Celebratio Mathematica — Atiyah

Papers on K-theory (1959–1970)

Works connected to Raoul H. Bott

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M. F. Atiyah, R. Bott, and A. Shapiro: “Clifford modules,” Topology 3 : Supplement 1 (July 1964), pp. 3–38. MR 0167985 Zbl 0146.19001 article

Raoul H. Bott	Related	Volume
Arnold Samuel Shapiro	Related

M. Atiyah and R. Bott: “On the periodicity theorem for complex vector bundles,” Acta Math. 112 : 1 (1964), pp. 229–247. MR 0178470 Zbl 0131.38201 article

The periodicity theorem for the infinite unitary group [Bott 1959] can be interpreted as a statement about complex vector bundles. As such it describes the relation between vector bundles over \( X \) and \( X\times S^2 \), where \( X \) is a compact space and \( S^2 \) is the 2-sphere. This relation is most succinctly expressed by the formula \[ K(X\times S^2) \simeq K(X)\otimes K(S^2), \] where \( K(X) \) is the Grothendieck group of complex vector bundles over \( X \). The general theory of these \( K \)-groups, as developed in [Atiyah and Hirzebruch 1961], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof.