Constraints on exact Lagrangians in cotangent bundles of manifolds fibred over the circle

TL;DR

This paper establishes topological restrictions on the existence of closed exact Lagrangian submanifolds in cotangent bundles of manifolds fibred over the circle, focusing on fundamental group properties.

Contribution

It provides new topological obstructions, specifically on the fundamental group structure, preventing certain Lagrangians from existing in these cotangent bundles.

Findings

Fundamental group cannot be a free product of two non-trivial groups.

The difference between generators and relations in any finite presentation is less than two.

Certain group-theoretic configurations are incompatible with the existence of such Lagrangians.

Abstract

We give topological obstructions to the existence of a closed exact Lagrangian submanifold in the cotangent bundle of a closed manifold M which is the total space of a fibration over the circle. For instance we show that the fundamental group of such a Lagrangian submanifold cannot be the free product of two non-trivial groups and that in any finite presentation of this group the difference between the number of generators and the number of relations is less than two.