Fibrations sur le cercle et surfaces complexes

Abstract

We give necessary and sufficient conditions for the realization of a given 3-manifold as the boundary of a degenerating family of complex curves, and for the realization of a given link in a 3-manifolds as the boundary of a germ of analytic function at a point of a normal complex surface. These results are based on a study of the topological objects given by these holomorphic maps: let M be a Waldhausen manifold and let L be a union of Seifert fibres, possibly empty, in a Waldhausen decomposition of M. We topologically classify the open-book fibrations Φ:M/L→𝕊 1 with binding L which are transverse to the Waldhausen decomposition of M. We give a necessary and sufficient condition for the existence of such a fibration in terms of a linear system with rational coefficients and we obtain an explicit description of all these fibrations from the topology of (M,L). If L≠∅, we show that there is only a finite number of them. If L≠∅, we show that there is only a finite number of them. If L=∅, we characterize the cases for which there exists an infinite number of such fibrations.