Let $\Sigma^3 \subset \mathbb{R}^7$ be a non-compact, orientable, smooth $3$-dimensional submanifold. Its normal bundle $N\Sigma \to \Sigma$ has rank $4$, and therefore (I think) has a non-vanishing section.
I would be happy to learn that $N\Sigma$ must, in fact, be trivial, but I'm guessing that's not true. So, are there simple examples — the simpler the better, really — of non-compact, orientable, smooth $\Sigma^3 \subset \mathbb{R}^7$ with non-trivial normal bundle?
(I'm guessing there are easy examples, but frankly, even if I were to start listing simple embeddings $\Sigma^3 \hookrightarrow \mathbb{R}^7$, I wouldn't know how to quickly compute the relevant characteristic classes (in fact, what would those even be here?).)
Best Answer
Every orientable 3-manifold is parallelizable, i.e. its tangent bundle is trivial. This implies that any normal bundle is stably trivial. So $N\Sigma$ is a stably trivial bundle of rank 4 over a 3-dimensional manifold.
Milnor proved that if a vector bundle has rank higher than the dimension of your CW complex, then the vector bundle is trivial, if and only if, the vector bundle is stably trivial. So we conclude that the normal bundle of any embedding of a 3-manifold into $\mathbb{R}^7$ (or any greater dimension), is trivial.