Local systems and vanishing Maslov class

TL;DR

The paper delivers a concise, generalizable proof that a closed exact Lagrangian $L\subset T^*X$ has vanishing Maslov class and is a homology equivalent copy of $X$, and extends the statement to Weinstein domains $W_X$ built by attaching sub‑critical handles to $D^*X$. It develops a robust algebraic framework using path and cosimplicial local systems, Floer theory with local coefficients, and Morita theory to relate different models and control moduli spaces via continuation maps and insulator/push‑off techniques. Key innovations include a coproduct‑style $\mu_2$ in Floer theory, a Morita equivalence between path and cosimplicial local systems, and explicit small local models that reduce global complexity while preserving essential structures. The argument culminates in a covering space analysis that forces the Maslov class to vanish and identifies $L$ with the zero section at the level of homology, yielding a solid bridge between symplectic topology and algebraic local systems with strong implications for the Fukaya category of these Weinstein domains.

Abstract

It is well known that closed exact Lagrangians in cotangent bundles of closed manifolds have vanishing Maslov class and are homotopy equivalent to the zero section. In this paper we greatly simplify the proof of vanishing Maslov class and generalize the proof to a slightly larger family of Weinstein domains.

Figures (6)

• Figure 1: Strip definition
• Figure 2: Floer co-product
• Figure 3: $h$
• Figure 4: Intersection of $K_t^-$, $L_t^+$ and $D_\delta^n$
• Figure 5: Projection of $L_t^+$

Theorems & Definitions (122)

• Theorem 1.1
• Definition 3.1
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Corollary 3.4
• proof
• Lemma 3.5
• proof