Let $X$ be a topological space. Let $ \rho : G \to H$ be a continuous homomorphism of topological groups. Let $ P \to X$ be a principal $ H $ bundle. Let's say that a lift of $P$ via $\rho$ is a pair $\left( Q , f \right)$ consisting of a principal $G$ bundle $Q$ and a bundle map $ f : Q \to P $ who lifts the identity and is equivariant, meaning that $ f(p\cdot h) = f(p) \cdot \rho h $. Two lifts $\left( Q , f \right)$ and $\left( \tilde Q , \tilde f \right)$ are equivalent if there exists an isomorphism of principal $G$ bundles $\Phi : Q \to \tilde Q$ such that $\tilde f \circ \Phi = f$.
My question is: what is the obstruction to the existence of such a lift? And what is the classification of such lifts?
Now let's give a bit of context. This question stems of course from the definition of a spin structure on a manifold. In that case, we have an oriented Riemannian manifold $(X,g)$, from which we get the principal $ SO (n)$ bundle $P$ of positively oriented orthonormal frames. A spin structure is simply an equivalence class of lifts of $P$ via the universal cover $Spin(n) \to SO(n)$.
I know how to get the obstruction to the existence of a spin structure, and how to classify spin structures, using Cech cohomology. Briefly, starting with an open cover $ \mathcal U = \{U_i\}$ of $X$ whose multiple intersections are all contractible, one picks sections $s_i : U_i \to P$ and gets associated transition maps $\phi_{ij} : U_{ij} \to SO(n)$. One then chooses lifts $\tilde \phi_{ij} : U_{ij} \to Spin(n)$ (these exist because $U_{ij}$ is contractible and $Spin(n) \to SO(n)$ is a covering map). The $\tilde \phi_{ij}$ do not necessarily define a $1$-cocycle, but $\gamma_{ijk} = \tilde \phi_{ij} \tilde \phi_{jk} \tilde \phi_{ki}$ takes values in the kernel of the representation, that is, $ \mathbb Z _2 $. One can check that it satisfies the cocycle condition, and therefore represents an element of $H^2 (X,\mathbb Z_2)$, which is the obstruction to the existence of a spin structure. Etcetera etcetera.
Now it seems to me that this construction can be repeated verbatim if $G \to H$ is a connected cover, so that:
- one can lift locally because of the lifting property of covering maps, and
- the kernel of the representation is abelian, and therefore the second cohomology group with coefficients in the kernel makes sense.
On the other hand, I know many more general instances where lifting the structure group is important, and where obstruction and classification are cohomological. In which generality can we play this game? Do you have references that discuss a general approach?
Thank you all.