Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be 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.


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.

  • $\begingroup$ I wish to add that Hirsch-Haefliger's result assumes $n>4$. The case $n=3$ has to be handled separately, and it was treated independently by Wall (see Rivin's answer below), or by Hirsch (quoted in the Hirsch-Haefliger's paper). $\endgroup$ – Igor Belegradek Jan 1 '16 at 23:43

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.