. Whitehead's CW-complexes, while Chapter VIII is concerned with the problem of determining the (n+1)-dimensional homotopy group of an (n−1)-connected CW-complex (n>2). This problem is considered mainly in order to illustrate the concepts of the previous chapters.
This book will certainly prove very helpful to students trying to learn homotopy theory. Perhaps the most noticeable omission is the lack of an account of the theory of obstructions to extensions and to homotopies of continuous maps. For example, nowhere in the book is Hopf's theorem on the classification of maps of an n-complex into an n-sphere even mentioned. It is also regrettable that the author did not include more discussion to motivate the introduction of the various new ideas and to explain their importance. (MathSciNet)