A Note On Lagrangian Vanishing Sphere Bundles

TL;DR

This paper generalizes the construction of Lagrangian spheres in symplectic manifolds via Lefschetz fibrations to Morse-Bott degenerations, leading to Lagrangian sphere bundles and exploring their topological restrictions through Floer homology.

Contribution

It introduces a Morse-Bott generalization of Lagrangian sphere construction and analyzes topological constraints on Lagrangian sphere bundles using Floer homology.

Findings

Generalization to Morse-Bott degenerations yields Lagrangian sphere bundles.

Topological restrictions on these bundles are derived from Floer homology.

The techniques provide new insights into Lagrangian topology in symplectic geometry.

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.