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 $m = [(3/2)n]$. \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]$-connected. 1 If we restrict our interest to manifolds which are$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$m = [(3/2)n]$. The conjecture is still open. Partial results are known: for example it's true when the manifold is$[n/4]\$-connected.