Homotopical Topology

This is a textbook on algebraic topology going back to the late 1960s curriculum at the famed Lomonosov Moscow University, from which both authors stem. The present book is a hot-off-the-presses second edition dating to this year.

Fomenko and Fuchs state at the very outset that their book was originally “conceived as a comprehensive course of homotopical topology, starting with the most elementary notions, such as paths, homotopies, and products of spaces, and ending with the most advanced topics, such as the Adams spectral sequence and K-theory.” They describe their rationale regarding the presence of the latter subject in a very amusing and pointed manner: they note, first, that it all began in 1895 with Poincaré’s analysis situs followed in the next half-century by the age of the titans, e.g., Alexander, Hopf, Kolmogorov, Whitney, and Pontryagin, and thereupon the age of homological algebra and spectral sequences (vive la France: it’s now the late 1940s and early 1950s with Serre, H. Cartan, A. Borel, and Leray). But

the progress of th[is] new algebraic topology was impressive but short-lived … [to wit:] The last big achievement of the algebraic topology … started by Serre was the Adams spectral sequence which … absorbed all major notions of [then] contemporary algebraic topology …

Happily, light dawned again with the entry onto the scene of Adams (all right, so he just stayed around for the next part of the play), Atiyah, Bott, and Hirzebruch, whose K-theory was responsible for ‘[r]eviving [the] ailing algebraic topology” by appropriately “strong means.” And then Fomenko and Fuchs continue on:

Developing K-theory was more or less completed in the mid-1960s [but c]ertainly it was not the end of algebraic topology. Very important results were obtained later; some of them, belonging to … Novikov, … Buchstaber, … Mishchenko, … Becker, … and Gottlieb, are discussed in the last chapter of this book.

So, given this historical orbit, we have before us a rather comprehensive book: not just an introduction, but a hell of a lot more. In fact, F & F say as much: “The book consists of an introduction [titled ‘The Most Important Topological Spaces’] and six chapters … [which] are divided into parts called ‘Lectures,’ … enumerated throughout the book from Lecture 1 to Lecture 44. Lectures are divided into sections … some [of which are] divided into subsections … [&c.]” and we see why this book adds up to 627 pages.

But what wonderful pages they are. Just look at the Chapters headings and the indicated subsections: under “Homotopy” and “Homology” (Chapters 1, 2) we find Lecture 5 dealing already with CW complexes, Lecture 9 with fibrations, Lecture 17 with homology and manifolds, and Lecture 19 with vector bundles and their characteristic classes. After that it’s “Spectral Sequences of Fibrations,” “Cohomology Operations,” and “The Adams Spectral Sequence,” and then there is the promised sixth chapter which includes, e.g., no less than Riemann-Roch, the Atiyah-Singer formula, and stuff on cobordisms. These last sections in Chapter 6 are necessarily compact and deal with highlights and snippets, but nonetheless there are theorems being proved and exercises being assigned.

All this is clearly quite sporty stuff: the authors’ goal is not just to introduce algebraic topology for general graduate students keen on passing qualifying examinations, but to go well beyond that and prepare fledgling topologists and serious fellow-travelers to become able to do some really serious things. Along these lines, they say

algebraic topology can now be seen as a completed domain, and it is possible to study it from the beginning to the end … not only possible, but highly advisable: [it] provides a necessary background for geometry, analysis, mathematical physics, etc. … This book is intended to help the reader achieve this goal.

Two last comments, one pedagogical, the other esthetic. First, Homotopical Topology contains approximately 500 exercises (ah yes, the Rodina), and it always bears repeating that doing them, or at least a decent number of them, is not just virtuous, it is needed for deeper understanding of, and comfort with, the material. You can’t learn, let alone do, mathematics without loving the daily grind of working very hard: no pain, no gain, as Arnold’s old saying goes (that’s Arnold, not Arnol’d, of course). A propos, with hope springing eternal, I keep telling my students this, but feel more and more like this. It’s a brave new world …

Finally, regarding esthetics, the book is chock-full of illustrations, most of them of the kind you’d expect a zealous topologist to draw, but that’s not all. Courtesy of Fomenko the reader is presented with a large number of idiosyncratic black-and-white drawings of a generally surreal feel. For example already on p. viii, there’s a version of Alexander’s horned sphere rendered as the intertwining of the two arms of a kneeling human figure with bent head: I guess any of us would feel that way if his limbs would turn into Alexander’s horned sphere … Then, on p. 353 we encounter something that looks like a combed space surmounted by a bridge containing a knotted construct, all situated in what looks like outer space (it’s perhaps part of a galaxy?), and sporting a pair of infinite sequences of pedestrians vanishing into the distance by virtue of perspective: on the bridge there are two figures overseeing the double parade below them. Quite a dystopic scene, really.

Well, let’s just say that Fomenko’s art lends a certain je ne sais quoi to the enterprise, which, with or without such surreal art, amounts to a truly fabulous text on algebraic topology. No wonder the mathematics department of Lomonosov Moscow University enjoys such legendary status.

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