A Note On Lagrangian Vanishing Sphere Bundles

Abstract

A classical way to construct a Lagrangian in a symplectic manifold $\Sigma$ is to let $\Sigma$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $\Sigma$ and these spheres, in turn, are representatives of the corresponding vanishing cycles in the homology of $\Sigma$. In this paper our aim is twofold: The first is to describe a generalization of the above mentioned construction to the "Morse-Bott" case. This leads, whenever such a degeneration exists, to the existence of Lagrangian sphere bundles rather than just spheres. In the second part of the paper we study the type of topological restrictions on such Lagrangian sphere bundles arising from the theory of Floer homology for Lagrangian intersections and to illustrate the techniques involved.