Let $S,M$ be manifolds which are stably parallelizable manifolds and suppose that $S \subset M$ is an embedded submanifold of $M$. When is the normal bundle of $S$ trivial? In particular, what is wrong with the following argument?
$\nu_S = TM|_S/TS \cong (TM|_S \oplus \underline{\mathbb{R}}^N)/(TS \oplus \underline{\mathbb{R}}^N)$ where $\underline{\mathbb{R}}^N$ is a trivial bundle of large enough rank so that both $TM$ and $TS$ are stabilized. So then, $\nu_S \cong \underline{\mathbb{R}}^{\dim M + N}/\underline{\mathbb{R}}^{\dim S + N} \cong \underline{\mathbb{R}}^{\dim M - \dim S}$ and hence, is trivial.
However, this argument must be faulty because Hsiang-Levine-Szczarba showed that there is a homotopy sphere $\Sigma^{16} \hookrightarrow \mathbb{R}^{29}$ which has non-trivial normal bundle. Homotopy spheres are stably parallelizable.