# Can a smooth domain in a sphere be a homology ball without being contractible?

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

• For your last point, you can argue $X$ must be simply-connected by using $X$ in a computation of the fundamental group of $S^n$, so it is contractible. This avoids Brown-Mazur. Oct 26, 2021 at 21:36

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.
• And you have a nice answer. I think this Zeeman theorem isn't as well known as it should be. It gives a clean prescription for how to smoothly embed once-punctured 3-manifolds in $S^4$, in terms of the standard language of 3-manifold theory and their automorphism groups. There is a follow-up theorem of Litherland's that extends Zeeman's construction maximally. Oct 26, 2021 at 22:04