Let $M$ be an $m$-dimensional smooth manifold with boundary $\partial M$, and let $G$ be a compact Lie group. When can a principal $G$-bundle with base manifold $\partial M$ be extended to a principal $G$-bundle with base manifold $M$?
If the general answer is unknown, then where can I find a good review of partial results?
I have a superficial/physics-level familiarity with the ideas of homotopy groups, classifying spaces, and bordisms, but I'm not yet fluent enough in the languages of topology to know exactly how to apply those ideas to this question.