Exact Lagrangians in the cotangent bundle of a sphere and a torus

TL;DR

This paper provides a Fukaya-theoretic proof that any closed, exact Lagrangian in the cotangent bundle of a sphere or torus shares the same homotopy type as the zero section, illustrating advanced symplectic geometry techniques.

Contribution

It offers a new proof for the homotopy equivalence of Lagrangians in cotangent bundles of spheres and tori using Fukaya categories, highlighting homological algebra methods.

Findings

Confirmed homotopy equivalence for spheres and tori

Demonstrated Fukaya-theoretic techniques in symplectic geometry

Provided a new proof approach for Lagrangian classification

Abstract

It is known that any closed, exact Lagrangian in the cotangent bundle of a closed, smooth manifold is of the same homotopy type as the zero section. In this paper, we give a Fukaya-theoretic proof of this fact for the sphere and torus to review and demonstrate some of the homological algebra techniques in symplectic geometry.