As a consequence of the famous Riemann Mapping Theorem, all simply connected domain are homeomorphic. I wonder if there is any higher dimensional analogue of this.
For example, if an open set of $\mathbb{R}^n$ has all homotopy(or homology) groups trivial, is it necessarily homeomorphic to $\mathbb{R}^n$?
This leads to another question: in the 2-dimensional case, does $H_1(U)$ being trivial imply $U$ is simply connected?
