K-theory PDF

Special Preface

K-theory is that part of linear algebra that studies additive or abelian properties (e.g., the determinant). Because linear algebra, and its extension to linear analysis, is ubiquitous in mathematics, K-theory has turned out to be useful and relevant in most branches of mathematics. Introduced first in Algebraic Geometry by Grothendieck, it was then transposed to topology by Hirzebruch and myself as expounded in these lecture notes. Subsequently it has wandered into several parts of analysis and, in a different guise, into number theory. It has also been studied systematically in a purely algebraic context. In each of these different fields there are of course different techniques which have to be employed and the relevant K-theory then rests on the appropriate linear algebra or analysis involved

Most of this development has taken place in the 20 years since this book first appeared and it may be helpful to provide at this stage a brief overview of these new manifestations of K-theory.

The first major application of topological K-theory was to the proof of the index theorem for elliptic differential operators by Singer and myself. K-theory enters into the theorem at two stages. First of all, it enters into the global analysis because of the connection with Fredholm operators explained in the Appendix. Secondly, it enters locally in terms of the symbol of the differential operator. Both of these have their antecedents in Grothendieck’s work, but both appearances of K-theory are purely algebraic.

The full development of index theory in its various generalizations makes extensive use of K-theory, and conversely index theory can be used to give proofs of the basic periodicity theorem. In fact, K-theory and index theory really become fused into a single theory in which it is hard to disentangle the topology from the analysis

Author: M. F. Atiyah