Lifting classes of principal bundles (Q1063303)

Consider a principal G-bundle \(\pi: P\to X\) and a group morphism with closed image \(\sigma\) : \(M\to G\). A lifting of P via \(\sigma\) is a principal M-bundle \({\tilde \pi}: \tilde P\to X\) equipped with a \(\sigma\)-equivariant strong principal bundle morphism \({\tilde \sigma}: \tilde P\to P\). Two \(\sigma\)-liftings \((\tilde P_ 1,{\bar \sigma}_ 1)\) and \((\tilde P_ 2,{\tilde \sigma}_ 2)\) are said to be \(\sigma\)- equivalent iff there exists a strong principal bundle isomorphism \({\tilde \phi}: \tilde P_ 1\to \tilde P_ 2\) such that \({\tilde \sigma}_ 1={\tilde \sigma}_ 2\circ {\tilde \phi}\). Supposing that \(\sigma\) has a central kernel C, the authors define a free operation of \(H^ 1(X,\underline C)\) (with coefficients in the sheaf of C-valued functions) on \({\mathcal L}(P,\sigma)\), the set of lifting classes of P. It is simple transitive, if in addition \(\sigma\) is an epimorphism; otherwise its orbits are classified by sections in the associated bundle \(P\times_ G(G/\sigma M)\), i.e., there exists an injective map \(\eta: {\mathcal L}(P,\sigma)/H^ 1(X,\underline C)\to \Gamma (X,P\times_ G(G/\sigma M))\).