Bott–Thom isomorphism, Hopf bundles and Morse theory

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