Let $U$ and $V$ be two bounded open regions in $\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes.
Assume $\partial U$ is homeomorphic to $\partial V$.
- Is $U$ be homeomorphic to $V$?
- A weaker question: if $U$ is contractible, is $V$ contractible?
For $n=1$, the answer to both is yes, since $\partial U\simeq \partial V$ are sets of $2i$-many points, which both divide $\mathbb{R}^1$ into $i$-many open intervals.
For $n=2$, the answer to both is yes, since $\partial U\simeq \partial V$ are sets of $i$-many topological circles, which both divide $\mathbb{R}^2$ into $i$-many open discs.