By Max Karoubi

*From the Preface: K*-theory used to be brought through A. Grothendieck in his formula of the Riemann- Roch theorem. for every projective algebraic kind, Grothendieck built a gaggle from the class of coherent algebraic sheaves, and confirmed that it had many great houses. Atiyah and Hirzebruch considered a topological analog outlined for any compact house *X*, a gaggle *K{X)* produced from the class of vector bundles on X. it really is this ''topological *K*-theory" that this booklet will study.

*K*-theory has develop into a massive instrument in topology. Using

*K*- idea, Adams and Atiyah have been in a position to supply an easy evidence that the single spheres which are supplied with

*H*-space constructions are S1, S3 and S7. in addition, it really is attainable to derive a considerable a part of strong homotopy concept from K-theory.