A History of Algebraic and Differential Topology, 1900–1960

I’m ecstatic to have, after over twenty years, my very own copy of A History of Algebraic and Differential Topology, 1900–1960, by the redoubtable Jean Dieudonné (1906–1992): I first became aware of this opus in 1989, the year of its original publication, via my university library, and I have had occasion to use the book early in the 1990s. My notes dating back to those days include references to §6B of Ch. IV (Part 1), “The Various Homology and Cohomology Theories,” §1D of Ch. V (Part 3), “Sophisticated Relations between Homotopy and Homology,” Ch. VI(Part 3), “Cohomology Operations,” and §3 of Ch. VII (Part 3), “Generalized Homology and Cohomology.” The indicated subsections’ titles, are, respectively, §6B/IV/1: “The Axiomatic Theory of Homology and Cohomology” (pp.107–113); §1D/V/3: “Homology and Cohomology of Groups (pp. 458–463); and §3/VII/3: “The Beginnings of K-Theory” (pp. 598–611 = the end of the book).

My focus in those days was the role played by low-dimensional (i.e. ≤ second) group cohomology in the theory of metaplectic covers, including in particular the work of C. C. Moore, Tomio Kubota, the team of Kahzdan and Patterson, and H. Matsumoto. To have Dieudonné’s homeric text at my disposal was a Godsend.

As the snippets cited above convey, the book under review (reissued by Birkhäuser in its “Modern Classics” series), is encyclopaedic in its scope. Its three main parts are titled, respectively, “Simplicial Techniques and Homology,” “The First Applications of Simplicial Methods and of Homology,” and “Homotopy and its Relation to Homology.” And the first part takes one from “The Work of Poincaré” to “Homological Algebra and Category Theory” (preceded by “Sheaf Cohomology,” as befits a scion of Bourbaki), the second takes one from “The Concept of Degree” (with §1 of Ch. I being “The Work of Brouwer”) to “Applications of Homology to Geometry and Analysis,” and the third takes one from “Fundamental Group and Covering Spaces” to the aforementioned “Generalized Homology and Cohomology.”

The sweep of this work has to be seen to be believed: its 611 pages appear to address literally everything in pre-1970s algebraic and differential topology, exclusive of low-dimensional topology. Regarding this omission, Dieudonné explicitly addressed the matter in his preface in the following words: “It was soon realized that some general tools could not give satisfactory results in spaces of dimension 4 at most, and, conversely, methods that were successful for those spaces did not extend to higher dimensions…” — shades of Thurston and the Poincaré conjecture à la Hamilton and Perelman.

The book gives  a necessarily compact treatment dripping with the measure of attention to detail and rigor Dieudonné was known for: it is for good reason that he was the primary scribe of Bourbaki. Indeed, it is fair to say, I think, that A History of Algebraic and Differential Topology, 1900–1960 is on a par, both stylistically and qua pedagogical intent to the Bourbaki volumes, at least to a large extent (after all, this work is not a Satz–Beweis affair), and appraisal this might provide something of a benchmark someone unfamiliar with Dieudonné’s works might use when considering what might lie ahead when he cracks the pages of this book: those who side with Bourbaki will love it, those who despise Bourbaki, well … enough said (I think it’s a theorem that there is no middle ground: either people love Bourbaki, or they love to hate Bourbaki).

Moreover, A History of Algebraic and Differential Topology, 1900–1960 can, and should, be compared to Dieudonné’s much shorter History of Algebraic Topology, dating to 1985, especially inasmuch as this work, too, seeks to convey the warp and woof of its title’s subject, fitted in a historical frame, while not neglecting to give a good deal of attention to the finer details of the subject. Both of these books serve beautifully as roadmaps for deeper studies and as orchestrations of the material under consideration.

There’s not much left to say, really. Dieudonné’s impeccable and unassailable scholarship shows forth on every page of the book. It belongs in every mathematician’s library, right next to Cartan–Eilenberg and Eilenberg–Steenrod. I very much look forward to using the book under review in my current and future researches which, happily, have taken on an even more emphatic algebraic and differential topological character. Dieudonné’s A History of Algebraic and Differential Topology, 1900–1960 was, or is, a wonderful bequest to the mathematical community.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.