Dynamical polysystems and vector bundles

AbstractLet Mn be a smooth manifold with smooth vector fields v1, v2. The 1-parameter groups defined by these vector fields combine to define an action of the free product R∗R on Mn. For suitable choice of v1, v2, the isotropy group L of some basepoint is of the same homotopy type as the loop space of Mn. Moreover, the natural linear representation of L into O(n) defined by the L-action on the tangent space at the basepoint deloops to the tangent bundle of Mn. This observation can be amplified: k-dimensional vector bundles over Mn are in 1-1 correspondence with equivalence classes of smooth representations of L into O(k). Consequently, for any CW complex C homotopy equivalent to a finite dimensional manifold, k-vector bundles over C may be identified with k-dimensional representations of L for some suitable subgroup L of R∗R