Bott periodicity, submanifolds, and vector bundles

Abstract

We sketch a geometric proof of the classical theorem of Atiyah, Bott, and Shapiro [3] which relates Clifford modules to vector bundles over spheres. Every module of the Clifford algebra Clk defines a particular vector bundle over Sk+1, a generalized Hopf bundle, and the theorem asserts that this correspondence between Clk -modules and stable vector bundles over Sk+1 is an isomorphism modulo Clk+1 -modules. We prove this theorem directly, based on explicit deformations as in Milnor’s book on Morse theory [8], and without referring to the Bott periodicity theorem as in [3].