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Handbook of K-Theory, 2 volume set (English and French Edition)

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K-theory is now a highly developed but esoteric subject, and touches many different areas in mathematics, including operator theory and functional analysis, algebraic geometry, and geometric topology. In addition, it has found its way into theoretical physics, thanks to the advent of string theory and its more modern metamorphosis M-theory. Everything about K-theory is fascinating, and this two-volume set gives a general overview of the subject from the standpoint of a collection of researchers who have been involved in its development. It is not written for those who are interested in learning K-theory, since it emphasizes developments lying in the frontier of the subject. Students of K-theory, and non-experts (such as this reviewer) can still gain a lot however from its perusal, due to the clarity exhibited in each article along with the copious references at the end of each. Historians of mathematics who want to trace the history of K-theory will also find the volumes of great interest.

K-theory has been developed in both a topological and algebraic context, with the former being more easily grasped for newcomers. It is in the context of algebraic geometry where research in K-theory has shown the greatest activity. Earlier developments in K-theory emphasized its role in the classification and study of vector bundles, and these developments led many to find suitable formulations for algebraic varieties and general schemes. What is now called `motive theory' involves the study of how well known constructions in algebraic topology can be carried over to algebraic geometry. One article in this handbook that gives a good motivation for this study is the one by Daniel Grayson on the motivic spectral sequence. In the article Grayson discusses different approaches to finding a `motivic' version of the Atiyah-Hirzebruch spectral sequence, the latter of which relates topological K-theory to singular cohomology. The trick is not only to find a suitable spectral sequence but also one that is computable. The author shows various ways in which spectral sequences can be constructed, such as the use of long exact sequences in homotopy theory and by using filtrations of a spectrum (such as the familiar Postnikov tower of a space). These are well known in algebraic topology, but for (nonsingular) varieties or (regular) schemes in algebraic geometry one needs another approach that respects as much as possible the general ideas in algebraic topology. One of the approaches discussed is actually fairly intuitive, since it relates K-theory to chain complexes, the latter of which are constructed from direct-sum Grothendieck groups of commuting automorphisms. This approach reflects the well-known strategy of studying the behavior of groups by relating them to the homotopy of a particular space (the mathematician Daniel Quillen used this idea to arrive at his definition of the higher K-groups). Grayson also discusses another approach to obtaining motivic cohomology by using the higher Chow groups, and the work of the mathematician Vladimir Voevodsky on using (affine) homotopy theory of schemes. Voevodsky's work is also motivated by a familiar idea in algebraic topology, namely that of a simplicial space. Voevodsky replaces the simplices by affine spaces over a field, along with the smooth varieties over this field and the colimits of diagrams between these varieties. The colimits are presheaves on these varieties, which are then made into sheaves in a topology called the Nisnevich topology (which is finer than the Zariski topology but coarser than the etale topology). The affine simplices are contractible, and allow the usual techniques of algebraic topology to be applied. In particular, spectra can be defined, called `motivic spectra', and the algebraic K-theory of these spectra results in the motivic spectrum. The Voevodsky construction of a motivic spectral sequence uses a suitable filtration of this motivic spectrum, called the `slice filtration.' The slice filtration involves taking suspensions of the suspension spectra of smooth varieties. Grayson discusses the viability of this approach via the conjectures that were made by Voevodsky, one of which was proved when the field is assumed to have characteristic zero.

Jonathan Rosenberg writes another interesting article in the handbook on the use of K-theory in geometric topology. One immediately thinks of vector bundles in this context and indeed Rosenberg outlines the role of K-theory in the study of flat vector bundles. The K-theory spectrum of the complex numbers arises here, in that every class in this spectrum arises from some flat vector bundle over a homology n-sphere. Also discussed, and definitely a more contemporary topic, is the Waldhausen A-theory, which is a variant of algebraic K-theory, and is highly complex in both its formulation and the proof of its main results. Rosenberg shows how to define the Waldhausen A(X) when X is a pointed space in terms of the infinite loop space whose homotopy groups are the stable homotopy groups of the loop group of X. With multiplication defined by concatenation of loops, this loop space is a `homotopy ring' and if X is path-connected there is map from the loop space to the group ring of the first homotopy group of X. A(X) is then the K-theory space of the ring up to homotopy. The advantage of A(X) according to Rosenberg is that there is essentially a linear map from it to the K-group of the first homotopy group ring, which in some cases is an equivalence. For the case of compact smooth manifolds and its space of pseudo-isotopies, A(X) gives information on higher Whitehead and Reidemeister torsion. Rosenberg ends the article with a very brief discussion of the application of K-theory to symbolic dynamics. In this area of chaotic dynamical systems one is interested in what transition matrices will give equivalent symbolic dynamics. One can define an equivalence relation between them, called `shift equivalence.' K-theory assists in the study of shift equivalence by defining a C*-algebra associated to the shift, and studying the zeroth K-group of this C*-algebra.