# Dimension in Whitney's theorem

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?

• 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. – Ryan Budney Feb 21 '16 at 1:36
• @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. – Mark Grant Feb 21 '16 at 10:00

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.