Importance: Outstanding ✭✭✭✭
Author(s): Ancient/folklore
Subject: Topology
Recomm. for undergrads: no
Posted by: rybu
on: November 7th, 2009
Problem   Is there a complete and computable set of invariants that can determine which (rational) homology $ 3 $-spheres bound (rational) homology $ 4 $-balls?

Determining which homology $ 3 $-spheres bound homology $ 4 $-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 $ 4 $-ball since the Rochlin invariant is non-trivial -- but $ M\#(-M) $ the connect-sum of Poincare Dodecahedral space $ M $ with its orientation-reverse does bound a homology 4-ball, and it has a simple construction: remove an open tubular neighbourhood of $ \{*\} \times I $ from $ M \times I $, this is the $ 4 $-manifold.

Standard invariants used to show homology $ 3 $-spheres do not bound homology $ 4 $-balls are various spin or spin^c cobordism invariants such as: the Rochlin invariant, Siebenmann's $ \overline{\mu} $-invariant, the Oszvath-Szabo $ d $-invariant, and there are many others.


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* indicates original appearance(s) of problem.


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