Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?

What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $k$-sphere smoothly embedded in $\mathbb{R}^n$ (I suspect it is not always trivial)?

What I know so far: For example, I can prove that the normal bundle of $S^1 \subseteq \mathbb{R}^n$, $n > 1$, a smoothly embedded circle in $\mathbb{R}^n$, is trivial. I also know that the normal bundles of tori and spheres are always trivial in the codimension-1 case, since these manifolds are orientable. I'm interested in a more general answer.

Bigger picture: More generally, I'm interested in techniques that can be used to prove normal bundles are trivial/nontrivial.


You should be able to prove that the normal bundles in codimension $2$ are trivial as well. This is a little harder than codimension $1$; you need to know that such bundles are determined by their Euler class. The older literature (see papers cited below) tells you that the normal bundles of spheres are trivial once the codimension is high enough.

The paper "On the normal bundle of a homotopy sphere embedded in Euclidean space" by Hsiang-Levine-Szczarba (Topology Volume 3, Issue 2, April 1965, Pages 173-181) quotes an unpublished result of A. Haefliger that there is an embedding of $S^{11}$ in $R^{17}$ with non-trivial normal bundle. Jerry Levine's paper "A classification of differentiable knots" (Annals 82 (1965) 15-50) determines (see proposition 6.2) the possible normal bundles in some range of codimensions in terms of maps in various exact sequences. It's hard to summarize the results, but he gives a table at the end dealing with relatively low dimensions, where calculations can be made explicit. The case $n=11$ and $k=6$ contains Haefliger's example; in fact there are exactly $5$ possible normal bundles among all embeddings.

The introductory sentence to that last section is a classic Jerry Levine understatement: "By strenuous use of Proposition (6.2), together with results of [1], [9], [10], [13], and [24], computation of many of the geometrically defined groups we have discussed can be carried out for low values of $n$."

It's reasonably likely (but you'd have to check) that some of those non-trivial normal bundles of spheres give rise to non-trivial normal bundles of tori, by taking the connected sum of a standard torus (with a trivial normal bundle) with one of those knotted spheres.

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  • $\begingroup$ I'm learning a lot from exploring the references in your answer. Thank you! $\endgroup$ – Matthew Kvalheim Apr 28 '16 at 13:56

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