Perhaps the ideological character of the mathematics of the second half of the twentieth century can be aptly conveyed by noting that algebraization was the order of the day — not algebraization in the sense of the algebraization of analysis at the hands of Cauchy and Weierstrass a century or so earlier, but algebraization in the sense of the systematic and large-scale transfer of geometrical and topological themes to the domain of algebra, principally group theory. As the paradigm par excellence of this evolution, algebraic topology came to the fore, following, e.g., Hopf, Cartan, Eilenberg, Pontryagin, and Mac Lane; and then algebraic geometry was dramatically transformed (into commutative algebra, so to speak) at the hands of, first, Zariski, and subsequently, and most spectacularly, Grothendieck.
In the new setting, a number of different flavors of cohomology (prefigured by Poincaré, of course), occupied central stage, while in the latter setting cohomology was quickly adapted to an environment containing varieties and schemes; consider in this connection Serre’s Faisceaux Algébriques Cohérents, where Cech cohomology was suitably transmogrified for varieties, and Grothendieck’s Sur Quelques Points d’Algèbre Homologique (a.k.a. “Tôhoku”), which featured a generalization of the functorial approach presented in Cartan-Eilenberg’s classic Homological Algebra, setting the stage for the characterization of sheaf coholology via right derived functors of the global sections functor.
Indeed, with Grothendieck algebraic geometry took on an unapologetically categorical character. It is still the case today that for many mathematicians (even the youngsters!) this is too much of a good thing: where have all the groups gone?
Interestingly, Grothendieck himself is also credited with another grand algebraic topological program, with groups occupying the limelight: K-theory. But let’s set the record straight: the very first line of the Preface to Complex Topological K-Theory, the book under review, reads thus: “Topological K-theory first appeared in a paper by Atiyah and Hirzebruch; their paper adapted the work of Grothendieck on algebraic varieties to an algebraic setting.” Fair enough. And it soon came to pass that K-theory became the province of a number of other trailblazers, covering a selection of different contexts: the names of Milnor, Bott, Quillen and Karoubi come to mind right away.
The subject is still going strong today, also having reached a form in which it can be safely and successfully introduced at a relatively early stage of a mathematical education. Efton Park’s book seeks to meet precisely this goal. Says Park: “No background in algebraic topology is needed; the reader need only have taken the standard first courses in real analysis, abstract algebra, and point-set topology.” Accordingly the book should be appropriate for a second or third year graduate student, and this is amplified by the fact that each of the book’s four chapters is followed by a solid set of exercises pitched at what looks to be exactly the right level.
The four chapters of Complex Topological K-Theory are titled, respectively, “Preliminaries,” “K-Theory,” “Additional Structure,” and “Characteristic Classes.” The first chapter properly focuses on complex vector bundles; the second is the heart of the book, addressing Bott periodicity and featuring a comparison between cohomology as such and K-theory; the third chapter includes discussions of the K-theory Mayer-Vietoris sequence, the Thom isomorphism theorem, and the Hopf invariant (and so takes the reader into the thick of things); and the fourth chapter starts with de Rham cohomology and ends with Chern classes.
Yes, Complex Topological K-Theory is well-written and pitched at the right level for the motivated reader. Park has included some very interesting material, making this book a terrific introduction to K-theory. The publishers may very well be exactly right in their back-cover description of the text as “the definitive book for a first course in topological K-theory.”
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.