On the Periodicity Theorem for Complex Vector Bundles

where K ( X ) is the Grothendleck group (2) of complex vector bundles over X. The generaI theory of these K-groups, as developed in [1], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory i t seems desirable to have an elementary proof of it, and it is the purpose of this paper t(~ present such a proof. Our proof will be strictly elementary. To emphasize this fact we have made the paper entirely self-contained, assuming only basic facts from algebra and topology. In particular we do not assume any knowledge of vector bundles or K-theory. We hope that , by doing this, we have made the paper intelligible to analysts who may be unacquainted with the theory of vector bundles but may be interested in the applications of Ktheory to the index problem for elliptic operators [2]. We should point out in fact tha t our new proof of the periodicity theorem arose out of an a t tempt to understand the topological significance of elliptic boundary conditions. This aspect of the mat ter will be taken up in a subsequent paper.(a) In fact for the application to boundary problems we need not only the periodicity theorem but also some more precise results tha t occur in the course of our present proof.