Which homology 3-spheres bound homology 4-balls?

Importance: Outstanding ✭✭✭✭
Author(s): Ancient/folklore
Subject: Topology
Recomm. for undergrads: no
Posted by: rybu
on: November 7th, 2009

\begin{problem} Is there a complete and computable set of invariants that can determine which (rational) homology $3$-spheres bound (rational) homology $4$-balls? % Enter your conjecture in LaTeX % You may change "conjecture" to "question" or "problem" if you prefer. \end{problem}

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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% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

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