In this voluminous book, Professor Dieudonné traces the development of algebraic and differential topology from the innovative work by Poincaré at the turn of the century to the period around 1960. He has given a superb account of the growth of these fields. As to his aims, the author writes: “I have tried to focus the history on the emergence of ideas and methods opening new fields of research, and I have gone into some details on the work of the pioneers, even when their methods were later superseded by simpler and more powerful ones. As Hadamard once said, in mathematics simple ideas usually come last.” The details are interwoven with the narrative in a very pleasant fashion.
The book is divided into three parts. The first is about simplicial techniques and homology, and is concerned with the development of the machinery of homology theory. There are, of course, several highlights here: the beginnings by Poincaré, the eventual clarification made possible by homological algebra, category theory and the axiomatic approach of Eilenberg and Steenrod, and the development of sheaf cohomology. The second part is devoted to some of the first applications of simplicial methods and of homology. The great innovator here was Brouwer, who reaped a bountiful harvest from the notion of the degree of a map. Brouwer’s arguments were terribly difficult to follow (and Poincaré often had not attempted rigorous proofs), so there was a great need for convincing arguments in the years ahead. There is a short chapter on local homological properties, and the suggestion that it would be worthwhile to rewrite R. L. Wilder’s book: Topology of manifolds [Am. Math. Soc. Colloq. Publ. No. 32 (1949; Zbl 0039.39602)] using modern techniques of algebraic topology. This part ends with a chapter on early applications of homology in geometry and analysis, especially Hodge theory and Morse theory.
The longest part by far is the third one, dealing with homotopy and its relation to homology. As expected, after a chapter on the fundamental group (Poincaré again!) and covering spaces, the work of H. Hopf receives pride of place, followed by a great many developments initiated by Hurewicz and J. H. C. Whitehead. Going beyond foundational homotopy theory, there are chapters on fibrations (ending with an account of the history of the construction of classifying spaces), homology of fibrations (characteristic classes and spectral sequences), sophisticated relations between homotopy and homology (e.g., Eilenberg-Mac Lane spaces, Postnikov towers, Serre’s contributions and Bott’s periodicity theorems), cohomology operations (including Adams’ solution of the Hopf invariant one problem by means of secondary cohomology operations) and finally generalized homology and cohomology (including details about unoriented and oriented cobordism, and an account of work by Kodaira and Spencer and by Hirzebruch in 1953 leading to Hirzebruch’s Riemann-Roch theorem). An impressive amount of beautiful mathematics is covered on the slightly over 300 pages of Part 3.
Thus, the coverage goes as far as Adams’ solution of the Hopf invariant one problem, and brief mention of the Adams spectral sequence, but not his solution of the vector field problem for spheres. As noted, Bott periodicity is covered, but one is not treated to Smale’s work on the h- cobordism theorem and the higher dimensional Poincaré conjecture.
Each part opens with an orderly introduction, but readers are urged to concentrate their attention to the main text in order to get the most benefit from the book. Readers might like to guess which four contributors are cited most often (consult the Index of Cited Names for the answer).
I find that the conventional wisdom and folklore of the field is quite accurate, and feel justified in having a poster of Poincaré in may office. Dieudonné quotes the opening sentence of Lefschetz’s Topology: “Perhaps on no branch of mathematics did Poincaré lay his stamp more indelibly than on topology.” It is good to read of the contributions of those such as W. Mayer and L. Vietoris, who were fine mathematicians but are today largely known for theorems bearing their names. Since I first came upon algebraic topology in about 1960, it is a treat to have the history up to that point made available.
There are a few points which some readers may find distracting. In the discussion of Ext-groups on pages 93-95, the order of the variables must be reversed. Formulas (90) and (91) on page 349 give results of C. H. Dowker [Am. J. Math. 69, 200–242 (1947; Zbl 0037.10101)] which require that \([X;S_n]\) be taken to mean uniform homotopy classes of maps from \(X\) to the \(n\)-sphere \(S_n\) (and that \(\dim X\le n\) if \(n>1)\). On page 362 just below (126), the citation should be to G. Whitehead’s survey article [Bull. Am. Math. Soc., New Ser. 8, 1–29 (1983; Zbl 0524.55002)]. The reader is warned that the quaternionic projective space \(P_{\infty}(H)\) is not an Eilenberg-Mac Lane space \(K(Z,4)\); cf. page 369. The history of K. Kodaira and D. C. Spencer’s work in 1953 is not quite as presented, and can be clarified by consulting their note [Proc. Natl. Acad. Sci. USA 39, 641–649 (1953; Zbl 0101.38302)] in which they first applied sheaf theory to the study of arithmetic genera of algebraic varieties. A vigilant reader will spot other small points of confusion, but should be able to clarify them.
Professor Dieudonné has previously written histories of functional analysis and of algebraic geometry, but neither book was on such a grand scale as this one. He has made it possible to trace the important steps in the growth of algebraic and differential topology, and to admire the hard work and major advances made by the founders. I hope a large audience will take the opportunity to profit from this fascinating historical essay.