Bott\u2013Thom isomorphism, Hopf bundles and Morse theory

Abstract

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 rightarrow 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}}}( mathord {{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.