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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?

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    $\begingroup$ The Riemann mapping theorem is about conformal maps, not just homeomorphisms. I think you need to narrow down your question or define it more precisely $\endgroup$
    – Yemon Choi
    Commented Nov 30, 2014 at 6:07
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    $\begingroup$ Yes I know the Riemann Mapping Theorem speaks about conformal maps, but here I am just interested in continuous case. $\endgroup$
    – Hesam
    Commented Nov 30, 2014 at 6:15
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    $\begingroup$ A Riemann mapping theorem that is not about conformal maps should not be called a Riemann mapping theorem. $\endgroup$
    – Yemon Choi
    Commented Nov 30, 2014 at 15:32

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This is not true. For example, Alexander horned sphere divides $S^3$ into two regions, both with trivial homology (Poincare duality), but the $\pi_1$ is nontrivial.

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The Whitehead manifold is a counterexample: it is a contractible subset of ${\mathbb R}^3$, but not homeomorphic to the open unit disk.

https://en.wikipedia.org/wiki/Whitehead_manifold

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Even in the holomorphic category, Riemann Mapping Theorem fails in high dimensions. For example, the unit ball is not biholomorphic to polydisks.

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