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 K_{0}, continuing with H. Bass's
definition of K_{1} motivated in part by questions in geometric topology, and
including J. Milnor's definition of K_{2} 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 K_{i} 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.

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Eric M. Friedlander <eric@math.northwestern.edu>

Daniel R. Grayson, editors <dan@math.uiuc.edu>