Is there any analogue for Continuous Riemann Mapping Theorem(!) in higher dimensions? For example is it true that every open subspace of $\mathbb{R}^3$ with zero homology in dimensions 1 and 2 is homeomorphic to interior of the unit disk?

Is there any analogue for Riemann Mapping Theorem(!) in higher dimensions?

Or a much simpler question, is it true that every open subset of $\mathbb{R}^3$ with zero homology in dimensions 1 and 2 is homeomorphic to interior of the unit disk? What about contractible subsets?