A model for embedding finite coverings defined by principal bundles into bundles of manifolds (Q1102580)

For any given finite group G and locally trivial fibration \(p: V\to X\) over a connected CW-complex with connected manifold of dimension \(\geq 2\) as fiber, a principal G-bundle \(P_ G: F_ G(V)\to C_ G(V)\) is constructed along with a locally trivial fibration \(p_ C: C_ G(V)\to X\). It is then shown that the sections of \(p_ C\) classify, via pullback of \(p_ G\), those principal G-bundles on X that admit an embedding into \(p: V\to X\). A special case of this construction (when p is the trivial complex line bundle) yields a classification of polynomial principal G- bundles. An algebraic characterization of such G-bundles is also given in terms of homomorphisms into braid groups.