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)?



Examples of homology sphere that bound a nonsmoothable 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.


Some keywords you might care about are: (integer or rational) homology cobordism group, homology cobordism invariants, Heegaard Floer correction terms (or variants thereof, like involutive Floer homology).


