**Problem**Is there a complete and computable set of invariants that can determine which (rational) homology -spheres bound (rational) homology -balls?

Determining which homology -spheres bound homology -balls is a long standing open problem in 3/4-manifold topology. Much effort has gone towards understanding the situation for the Brieskorn homology spheres. For example, the Poincare Dodecahedral space is known not to bound a homology -ball since the Rochlin invariant is non-trivial -- but the connect-sum of Poincare Dodecahedral space with its orientation-reverse does bound a homology 4-ball, and it has a simple construction: remove an open tubular neighbourhood of from , this is the -manifold.

Standard invariants used to show homology -spheres do not bound homology -balls are various spin or spin^c cobordism invariants such as: the Rochlin invariant, Siebenmann's -invariant, the Oszvath-Szabo -invariant, and there are many others.

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* indicates original appearance(s) of problem.