K-Theory: An Introduction

I had the great good fortune of being a graduate student at UCSD in the early 1980s when Max Karoubi was visiting there. I recall vividly how elegant his lectures were, even though they were largely out my reach at the time. Not that I gave it much of a try: as someone busy writing his thesis on modular forms and L-functions, I only attended Karoubi’s course for cultural reasons. But I was a fool not to have taken it more seriously. It testifies to my immaturity and shortsightedness that I let such a fantastic opportunity pass me by: I should have worked my fingers to the bone to learn what he was teaching at the time.

Still, there was obviously a measurable effect, resonating even over two decades later, and it is now my great good fortune to have a chance to review Karoubi’s classic K-Theory, An Introduction, whose stated purpose it is “to provide advanced students and mathematicians in other fields with the fundamental material in this subject.” With an unabashedly topological orientation, the book should be accessible to readers with “familiar[ity] with the basic definitions of homotopy theory: homotopy classes of maps and homotopy groups. … Ordinary cohomology theory is used, but not until the end of Chapter V [the last chapter of the book]. Thus this book might be regarded as a fairly self-contained introduction to a ‘generalized cohomology theory.’”

Indeed, himself a student of Grothendieck (and H. Cartan), Karoubi points out that “K-theory was introduced by Grothendieck in his formulation of the Riemann-Roch theorem,” for which he directs us en passant to none other than Borel-Serre (i.e. their famous 1958 paper in Bull. Soc. Math. France). But he goes on to note that whereas Grothendieck worked in the category of coherent analytic sheaves, “this book will study” a “topological analog” dealing with the category of vector bundles on a compact space. This “topological K-theory” is associated with Atiyah-Hirzebruch.

Then Karoubi goes on to cite (in characteristically terse but crystal-clear prose) topological K-theoretical applications by Adams-Atiyah, Adams, Atiyah-Singer, Bass, and Quillen. He states, tellingly, that “[a] key factor in these applications is Bott periodicity.” This amounts to a foreshadowing of what lies ahead, and it is exciting to see, right off the bat, the master’s touch in the presentation of this exciting subject: the above players, whom Karoubi brings in already in the Foreword (dating to 1977), are the pathfinders and early explorer themselves, and Karoubi is obviously keen on setting the stage as effectively as possible, going at the same time for maximal depth, so to speak.

The topological or geometrical setting, then, in which everything takes place ab initio, is that of vector bundles (or locally free sheaves of finite type, although Karoubi says nothing explicit about sheaves in the 300 pages that comprise this wonderful book), and the first 50 or so pages of the book under review deal with this important topic. The second chapter is titled, “First notions of K-theory,” while the third (and obviously central) chapter is “Bott periodicity.” Then, the last two chapters present, respectively, “Computation of some K-groups,” and “Some applications of K-theory.” Thus, the entire orbit from the fundamentals of the subject to how it is used by the experts (if only at a relatively fundamental and accessible level) is covered.

And, to be sure, there are many gems to be mined throughout, what with Thom isomorphism (in the setting of complex vector bundles) featured in Chapter IV and Chapter V including discussions of the vector field problem on the sphere (cf. p. 277: Adams’ theorem giving the maximal number of linearly independent vector fields on a sphere), characteristic classes and the Chern character, and Riemann-Roch. All exceedingly beautiful mathematics!

The book also comes provided with serious exercises the reader, or student, should do if true understanding is the goal. As already indicated, the writing style is marvelous: compact and precise, and very, very clear. Karoubi has also done us the service of appending brief but informative historical notes to each chapter, presenting both indications for further studies along the indicated lines, and interesting insiders’ perspectives (for example: “the introduction of Clifford algebras in real K-theory, which is a key in our proof of the periodicity theorems is due to Atiyah, Bott, and Shapiro [in their 1964 paper on Clifford modules]” (p. 179)).

In closing, then, K-Theory, An Introduction is a phenomenally attractive book: a fantastic introduction and then some. Only a master like Karoubi could have written the book, and it will continue to be responsible for many seductions of fledglings to the ranks of topological K-theorists as well as serve as a fundamental reference and source of instruction for outsiders who would be fellow travelers.

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