TITLE: Can a smooth domain in a sphere be a homology ball without being contractible?
QUESTION [12 upvotes]: Suppose $\Omega\subset \mathbb{S}^n$ is an open set with $\Sigma=\partial \Omega$ a smooth hypersurface.  If $\Omega$ is a homology ball, in the sense that $H_i(\Omega)=H_i(\mathbb{D}^n)$ for $0\leq i \leq n$, then is it possible for $\Omega$ to be non-contractible?
Some (possibly incorrect) observations:

When $n=1,2$ it is clear any such $\Omega$ is contractible.
For all $n>1$, I believe $\Sigma$ is a homology sphere (via Poincare duality) so when $n=3$ there is no such non-contractible $\Omega$ (using the classification of surfaces and Alexander's theorem -- is there a simpler argument?)
By removing a small ball from a homology sphere one can produce a manifold with boundary, $X$, that is a homology ball that is not contractible.  However, by the generalized Schoenflies conjecture of Brown and Mazur, such an  $X$ can't embed in $\mathbb{S}^n$.

EDIT:
For what it is worth, this was essentially asked (and answered) previously in this question

REPLY [17 votes]: Yes, such domain exists. Let $P$ be Poincare homology 3-sphere. And $X= P\times I - D^3\times I$, then $X$ can be smoothly embedded in $S^4$(mostly the double of $X$ is $S^4$) . Let $\Omega$ be a small open neighbourhood of $X$. Then notice that $\Omega$ deformation retract onto $X$, but $\Omega$ cannot be contractible as $X$ is not.
There is a deeper reason why a homology ball that $P\# -P$ bounds cannot be contractible, and it is essentially follows from Taube's periodic end theorem.
Here is one of my attempt to draw a picture.