There are some interesting examples of (classes of) manifolds for which I suspect that the Whitney theorem can be strenghtened. For example it is known that every (smooth) $n$-dimensional manifold can be embedded into $\mathbb{R}^{2n-1}$ for example if $M$ is orientable or is not of dimension of the form $2^k$. In particular I'm interested what is the minimal number $emb(n)$ with the property that each $n$-dimensional manifold can be embedded into $R^{emb(n)}$ if we assume additionally that $M$ is:

a) simultaneously orientable and not of dimension of the form $2^k$,

b) a Lie group, and

c) the boundary of some other manifold?

  • 2
    $\begingroup$ This is a well-known wide-open problem. The answer is not known even if you restrict to $M=\mathbb RP^n$. I don't think there is much reason to anticipate any success at the level of generality you are interested in. $\endgroup$ – Ryan Budney Feb 21 '16 at 1:36
  • 1
    $\begingroup$ @RyanBudney: Is it possible that you misread the question? Most (well, at least half of) real projective spaces don't fit under any of the categories a) to c) mentioned in the question. I think it makes perfect sense to ask if Whitney's Theorem can be improved for Lie groups, say, which are parallelizable. $\endgroup$ – Mark Grant Feb 21 '16 at 10:00

The following paper of Elmer Rees addresses precisely your question for Lie groups:

Rees, Elmer Some embeddings of Lie groups in Euclidean space. Mathematika 18 (1971), 152–156.

The main result is that if a compact connected Lie group $G$ of rank $\ell$ admits a faithful representation on $\mathbb{R}^k$, then $G$ admits an embedding into Euclidean space whose codimension is $k-\ell$. This improves on Whitney's Theorem when $k-\ell<\operatorname{dim}G$, which is the case for many of the classical families of Lie groups.


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.