New York Journal of Mathematics
Volume 17 (2011) 383-435


Paul J. Kapitza

On small geometric invariants of 3-manifolds

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Published: June 27, 2011
Keywords: Heegaard genus, extended Nielsen genus, Boileau-Zieschang manifolds
Subject: Primary 57M20; Secondary 57M50, 57R65

A small geometric invariant is a nonnegative integer invariant associated with a 3-manifold whose value is bounded above by the Heegaard genus of the manifold.

Craggs has studied techniques to detect for a given 3-manifold M3, whether the double 2M=Bd(M× [-1,1]) bounds a 4-manifold N that has the same 3-deformation type as the complement of the interior of a 3-ball in M and has a handle presentation with, in some sense, a minimal number of 1-handles. Here, M is obtained from M by removing an open ball. He exhibits a pair of surgery obstructions, whose vanishing is sufficient for the existence of this type of 4-manifold N and minimal handle presentation.

We show that for the double of one of the Boileau-Zieschang manifolds, there is a certain handle presentation which, in the absence of the obstructions studied by Craggs, is reducible to this minimal number of 1-handles and we provide an explicit construction. For this case, the question of the existence of a minimal handle presentation is reduced to a study of the obstructions defined by Craggs.

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Berry College