Is $TS^n$ diffeomorphic to an open subset of $\mathbb{R}^{2n}$ For what values of $n \neq 1,3,7$ is the tangent bundle $TS^n$ of the $n$-sphere diffeomorphic to an open subset of $\mathbb{R}^{2n}$?
 A: There are no sphere's with non-trivial normal bundle in that dimension. As far as I know, this is  originally a theorem of Massey. See http://www.ams.org/journals/proc/1959-010-06/S0002-9939-1959-0109351-8/S0002-9939-1959-0109351-8.pdf for details.
A: Take $n$ odd; this is never possible for $n$ even. Suppose the unit disc bundle of $TS^{n}$ embeds in $S^{2n}$. One may calculate the relative homology of its complement (relative to the boundary) to be supported in degrees $n+1$ and $2n$, and so after appropriate handle cancellation can be obtained by attaching precisely one handle in each of those degrees. But where are we attaching the $(n+1)$-handle on $T^1 S^n$? Its attaching sphere homologous to a section of the bundle; I claim that any sphere in that homology class is isotopic to a section. This is at least true in a sufficiently stable range $(n \geq 6)$ so that homotopy classes of $n$-spheres only contain a single isotopy type. But you can identify the normal bundle of the section with the subbundle of $TS^n$ your section splits off; if this is trivial, then your tangent bundle itself must have been trivial, so $n = 1, 3, 7$ (or I suppose $5$, because I don't know whether there are some extra isotopy classes I don't want for some reason).
