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.
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1$\begingroup$ With contractible manifolds it's the "ends" of the manifolds that are interesting so configuration spaces in the manifold shouldn't see anything interesting unless you choose some funny notion of configuration space -- I imagine configuration spaces aren't "cofinal enough" to see ends. Off the top of my head I'm not sure what topology I'd want to put on a space of embeddings of a contractible manifold into a euclidean space. But with whatever definition there should be many embeddings -- when constructing the Whitehead manifold you can use various "twist" variations of the Whitehead link. $\endgroup$– Ryan BudneyOct 14, 2010 at 4:05
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3$\begingroup$ Do you expect the set of connected components to be uncountable for the space of embeddings of the Whitehead manifold in R^3? $\endgroup$– VictorOct 14, 2010 at 6:32
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(This is by far not a complete answer, just an example.) In dimension 4, a paper of Livingstone (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.
answered Oct 14, 2010 at 7:15