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Based on Morse theory for the energy functional on path spaces we develop adeformation theory for mapping spaces of spheres into orthogonal groups. Thisis used to show that these mapping spaces are weakly homotopy equivalent, in astable range, to mapping spaces associated to orthogonal Cliffordrepresentations. Given an oriented Euclidean bundle $V \to X$ of rank divisibleby four over a finite complex $X$ we derive a stable decomposition result forvector bundles over the sphere bundle $\mathord{\mathbb S}( \mathbb{R} \oplusV)$ in terms of vector bundles and Clifford module bundles over $X$. Afterpassing to topological K-theory these results imply classical Bott-Thomisomorphism theorems.

Bott-Thom isomorphism, Hopf bundles and Morse theory