# Topology of (small) Lagrangian cobordisms

## TL;DR

This paper investigates how the topology of small Lagrangian cobordisms in symplectic topology is largely determined by their boundary, leading to homological uniqueness results and connections to Fukaya categories and characteristic classes.

## Contribution

It establishes homological uniqueness for small Lagrangian cobordisms and links these results to Fukaya categories and Lagrange characteristic classes.

## Key findings

- Homological uniqueness of small Lagrangian cobordisms.
- Connection between small cobordisms and Fukaya category operations.
- Most known Lagrangian cobordisms are small in the studied sense.

## Abstract

We study the following quantitative phenomenon in symplectic topology: In many situations, if a Lagrangian cobordism is sufficiently small (in a sense specified below) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms which, by Biran-Cornea's Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev's theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.