  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 $\S^{k+1}$, a generalized Hopf bundle, and the theorem
asserts that this correspondence between $Cl_k$-modules and stable vector
bundles over $\S^{k+1}$ 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}.
