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The wrapped Fukaya category of plumbings

Dogancan Karabas, Sangjin Lee

Abstract

Plumbing spaces have drawn significant attention among symplectic topologists due to their natural occurrence as examples of Weinstein manifolds. In our paper, we provide a general formula for the wrapped Fukaya category of plumbings (with arbitrary grading structure) of cotangent bundles along any quiver. Our approach relies on "local-to-global" computations. Specifically, we compute the wrapped Fukaya category of "plumbing sectors" that serve as local models for the singularities of Lagrangian skeletons of plumbing spaces. As corollaries, we fully describe the wrapped Fukaya category of plumbing spaces in dimension $4$ and plumbings of $T^*S^n$ for $n \geq 3$. We show that any Ginzburg dg algebra/category of a graded quiver without potential is equivalent to the wrapped Fukaya category of a plumbing of $T^*S^n$ (with the corresponding grading structure).

The wrapped Fukaya category of plumbings

Abstract

Plumbing spaces have drawn significant attention among symplectic topologists due to their natural occurrence as examples of Weinstein manifolds. In our paper, we provide a general formula for the wrapped Fukaya category of plumbings (with arbitrary grading structure) of cotangent bundles along any quiver. Our approach relies on "local-to-global" computations. Specifically, we compute the wrapped Fukaya category of "plumbing sectors" that serve as local models for the singularities of Lagrangian skeletons of plumbing spaces. As corollaries, we fully describe the wrapped Fukaya category of plumbing spaces in dimension and plumbings of for . We show that any Ginzburg dg algebra/category of a graded quiver without potential is equivalent to the wrapped Fukaya category of a plumbing of (with the corresponding grading structure).
Paper Structure (30 sections, 35 theorems, 187 equations, 6 figures)

This paper contains 30 sections, 35 theorems, 187 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminaries
  4. Dg categories
  5. Weak equivalence
  6. Semifree dg category and homotopy colimit
  7. Symplectic topology
  8. Plumbing spaces
  9. Weinstein pair and sector
  10. Gluing of Weinstein pairs/sectors
  11. Plumbing data and plumbing sector
  12. Construction of plumbing spaces
  13. Collection of Weinstein sectors
  14. Gluing information
  15. Equivalent plumbing spaces
  16. ...and 15 more sections

Key Result

Theorem 1.1

Fix $n\geq 2$. Let $P$ be a plumbing of cotangent bundles of connected, oriented $n$-manifolds (with or without boundary) along any quiver with or without negative intersections. In other words, there is an arbitrary plumbing data $(Q,M,\mathrm{sgn})$ such that $P = P(Q,M,\mathrm{sgn})$. Then, the w where $\mathcal{P}_n$ is a semifree dg category given as follows: where $\eta_v(z)\in C_{-*}(\Omeg

Figures (6)

  • Figure 1: The skeleton of $\Pi_n$
  • Figure 4: The left is the Lagrangian skeleton of $\Pi_n$, i.e., two disks transversely intersecting at one point. The red, green, blue geodesics correspond to $z, x, y$ respectively. The right figure is one of two disks of the Lagrangian skeleton, and the geodesics on the disk are drawn in the same colors. One can see that there exists a triangle bounded by red, green, blue geodesics on the right figure.
  • Figure 6: Perturbation $\Lambda\subset S^*D^n$ of $S^*_0D^n$ by the reverse Reeb flow for the case $n=2$
  • Figure 7: A decomposition of $\Sigma_{g,m}$ as a union of two subsets $X$ and $Y$.
  • Figure 8: The upper picture is the decomposition of $X=\Sigma_{g,1}$ into four subsets $X_1, \dots, X_4$. The lower picture is describing $j^{\text{th}}$ connected component of $X_1$ and $X_2$.
  • ...and 1 more figures

Theorems & Definitions (96)

  • Theorem 1.1: Theorem \ref{['thm:wfuk-plumbing-general']}
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: hocolim
  • Proposition 2.5: hocolim
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 86 more