Bott Periodicity, Submanifolds, and Vector Bundles

We sketch a geometric proof of the classical theorem of Atiyah, Bott, and Shapiro \cite{ABS} which relates Clifford modules to vector bundles over spheres. Every module of the Clifford algebra $Cl_k$ defines a particular vector bundle over $§^{k+1}$, a generalized Hopf bundle, and the theorem asserts that this correspondence between $Cl_k$-modules and stable vector bundles over $§^{k+1}$ is an isomorphism modulo $Cl_{k+1}$-modules. We prove this theorem directly, based on explicit deformations as in Milnor's book on Morse theory \cite{M}, and without referring to the Bott periodicity theorem as in \cite{ABS}.

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