What is known about spaces of embeddings of contractible manifolds into Euclidean space? I am also curious about the case of small codimension (or even codimension 0). The same question about the configuration spaces in such manifolds. 



Embedding theory for contractible manifolds
(This is by far not a complete answer, just an example.) In dimension 4, a [paper of Livingstone](http://nyjm.albany.edu:8000/PacJ/p/2003/209-2-8.pdf) (build on previous work of Lickorish) constructs some (compact with boundary) contractible 4-manifold which embeds in $\mathbb R^4$ in infinitely many (countable) distinct ways. These are distinguished by the fundamental group of the complement. 


