criterion to show manifold only admits trivial principal bundle

Suppose $M$ a $n-$ dimension manifold and $N$ a $n-$ dimension manifold with boundary. If there exist an embedding $N\to M$ and $M$ only admits trivial principal $G$ bundle. Can we show that $N$ only admits the trivial principal $G$ bundle?

My motivation is trying to proof that every compact $3$ manifold only admits trivial $SU(2)$ bundle using the Heegaard spiltting. What I want to do is showing that every handlebody only admits the trivial $SU(2)$ bundle and using the clutching function to paste two handlebody with trivial $SU(2)$ bundle together. Just like the proof that every $SU(2)$ bundle over $\mathrm{S}^3$ is trivial.

Given a handlebody $H$ and a principal $SU(2)$ bundle $P$ on it. To prove $P$ is trivial, I notice that I can find another handlebody and glue them together to obtain a sphere, then we can use the fact that every $SU(2)$ bundle over sphere is trivial. So what I need to do is just find a suitable principal bundle of the another handlebody such that this bundle together with $P$ give a $SU(2)$ bundle over $\mathrm{S}^3$. But I fail to construct this kind of bundle. So I come here to ask if there is a construction of such bundle, or the question I asked at the first paragraph has a positive answer.

Solution 1:

The answer to the question of the first paragraph is no: If $M$ is the Cartesian plane and $N$ is a closed annulus, then

• Every principal bundle over $M$ is trivial because $M$ is contractible;
• There exists a non-trivial $O(1) = \{1, -1\}$ principal bundle over $N$.

This doesn't address your motivating question, but does seem to say you need to consider the specifics of $SU(2)$-bundles over three-manifolds. Since clutching functions of an $SU(2)$ principal bundle over a three-dimensional CW complex may be viewed as $SU(2)$-valued functions on points, curves, or surfaces, you can argue (modulo details) they're homotopic to constant maps since $SU(2)$ minus a point is contractible.

Solution 2:

Let me answer the actual question you are interested in. Suppose that $M$ is a (connected) 3-dimensional manifold (possibly with boundary, possibly noncompact, it does not matter). For every Lie group $G$, principal $G$-bundles over $M$ are classified by homotopy classes of maps $M\to BG$. In the case of interest, $G=SU(2)$ and, thus, $BG={\mathbb H}P^\infty$, see Tyrone's answer here, that deserves another upvote. Hence, all homotopy groups of $BG$ vanish in degrees $\le 3$. Since $M$ is 3-dimensional, it follows that every map $M\to BG$ is null-homotopic. Hence, every principal $SU(2)$-bundle over $M$ is trivial.