A topological model for the Fukaya categories of plumbings

TL;DR

This paper establishes a topological model linking the algebra of singular cochains to the A-infinity structures in Lagrangian Floer theory for certain cotangent bundle embeddings, providing a new perspective on Fukaya categories.

Contribution

It constructs a differential graded category from singular cochains that is equivalent to the Fukaya category of certain Lagrangian submanifolds, revealing a topological model for these categories.

Findings

Proves equivalence between singular cochain algebra and Floer A-infinity structure.

Constructs a dg-category from singular cochains matching the Fukaya category.

Provides a topological framework for understanding Fukaya categories of plumbings.

Abstract

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddings of a submanifold B with isomorphic normal bundles, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A-infinity category obtained by considering exact Lagrangian embeddings intersecting cleanly along B.