Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

TL;DR

This paper introduces a spectral sequence for exact Lagrangians in cotangent bundles, providing a simplified proof that such Lagrangians with vanishing Maslov class are homology equivalent to the base, extending previous results.

Contribution

It offers a concise proof that exact Lagrangians with vanishing Maslov class are homology equivalent to the base, and establishes conditions for homotopy equivalence based on fundamental groups.

Findings

Exact Lagrangians with vanishing Maslov class are homology equivalent to the base.

The induced map on fundamental groups is an isomorphism.

Homotopy equivalence follows when the base's fundamental group is pro-finite.

Abstract

We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also obtained by Nadler). The ideas in that paper were extended by Abouzaid who proved that vanishing Maslov class alone implies homotopy equivalence. In this paper we present a short proof of the fact that any exact Lagrangian with vanishing Maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism. When the fundamental group of the base is pro-finite this implies homotopy equivalence.