# Examples of homology sphere that bound a nonsmoothable contractible 4-manifold

Freedman’s theorem shows that all 3-dimensional homology spheres bound topologically a contractible 4-manifold. It is well known that the Poincaré homology sphere does not bound a smooth contractible 4-manifold. Is there other explicit examples or is there an invariant of homology sphere characterizing this property(whether or not smoothly bound a contractible 4-manifold)?

There are certainly lots of obstructions coming from gauge theory, for instance Frøyshov's $$h$$-invariant or Ozsváth and Szabó's $$d$$-invariant. If an integer homology sphere $$P$$ has $$d(P) \neq 0$$, then $$P$$ does not even bound a rational homology ball. There are $$\mathbb{Z}/2\mathbb{Z}$$ refinements (more precisely, obstructions to bounding spin rational homology balls).
In addition to this, and closer to your question: using periodic ends and Seiberg–Witten theory, Taubes proved that if $$Y$$ is a homology 3-sphere that bounds a negative definite 4-manifold with non-diagonal intersection form (e.g. $$Y$$ is the Poincaré sphere, which bounds the $$E_8$$-plumbing of spheres), then $$Y \# {-Y}$$ does not bound a contractible 4-manifold. (It does bound an integer homology 4-ball, namely $$(Y\setminus B^3)\times [0,1]$$.) As far as I know, there is no single invariant capturing this information. Certainly there is no single invariant capturing whether a 3-manifold bounds an integer homology 4-ball.
Freedman proved that any integral homology 3-sphere topologically bounds a contractible, compact 4-manifold. If we start with an integral homology 3-sphere $$\Sigma$$ and double the associated contractible 4-manifold along $$\Sigma$$, then the resulting space is a homotopy 4-sphere, which is homeomorphic to $$S^4$$ (due to the topological 4D Poincare conjecture). So I think the core of the posted question is whether $$\Sigma$$ smoothly embeds in $$S^4$$. (It's easy to see that every homology sphere admits a locally flat topological embedding in $$S^4$$) There are Brieskorn homology spheres which smoothly embeds but some Brieskorn homology spheres do not embed. If I remember correctly, it's still unclear which Brieskorn homology sphere embeds and which does not. On P31 in Budney-Burton, they listed several Brieskorn homology spheres which do not embed in $$S^4$$ since the Rochlin invariant or d-invariant (which is mentioned in Marco Golla's answer) is not zero. However, Budney-Burton also provided Brieskorn homology spheres do embed in $$S^4$$ smoothly. See Theorem 2.13 or P24. I don't know if there are other classified Brieskon homology spheres which are not listed in their survey.