  Based on Morse theory for the energy functional on path spaces we develop a
deformation theory for mapping spaces of spheres into orthogonal groups. This
is used to show that these mapping spaces are weakly homotopy equivalent, in a
stable range, to mapping spaces associated to orthogonal Clifford
representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible
by four over a finite complex $X$ we derive a stable decomposition result for
vector bundles over the sphere bundle $\mathord{\mathbb S}( \mathbb{R} \oplus
V)$ in terms of vector bundles and Clifford module bundles over $X$. After
passing to topological K-theory these results imply classical Bott-Thom
isomorphism theorems.
