Maslov class of exact Lagrangians and cylindrical handles

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.