Maslov class of exact Lagrangians and cylindrical handles

TL;DR

This paper demonstrates that the Maslov class can be non-zero for certain exact Lagrangians in Weinstein domains, contrasting with previous results for cotangent bundles, by explicitly constructing such examples and introducing cylindrical handles.

Contribution

It provides explicit constructions showing the Maslov class does not always vanish in Weinstein domains and introduces cylindrical handles as a generalization of Weinstein handles.

Findings

Maslov class can be non-zero in Weinstein domains

Explicit construction of non-vanishing Maslov class examples

Introduction of cylindrical handles as a generalization

Abstract

A fundamental and deep result in symplectic topology due to Abouzaid and Kragh states that the Maslov class vanishes for closed exact Lagrangians in cotangent bundles of closed manifolds. In this article we prove by an explicit construction that the Maslov class does not vanish in general for closed exact Lagrangians in Weinstein domains obtained by performing a critical handle attachment to a cotangent bundle. We also define cylindrical handles as a generalization of Weinstein handles.