Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be smoothly embedded in $\mathbb R^{2n-1}$?


I know these cannot exist for $n=1$, i.e. $S^1$. If we ignore orientability, then if we take $n=2^r$, $\mathbb RP^n$ cannot be embedded in $\mathbb R^{2n-1}$.


While searching on the internet, I found some sharper results [here](https://en.wikipedia.org/wiki/Whitney_embedding_theorem#Sharper_results).


So can we say anything when $\dim(M)= 2^r$ and $M$ is orientable?


It would be very helpful if someone could provide me with some more references. Thanks in advance.



Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^{2n-1}$?
A closed smooth $n$-manifold embeds into $\mathbb R^{2n-1}$ if and only if the normal $(n-1)$th Stiefel-Whitney class vanishes. This is due to Hirsch-Haefliger in dimensions $\neq 4$ and to Fang in dimension $4$. Massey showed that if the normal $(n-1)$th Stiefel-Whitney class is nonzero, then $M$ is non-orientable and $n$ is a power of $2$. Thus any smooth closed orientable $n$-manifold smoothly embeds into $\mathbb R^{2n-1}$.
For references see [here](http://www.sciencedirect.com/science/article/pii/0040938363900284) and [here](http://www.sciencedirect.com/science/article/pii/0040938394900221).


