K-THEORY. An elementary introduction

The purpose of these notes is to give a feeling of “K-theory”, a new interdisciplinary subject within Mathematics. This theory was invented by Alexander Grothendieck 1[BS] in the 50’s in order to solve some difficult problems in Algebraic Geometry (the letter “K” comes from the German word “Klassen”, the mother tongue of Grothendieck. This idea of K-theory has invaded other parts of Mathematics, for example Number Theory [Ga], Topology [Bott] (AH1] and Functional Analysis [Connes]. Among many successes of K-theory, one should mention the solution of classical topological problems ([K] chapter V), the Atiyah-Singer index theorem [AS], its wide generalization to the new subject called “Noncommutative geometry” [Connes] and many algebraic applications [MS] [Sr].