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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?).)

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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.

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  • $\begingroup$ I should have known this Milnor result, but I don't recall knowing it. Do you have an easy reference? $\endgroup$ – Ted Shifrin Jan 25 at 21:02
  • $\begingroup$ @TedShifrin A procedure for killing homotopy groups of differentiable manifolds - lemma 4. maths.ed.ac.uk/~v1ranick/papers/milnorsurg.pdf $\endgroup$ – Connor Malin Jan 25 at 21:21
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    $\begingroup$ Wow, many thanks! $\endgroup$ – Ted Shifrin Jan 25 at 21:29
  • $\begingroup$ Thank you very much for this answer; it's extremely helpful. If you have a moment, I'm wondering about two generalizations: (i) Is this still true for immersed submanifolds (or only embedded ones), and (ii) Is this still true if we replace $\mathbb{R}^7$ by any $7$-manifold? (i.e.: Is $N\Sigma$ still stably trivial?) $\endgroup$ – Jesse Madnick Jan 26 at 8:35
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    $\begingroup$ @JesseMadnick Yes, with the same dimension restriction the result holds for immersions. The other is false, in fact any vector bundle is realized as a normal bundle because the normal bundle of the inclusion of $\Sigma$ as a zero section is the vector bundle itself. $\endgroup$ – Connor Malin Jan 26 at 13:07

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