Dynamical polysystems and vector bundles (Q1193420)

In an earlier paper [Topology 15, 55-67 (1976; Zbl 0335.57013)] the author introduced the notion of a \(w\)-group which is a modification of a topological group. A relevant example of a \(w\)-group is the free product \(\mathbb{R}*\mathbb{R}\). Each \(w\)-group \(G\) has a classifying space \(BG\), and the author showed there that each smooth manifold \(M\) has the homotopy type of \(BG\) for some \(w\)-group \(G\). This old work is continued in the present paper by showing the following: Assign to a representation \(\rho: G\to GL(n,\mathbb{R})\) of \(G\) the vector bundle \(\xi(\rho)\) over \(M\) classified by the map \(M\to BGL(n,\mathbb{R})\) which is the composition of the homotopy equivalence \(M\to BG\) with \(B\rho\). Then \(\xi\) induces a bijection between equivalence classes of representations of \(G\) and equivalence classes of vector bundles on \(M\). The proofs use a generic pair of vector fields on \(M\) and a resulting operation of \(\mathbb{R}*\mathbb{R}\) on \(M\).