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Modern Classical Homotopy Theory

Jeffrey Strom
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 127
[Reviewed by
Michael Berg
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Jeffrey Strom’s big book (over 800 pages) on the subject of homotopy theory, more precisely, Modern Classical Homotopy Theory, sports a useful blurb on its back-cover, which goes a long way to explaining what at first glance seems like a contradiction, or at best a stretch, viz. something that’s both classical and modern:

The core of classical homotopy is a body of ideas and theorems that emerged in the 1950s and was later codified in the notion of a model category … Brown’s representability theorems show that homology and cohomology are also contained in classical homotopy theory …

So what we’re dealing with is really the material situated at the very heart of modern algebraic topology, taking note of the fact that the subject is now old enough to support the distinction of possessing a classical period, so to speak. It also betrays a fast evolutionary trajectory in that what we’re dealing with here is, indeed, traceable to major, if senior, players in topology who made their mark in the 1950s and 1960s, i.e., scholars of only a generation or two ago.

But what is clear as vodka, at least if your exposure to homotopy (as well as (co)homology) dates to ca. the 1970s, is that a very different flavor pervades the book under review than is the case for, say, Spanier’s classic book on Algebraic Topology or Albrecht Dold’s Lectures on Algebraic Topology. There is a far greater emphasis on category theory and the unified view of things it affords.

Dold’s book only spends a dozen introductory pages on “Preliminaries on Categories, Abelian Groups and Homotopy,” after which it all begins for real with homology. It’s not till over a hundred pages later that we get to “Functors of Complexes.” Not dissimilarly Spanier starts off with only eight pages on categories and functors, and then races ahead to homotopy.

In sharp contrast, Strom painstakingly (and, to my mind, quite correctly) goes into significant detail vis à vis “The Language of Categories,” devoting the entirety of his Part 1 to this material, systematically developing it up to and including a discussion of limits and colimits replete with diagrams galore. The fact that this latter theme is covered in only a page (p.18) in Spanier’s book again suggests in no uncertain terms that the contemporary approach to this part of topology is rather different, pointing toward a shift in the thrust of the subject.

Indeed, in his Part 2, “Semi-Formal Homotopy Theory,” Strom starts off with a full chapter titled “Categories of Spaces,” and doesn’t get to homotopy per se till around p.70: we meet CW complexes and pointed spaces first, and, once we get to homotopy, it’s all done very systematically and carefully, but not so as to retard progress toward very important results relatively early on. At this stage in the proceedings, when we’re really just beginning to climb to cruising altitude, Strom gives us “Groups and Cogroups in the Homotopy Category,” a cogroup object being dual to a group object (as was discussed earlier in the book, in the indispensable Part 1).

It is fair to say that Part 2 is where the rubber hits the road in this huge book, as we encounter in its two hundred pages or so topics ranging from, of course, homotopy, to fibrations and cofibrations, homotopy limits and colimits, homotopy pullbacks and pushouts, as well as Mayer-Vietoris, stuff by George Whitehead, as well as L(y)usternik-Schnirelman. I was particularly pleased to see a separate section in Strom’s ninth chapter devoted to “Unitary Groups and Their Quotients,” the focus naturally falling on, among other things, the Grassmannian.

Well, on and on it goes: we’re flying high now, but at cruising speed — no need to go supersonic. Strom takes his time, he is very thorough, he peppers the book with problems for the student (indeed, his book is, in a way, all problems: see below), and he uses a colloquial style. It all works quite well.

Part 3 concerns “Four Topological Inputs,” and includes coverage of, e.g., dimension theory, subdivision of disks (including Seifert-Van Kampen), fibrations done locally, and pulling back cofibrations; it is here that we encounter such important themes as locally trivial bundles, covering spaces (and group actions), and Serre fibrations. Part 4 is heavily categorical in nature: “Targets as Domains, Domains as Targets,” but we are of course heavily invested in algebraic topology proper at this stage: Postnikov, Moore paths and loops, Eilenberg-Mac Lane spaces, Moore spaces, and so on.

Parts 5 and 6 are titled, respectively, “Cohomology and Homology,” and “Cohomology, Homology, and Fibrations.” We find the usual suspects here, of course, ranging from Künneth (on p.487), about which Strom says that “[i]n algebraic topology, any theorem that gives a formula for the cohomology (of whatever kind) of a product of spaces is referred to as a Künneth theorem,” to Leray-Serre. He also presents Brown’s Representability Theorem here, saying.

When faced with an abstract cohomology theory … wouldn’t it be nice to know that there were [natural] spaces and natural isomorphisms … [that] would allow us to apply all that he have learned about computing and analyzing homotopy sets to understand the otherwise utterly abstract functors? The Brown Representability Theorem tells us that there are such spaces, though the natural transformations are only guaranteed to be isomorphisms for CW complexes …

In accord with our earlier remarks, this is of course the raison d’être of much of Strom’s fine book: it’s a means of capturing a lot of (co)homology under the umbrella of “classical homotopy theory.” (For me, as a number theorist with a number theorist’s idiosyncratic needs and uses for topology, particularly cohomology theories of different types, not generally stressed by topologists as such, this is in itself quite interesting: Strom’s perspective is very tantalizing.)

So Bott periodicity appears on p.681, the aforementioned Leray-Serre spectral theorem appears ca. p.700 (Wow!), and, to be sure, homotopy groups of spheres are touched on starting on p.711 with the following apposite introduction:

The computation of homotopy groups is quite difficult, so results that apply to all the homotopy groups of a sphere are surprising and wonderful. In this section you [!] will establish a variety of ‘qualitative’ global results about the homotopy groups of spheres … [s]pecifically, … exactly which groups πk(Sn) are finite and which are infinite, and you will find the first few nontrivial p-torsion groups for each prime p…

This snippet illustrates Strom’s prevailing pedagogical approach throughout his book, as well as the reason for its huge size. Having laid the scaffolding for what he now asks his reader to do, Strom parses the pending big results into a sequence of sub-problems each of which is accessible to the prepared player, modulo the obvious stipulation that the latter has indeed carefully covered what came before. This is clearly a superb approach: learn hard mathematics by getting your hands really, really dirty. There’s really no other way, of course.

In fact, this is ultimately a rather unusual feature of Modern Classical Homotopy Theory, and perhaps something of a possible controversy; in Strom’s own words:

Many authors of textbooks assert that the only way to learn the subject is to do the exercises. I have taken this to heart, and so there are no outright proofs in the book. Instead, theorems are followed by multi-part problems that guide the readers to find the proofs for themselves. To the expert, these problems will read as terse proofs, perhaps suitable for exposition in a journal article. Reading this text, then, is a preparation for reading journal articles…

And for doing mathematics yourself, at a pretty high level. It is bound to work, at least for the student who has the requisite perseverance for this experience.

Finally, regarding Strom’s teaching recommendations; he proposes that Parts 1 to 4 cover the first semester of an introductory graduate course in algebraic topology, and Parts 5 to 6, the second. Part 7, titled, “Vistas,” is proposed under the heading “if time permits.”

Obviously the book was a labor of love for its author: this is visible on every page. The coverage of the material is, in a word, amazing, even to an outsider like me. The book is well-written, as I have already indicated, and Strom’s “problems first-and-foremost” approach is bound to be a big pedagogical hit for those who can handle it, both in front of the class and in it. After all, there is already something of a famous precursor available in the area of analysis, in the form of Pólya-Szegö, Aufgaben und Lehrsätze aus der Analysis, and its pedagogical successes are legendary. The book under review is a wonderful contribution indeed.

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