Handbook of K-theory, by Eric M. Friedlander and Daniel R. Grayson, editors

This is the Handbook of K-theory, published in 2005 by Springer, and now released online with permission. Use the URL http://k-theory.org/handbook/ for easy access.

Here are the first two paragraphs from the preface:

This volume is a collection of chapters reflecting the status of much of the current research in K-theory. As editors, our goal has been to provide an entry and an overview to K-theory in many of its guises. Thus, each chapter provides its author an opportunity to summarize, reflect upon, and simplify a given topic which has typically been presented only in research articles. We have grouped these chapters into five parts, and within each part the chapters are arranged alphabetically.

Informally, K-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces. Thus, in some sense, K-theory can be viewed as a form of higher order linear algebra that has incorporated sophisticated techniques from algebraic geometry and algebraic topology in its formulation. As can be seen from the various branches of mathematics discussed in the succeeding chapters, K-theory gives intrinsic invariants which are useful in the study of algebraic and geometric questions. In low degrees, there are explicit algebraic definitions of K-groups, beginning with the Grothendieck group of vector bundles as K0, continuing with H. Bass's definition of K1 motivated in part by questions in geometric topology, and including J. Milnor's definition of K2 arising from considerations in algebraic number theory. On the other hand, even when working in a purely algebraic context, one requires techniques from homotopy theory to construct the higher K-groups Ki and to achieve computations. The resulting interplay of algebra, functional analysis, geometry, and topology in K-theory provides a fascinating glimpse of the unity of mathematics.

Eric M. Friedlander <eric@math.northwestern.edu>
Daniel R. Grayson, editors <dan@math.uiuc.edu>