Nearby Lagrangians with vanishing Maslov class are homotopy equivalent

TL;DR

This paper proves that all closed exact Lagrangians with vanishing Maslov class in a cotangent bundle are homotopy equivalent to the zero section, using advanced Fukaya category techniques and geometric arguments.

Contribution

It introduces an extended Fukaya category with local systems, demonstrating generation by cotangent fibers, and proves homotopy equivalence of Lagrangians in general cotangent bundles.

Findings

Homotopy equivalence of Lagrangians to the zero section

Generation of Fukaya category by local systems over cotangent fibers

Universal cover of Lagrangians is trivial in simply connected cases

Abstract

We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study an extension of the Fukaya category in which Lagrangians equipped with local systems of arbitrary dimension are admitted as objects, and prove that this extension is generated, in the appropriate sense, by local systems over a cotangent fibre. Whenever the cotangent bundle is simply connected, this generation statement implies that the universal covering of every closed exact Lagrangian of vanishing Maslov index is trivial. Finally, we borrow ideas from coarse geometry to develop a Fukaya category associated to the universal cover, allowing us to prove the result in the general case.