# Which homology 3-spheres bound homology 4-balls?

 Importance: Outstanding ✭✭✭✭
 Author(s): Ancient/folklore
 Subject: Topology
 Keywords: cobordism homology ball homology sphere
 Posted by: rybu on: November 7th, 2009
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.

## Bibliography

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[L] Lisca, Paolo Sums of lens spaces bounding rational balls. Algebr. Geom. Topol. 7 (2007), 2141--2164.

* indicates original appearance(s) of problem.