Given a manifold $M$, we can always embed it in some Euclidian space (general position theorem). Hence we can define the minimal embedding space of $M$ to be the smallest euclidean space that we can embed $M$ in. My question is, will this depend on the category of $M$ (piece-wise linear or smooth)? I am not an expert in this area and I know the difference between these two categories can be subtle. Any pointer is very appreciated.



minimal embedding space of a manifold in smooth and PL case
Yes, it depends on the category of manifolds you are considering.


For example, by Corollary 1.4 of [Hsiang-Levine-Szczarba](https://www.sciencedirect.com/science/article/pii/0040938365900418), the 16-sphere with non-standard smooth structure does not admit a smooth embedding into $\mathbb{R}^{19}$, and hence also not in any Euclidean space of smaller dimension. However, as it is PL-homeomorphic to the standard 16-sphere it does admit a PL-embedding into $\mathbb{R}^{19}$.


