TITLE: Is it possible to improve the Whitney embedding theorem?
QUESTION [37 upvotes]: Edited to fix the example, as per Zack's suggestion. 
Edit 2: So  it turns out that when I think 'manifold' I tend to assume the nicest possible object. As I believe is standard, I would like to assume that all manifolds are 2nd countable and Hausdorff. Furthermore, let's say that our manifolds are connected and closed.
The Whitney embedding theorem states that any smooth $n$-manifold may be smoothly embedded into $\mathbb{R}^{2n}$. If we consider embeddings into more general $k$-dimensional manifolds, is it possible find a '$n$-universal' manifold of dimension less than $2n$?
For example, a non-orientable 2-manifold cannot be embedded into $\mathbb{R}^3$, demonstrating the sharpness of the Whitney embedding theorem. 
However, there are 3-manifolds into which we can embed any surface, such as $M = \mathbb{RP}^3 \sharp \mathbb{RP}^3$. Indeed, by the classification of surfaces we know that any surface may be decomposed as a connected sum of copies of $\mathbb{RP}^2$ and tori. In fact, by the monoid structure of closed surfaces under connected sums we may take this sum to have at most 2 copies of the projective plane. Now, embed 2 disjoint copies of the projective plane into $M$ and arbitrarily many copies of the torus. Taking the connected sum of these we see that any closed surface is embeddable into $M$.
Can we do something similar in higher dimensions?

REPLY [10 votes]: If we restrict our interest to manifolds which are $k$-connected, then Wall proved that any $n$-manifold (closed) $M$ admits a locally flat PL embedding in $\Bbb R^{2n-k}$, thereby improving on Whitney by $k$ dimensions. If in addition we assume the metastable range condition $2k < n$, then we can even take the embedding to be smooth. The latter theorem was also known to Haefliger and Hirsch and is historically earlier.
One further thing worth mentioning: the Hirsch Conjecture says that a stably parallelizable $n$-manifold is supposed to embed in $\Bbb R^m$, where $m = \lceil (3/2)n\rceil $. The conjecture is still open. Partial results are known: for example it's true when the manifold is $[n/4]$-connected.