Imbeddings of $n$-dimensional PL-manifolds in $2n$-euclidean space Does anyone know how to prove that any $n$-dimensional PL-manifold has an imbedding in euclidean space of dimension $2n$ ? Is this done via dimension theory ? Are there any references ?
Thanks a lot for your help !!!
Cheers
 A: The sketch of the idea is here: http://en.wikipedia.org/wiki/Whitney_embedding_theorem
it's called the Strong Whitney Embedding Theorem. 
Getting the embedding into $\mathbb R^{2n+1}$ is just general position.  Similarly, you can construct an immersion in $\mathbb R^{2n}$ by general position.  
At that point, you perturb the immersion so that all double points are regular, meaning intersecting like $\mathbb R^n \times \{0\}^n$ and $\{0\}^n \times \mathbb R^n$ in $\mathbb R^{2n}$. 
Then you try to apply the "Whitney Trick".  This only works if the manifolds dimension $n$ is at least $3$.  When $n=1,2$ you have to resort to other methods.  $n=1$ is simple, since the circle is the only compact connected boundaryless 1-manifold.  $n=2$ could use the classification of surfaces. 
The basic idea of the Whitney trick is explained on the webpage.  You try to `cancel' opposite double-points by finding a local model where you can construct the cancelling motion (regular homotopy it tends to be called in that literature -- or a 1-parameter family of immersions).  
Sometimes you can't, and for that you modify the immersion by adding a local double point.  That's what the function $\alpha_m$ describes on the Wikipedia page, the local model for the double-point introduction. 
