Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^{2n-1}$?

Question

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.

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.

Answer 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 and here.

Answer 2

Every closed orientable (smooth) $n$-manifold (smoothly) embeds in $\mathbb{R}^{2n-1}.$ This is true for $n> 4$ by Haefliger-Hirsch, $n=3$ by C.T.C. Wall (Wall's paper has the reference to Haefliger-Hirsch)

Wall, C. T. C.
All 3-manifolds imbed in 5-space. 
Bull. Amer. Math. Soc. 71 1965 564–567. 

For $n=4$ this is true in the topological category, and is open (as far as I know) in the smooth category [which means that no counterexample is known] - see Bruno Martelli's answer to this question.

Answer 3

For dim $n=4$ it has proven recently by Ghanwat and Pancholi (https://arxiv.org/pdf/2002.11299.pdf) that every closed oriented 4-manifolds smoothly embeds in $\mathbb R^7$.

The (beautiful) key idea of their proof is if we have a closed oriented smooth 4-manifold $M$ such that there exists two smoothly embedded 2-spheres $S^2_a, S^2_b$ that transversally intersect at one point and represent non-trivial element in $H_2(M)/Tor$, then any smooth closed oriented 4-manifold $X$ smoothly embed in $M\times CP^1$. This telling us in particular $X$ embeds smoothly in $S^2\times S^2\times S^2= \partial (S^2\times S^2\times D^3)$ which embeds in $\mathbb R^7$.