Suppose $\alpha:M\to N$ is a homomorphism of compact Lie groups and $\eta=(E,B,N)$ is an $N-$principal bundle over a compact manifold. Give a condition of $\alpha$ such that $\eta$ can lift as an $M-$principal bundle through $\alpha$. Can $\alpha$ be represented by a characteristic class of $\eta$?
I do not have any ideas about this. Could anyone give me some hints?
So we have $$\eta:E\longrightarrow B\ .$$ The morphism $\alpha$ always gives a $M$-bundle structure on $E$, by $$m\cdot e := \alpha(m)\cdot e\ .$$ If we want this action to be free, then we need that for every distinct $m_1,m_2\in M$ and any $e\in E$ we have $$\alpha(m_1)\cdot e\neq\alpha(m_2)\cdot e\ .$$ In particular, $\alpha$ must be injective. Similarly, if we want the action to be transitive on the fibers, then $\alpha$ must be surjective. Therefore, the only case where $\eta$ can lift to a principal $M$-bundle through $\alpha$ is when $\alpha$ is an isomorphism of Lie groups.