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) simoultaneously orientable and not of dimension of the form $2^k$
b) a Lie group
c) the boundary of some other manifold?

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?