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

Abstract

We give topological obstructions to the existence of a closed exact Lagrangian submanifold $L\hookrightarrow T^{\ast }M$, where $M$ is the total space of a fibration over the circle. For instance, we show that ${\pi }_1(L)$ cannot be the free product of two non-trivial groups and that the difference between the number of generators and the number of relations in a finite presentation of ${\pi }_1(L)$ is less than two.