\(K\)-theory. An introduction. With a new postface by the author and a list of errata
Topological \(K\)-theory, \(K^*(X)\), is a generalized cohomology theory constructed using stable equivalence classes of vector bundles on a topological space \(X\). It was originally introduced in 1961 by Atiyah and Hirzebruch [Zbl 0108.17705] who developed it from a similar construction in algebraic geometry made by Grothendieck. It rapidly became one of the most important tools in the topologist's tool kit, giving new results and simplifying proofs of existing theorems such as Adam's Hopf invariant \(1\) theorem. Among the many introductions to the subject which have appeared, one of the most popular has been this volume by Max Karoubi which originally appeared in 1978 [Zbl 0382.55002]. It has now been reprinted with a brief list of errata and a page of comments on more recent developments. It begins with a careful development of the properties of vector bundles and of constructions on bundles. \(K(X)\) is defined as the Grothendieck group of the category of vector bundles on \(X\) and its basic properties are described including the cup product. The Bott periodicity theorem is established both in the complex and the real case, the latter making use of some properties of Clifford algebras which are developed in a separate section. The \(K\)-theory of projective spaces and flag and Grassmann bundles are computed and the Thom isomorphism theorem is established. A final chapter on applications describes some of the many successes \(K\)-theory has had.
scientific article; zbMATH DE number 5306304