Is it possible to do mathematics (in the proper sense of the word: the business of making new discoveries and proving new theorems) without knowing a decent amount of algebraic topology? When I was young, say, into my graduate school days and even slightly beyond, I figured it was possible. I picked up a bit of algebraic topology en passant, e.g. in the context of algebraic geometry, and even in number theory (my specialty) a bit of algebraic topology entered into, for instance, the business of explaining the magic role of the number of 12 in the theory of elliptic modular forms. But I had in truth opted to avoid true curricular studies in algebraic topology in order to pursue my extramural interest in mathematical logic. I recall, also, attending Karoubi’s course of lectures on K-theory in the wild hope that I’d be able to grasp at least some of what was being presented without algebraic topology in my arsenal, and this turned out not to be possible. It was doubly tragic, really, because Karoubi very kindly provided me with some notes on connections with number theory that were of course altogether wasted on me. So the handwriting actually already began to appear on the wall at that stage. As time went on it became more and more obvious that I needed to fill in this untenable gap in my education. It became crystal clear to me that, at least for me, the answer to the above question is an emphatic and resounding no: algebraic topology is centrally important and should not be skipped.
So, over the years I have had the pleasure of reading and studying a number of books on the subject in question, including one that is in fact an absolute favorite, namely, Homological Algebra, by Henri Cartan and Samuel Eilenberg. It has all been great fun, and while my grasp of the subject still leaves something to be desired (meaning that the studies continue), I am happily considerably more at home in the subject.
But there are different facets to algebraic topology, so that reading, say, J. Peter May’s fantastic A Concise Course in Algebraic Topology is an altogether different pedagogical experience from reading Raoul Bott’s and Loring Tu’s marvelous Differential Forms in Algebraic Topology. And what about the time-honored text by E. H. Spanier, Algebraic Topology, compared with, say, J. F. Adams’ Algebraic Topology, A Student’s Guide? Two very different approaches, but both distinguished by very broad success. Well, perhaps it’s reasonable to approach the matter as the sort of thing that requires two complementary approaches: to learn something like this, especially working on one’s own, it’s important (1) to have the structure under control (Sätze und Beweise) and (2) to have a decent intuition about what’s going on, i.e. the view of things usually vouchsafed to the insiders — perhaps even a fellow traveler can get some feeling for the subject that, so to speaks, adds the orchestration and makes it communicable (especially in lectures). It is on the latter count that the book under review succeeds in spades. When I discovered it some years back I proceeded to devour it, and it has served me well ever since.
What Sato’s Algebraic Topology: An Intuitive Approach does is to present a sweeping view of the main themes of algebraic topology, namely, homotopy, homology, cohomology, fibre bundles, and spectral sequences, in a truly accessible and even minimalist way, by requiring the reader to rely on geometrical intuition, by sticking to the most easily presented forms of these subjects (homology of cell complexes and cohomology of simplicial complexes, yes; satellites and derived complexes, no), by explicating a number of (again, very accessible) examples, and by inviting the reader to do a decent number of very evocative exercises. Accordingly, the book can be used profitably in a number of ways. In my own approach, as I already indicated above, it serves to give shape to otherwise very abstract notions, and therefore makes a fine supplement to any of the earlier texts. In another approach, and one that is particularly relevant for teaching purposes, the book (which is short and cheap!) can be used as a primary text even at the advanced undergraduate level, the proviso being that more expansive (and expensive) sources are to be consulted down the line. Actually, in the latter connection, I would go to the aforementioned book by Adams, which is a compendium of original papers on algebraic topology covering pretty much the entire spectrum, with added commentary by Adams himself in the form of brief but pithy snippets. It would be a terrific course to teach!
In any case, Sato’s book is a gem, and I am happy to recommend it in very enthusiastic terms.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.