Can you exhibit an example of a noncontractible domain in R^n with the dth cohomology groups trivial for all d greater or equal to 1? Thank you
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There are a million examples. You should google "acyclic space". Here is one: if you remove a point from a homology sphere you get a manifold whose cohomology is trivial in positive degrees. Take a tubular neighbourhood of it in some $\mathbf R^n$ and you get an open domain. 

