Quantization of conic Lagrangian submanifolds of cotangent bundles

TL;DR

This paper establishes a canonical sheaf associated with a conic Lagrangian submanifold in cotangent bundles, leading to new insights into the Maslov class and homotopy group isomorphisms.

Contribution

It introduces a canonical sheaf construction for conic Lagrangian submanifolds, providing new proofs of properties like Maslov class vanishing and homotopy equivalences.

Findings

Maslov class of Λ is zero

Projection induces isomorphisms between homotopy groups

Existence of a canonical sheaf with specified microsupport

Abstract

Let $M$ be a manifold and $\Lambda$ a compact exact connected Lagrangian submanifold of $T^*M$. We can associate with $\Lambda$ a conic Lagrangian submanifold $\Lambda'$ of $T^*(M\times R)$. We prove that there exists a canonical sheaf $F$ on $M\times R$ whose microsupport is $\Lambda'$ outside the zero section. We deduce the already known results that the Maslov class of $\Lambda$ is $0$ and that the projection from $\Lambda$ to $M$ induces isomorphisms between the homotopy groups.