Which homology 3-spheres bound homology 4-balls?

Definitions and mathematical statement

Fix coefficients $R=\mathbb Z$ or $R=\mathbb Q$; these give two separate versions. A closed connected oriented smooth three-manifold $Y$ is an $R$-homology sphere if $H_i(Y;R)\cong H_i(S^3;R)$ for every $i$. It bounds an $R$-homology ball if there is a compact connected oriented smooth four-manifold $W$ with oriented boundary $\partial W\cong Y$ and \[ H_i(W;R)\cong H_i(D^4;R)\qquad\text{for all }i. \] In particular the positive-degree homology groups vanish, but contractibility or trivial fundamental group is not required.

The problem asks for complete computable invariants deciding, from a finite triangulation or equivalent finite description of $Y$, whether such a smooth $W$ exists. Completeness means a necessary and sufficient criterion; computability means an effective evaluation and decision procedure that terminates on every allowed input. A list of obstructions with no terminating sufficiency test does not yet provide the requested decision.

The smooth convention follows the source's four-manifold examples and smooth cobordism obstructions; replacing it by arbitrary topological fillings changes the problem. In the rational version torsion in integral homology is allowed. An integral homology ball is automatically a rational one, but the converse need not hold. Both the coefficient ring and the filling category must remain fixed throughout an instance.

Short English statement

Can a complete effective criterion decide which integral or rational homology three-spheres bound smooth homology four-balls with the same coefficients?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3414 (OPG-37145), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Which homology 3-spheres bound homology 4-balls?. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/which_homology_3_spheres_bound_homology_4_balls.