Double bubble plumbings and two-curve flops

TL;DR

This paper explores the symplectic topology of certain Stein manifolds formed by plumbing spheres, classifies spherical objects using contraction algebras, and describes homology classes of Lagrangians.

Contribution

It introduces a classification of spherical objects and topological insights for Stein manifolds related to floppable curves via contraction algebras.

Findings

Classification of spherical objects on the B-side.

Complete description of homology classes of graded exact Lagrangians.

Connection between plumbing constructions and local threefolds.

Abstract

We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $(-1,-1)$-curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.