Handlebody decompositions of 4-manifolds and torus fibrations

Abstract

In this theorem, a and b are the numbers of 2 and 3 handles, respectively. The type of the singular fiber in the above fibration is not necessarily 'good' in the sense of [7]. Theorem A was first announced in 1982 in [6]. (See also [7], [8].) The main reason for the long delay in publishing the proof is, of course, the author's laziness. But a reason was partly because the author was not fully convinced of the usefulness of the result; the variety of the singular fibers appearing in the construction seemed quite uncontrolable. As a matter of fact, such wide variety was a key to the proof of the existence theorem. Recently, the author received an enquiry from Daniel Ruberman about the proof. In trying to answer him, the author found a new example of a smooth torus fibration of S over S 2 applying the general construction in this paper to S. Also, he found that, if H2(M;Z) φ {0}, we can arrange so that the general fiber is not homologous to zero in M (Theorem B in Section 3). He hopes that these improvements might justify this late publication of the proof. The author thanks D. Ruberman, whose enquiry gave him an opportunity to publish this paper.