On small geometric invariants of 3-manifolds

摘要

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 certainhandle 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.