Lagrangian cobordisms and Lagrangian surgery

TL;DR

This paper studies how Lagrangian cobordisms can be decomposed into simpler pieces using Lagrangian surgery, linking topological modifications with holomorphic and Floer-theoretic structures, and provides computational methods for bounding cochains.

Contribution

It proves that any Lagrangian cobordism can be homotoped to a composition of surgery traces and suspensions, with applications to Floer theory and explicit examples.

Findings

Lagrangian cobordisms are homotopic to concatenations of surgery traces and suspensions.

Each surgery trace bounds a holomorphic teardrop linking Morse and Floer cochains.

Algorithmic construction of bounding cochains using these decompositions.

Abstract

Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection. Associated to a $k$-surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. This exact homotopy can be chosen with as small Hofer norm as desired. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated with the handle attachment to the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds. In an appendix, we describe a 2-ended embedded monotone Lagrangian cobordism which is not the suspension of a Hamiltonian isotopy following a suggestion of Abouzaid and Auroux.