A Homotopical Invariant of Weinstein Surfaces

TL;DR

This paper constructs a new homotopical invariant for Weinstein surfaces using microlocal sheaf categories, providing explicit models, invariance proofs, and combinatorial presentations for all topological surfaces.

Contribution

It introduces a novel homotopical invariant for Weinstein surfaces based on microlocal sheaf categories, with explicit models and invariance under Weinstein homotopies.

Findings

Constructed explicit models for homotopy limits of microlocal sheaf categories.

Proved invariance of the invariant under Weinstein homotopies.

Provided combinatorial presentations for all topological surfaces.

Abstract

One generally expects that the techniques of arboreal singularities and gluing of local differential graded categories will result in a useful global invariant for all Weinstein manifolds. In this paper we construct explicit models for the homotopy limits of diagrams of microlocal sheaf categories which arise from Weinstein surfaces with arboreal skeleta. This is done by characterizing all relevant Reedy model structures on the categories of diagrams that we care about. We prove invariance using a complete set of moves for Weinstein homotopies in this setting. Finally we give combinatorial presentations of the invariant for all topological surfaces.