Generalized local systems and the Maslov class of exact Lagrangians

This PhD-thesis in symplectic topology consists of an introduction and two research papers. The goal of the thesis is to investigate the Maslov class for exact Lagrangians in Weinstein domains. In the introduction we give some history, introduce the research area and explain everything that is needed to understand the main results of the papers. Afterwards we give a short summary of the papers and their main results. Finally we state some open questions that the author was not able to answer during his time as a student. In Paper I we reprove with new methods the well known fact that the Maslov class vanishes for closed exact Lagrangians in cotangent bundles of closed manifolds. We also further extend that result to closed exact Lagrangians in slightly more general Weinstein domains built by attaching sub-critical handles in a certain way to cotangent bundles. The proof uses Floer theory with coefficients in path local systems which is a theory developed in the paper that builds on previous works by Abouzaid, Barraud and Cornea. In paper II we show that the main result of Paper I does not hold if we allow the handles that build the Weinstein domains to be critical. For instance we show that to any Weinstein domain one can attach a one-handle and a critical handle and then find a closed exact Lagrangian with non vanishing Maslov class. In the proof we construct the Lagrangians with explicit formulas and explicitly compute their Maslov class.