Lifting a principal G-bundle to a principal bundle with structure group a covering of G

Question

Let $P\to $ be a principal $G$-bundle. Suppose $U$ covers $G$. What do we mean by a lift of $P$ with respect to $U$? Can we take $P,M,G,U$ such that no lift exists?

Answer 1

If $f : G \to H$ is a morphism of topological groups, there's a functor from principal $G$-bundles over $X$ to principal $H$-bundles over $X$ given explicitly by applying $f$ to Cech cocycles, and abstractly by composing the classifying map $X \to BG$ of a principal $G$-bundle (where $BG$ is the classifying space) of $G$) with the induced map $Bf : BG \to BH$.

If $X \to BH$ is a principal $H$-bundle, a lift of it to a principal $G$-bundle is a bundle $X \to BG$ which, after applying the above functor $Bf$, reproduces the original bundle $X \to BH$. There are cohomological obstructions to such lifts existing; for example, if $H = SO(n), G = Spin(n)$, the question is when an oriented $n$-dimensional real vector bundle admits a spin structure, and this occurs if and only if the second Stiefel-Whitney class $w_2 \in H^2(X, \mathbb{F}_2)$ of the bundle vanishes. You can calculate, for example, that $w_2$ of the tangent bundle of $\mathbb{CP}^2$ doesn't vanish, so $\mathbb{CP}^2$ doesn't admit a spin structure.