Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Homotopical Invariant of Weinstein Surfaces

Shanon J. Rubin

TL;DR

The paper develops a robust homotopical framework for dg-categories to construct a global invariant of Weinstein surfaces from arboreal skeleta. It defines $\mathcal{L}(W)=\operatorname{holim} D$ using explicit local models and shows invariance under a complete set of arboreal moves that realize Weinstein homotopies, achieving a combinatorial handle on surface invariants. A general holim formula for diagrams of dg-categories is provided, with concrete representations via path objects and explicit lemmas, enabling stepwise computations and reductions to graph data. The results connect microlocal sheaf theory with a topological Fukaya-category-like invariant, offering concrete computations for all topological surfaces and setting the stage for higher-dimensional generalizations and links to broader categorical frameworks.

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.

A Homotopical Invariant of Weinstein Surfaces

TL;DR

The paper develops a robust homotopical framework for dg-categories to construct a global invariant of Weinstein surfaces from arboreal skeleta. It defines using explicit local models and shows invariance under a complete set of arboreal moves that realize Weinstein homotopies, achieving a combinatorial handle on surface invariants. A general holim formula for diagrams of dg-categories is provided, with concrete representations via path objects and explicit lemmas, enabling stepwise computations and reductions to graph data. The results connect microlocal sheaf theory with a topological Fukaya-category-like invariant, offering concrete computations for all topological surfaces and setting the stage for higher-dimensional generalizations and links to broader categorical frameworks.

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.

Paper Structure

This paper contains 22 sections, 54 theorems, 6 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Symplectic results
  4. Homotopical results
  5. Fibrancy recognition
  6. Categorical preliminaries
  7. Reedy structures on graphic quivers
  8. Checking \ref{['p1']} for homotopy limits
  9. Finding adequate Reedy structures
  10. Homotopy limits via path objects
  11. Homotopy limits in steps
  12. Homotopy limits of dg-categories
  13. Differential graded categories
  14. Explicit formulas
  15. Local moves
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix a strongly pre triangulated dg-category $\mathcal{A}$ with 2-periodic hom complexes. Given a Weinstein surface $W$, let $\mathfrak X^{\mathrm{arb}}$ be the skeleton of any arborealization of $W$. Let $D=D_{\mathcal{A}}(\mathfrak X^{\mathrm{arb}})$ be the diagram constructed from $\mathfrak X^{\m

Figures (25)

  • Figure 1: This schematic indicates with a mark those index 1 bones which are assigned a shift in our $\mathbb Z/2$-graded diagram. The markings are determined by how the joints interact with a fixed choice of (ultimately auxiliary) orientations on the index 0 bones. See \ref{['invariantdef']} for more details.
  • Figure 2: We use the convention in \ref{['graphconvention']}. The graph on the left is the arboreal graph associated to an arboreal surface. The graph in the middle is the arboreal graph associated to a non-generic arboreal surface. The graph on the right is not arboreal, but by \ref{['caveat']} it will still encode a valid diagram when we come to \ref{['arborealdiagramchapter1']}.
  • Figure 3: The four arboreal moves on arboreal graphs, representing isotopy around a boundary, handle cancellation, isotopy of two 1-handles, and handle slide. The arboreal graph data is implied by \ref{['graphconvention']}.
  • Figure 4: Isotopies realize all ways to arrange 1-handles attached to a fixed 0-bone. The first row is $M_{s_{i}}$, the second is three possible choices for $W^{(i)}$, and the third shows how the associated arboreal graphs $G(W^{(i)})$ are arboreal equivalent via Moves 0 and 2.
  • Figure 5: The two types of handle slides, each shown first on Morse skeleta, then arboreal skeleta, and finally on the associated arboreal graphs via Move 3.
  • ...and 20 more figures

Theorems & Definitions (120)

  • Theorem 1.1: technical statement in \ref{['final']}
  • Lemma 2.1: technical statement in \ref{['geo']}
  • Theorem 2.2
  • Theorem 3.1: technical statement in \ref{['Reedy']}
  • Proposition 3.2: \ref{['pathresult']}
  • Lemma 1.1.1
  • proof
  • Lemma 1.1.2
  • proof
  • Definition 1.2.1
  • ...and 110 more