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Relating categorical dimensions in topology and symplectic geometry

Andrew Hanlon, Jeff Hicks, Oleg Lazarev

TL;DR

The paper investigates how categorical dimensions, notably the Rouquier dimension $\mathrm{Rdim}$ and diagonal dimension $\mathrm{Ddim}$, of wrapped Fukaya categories associated to topology and Weinstein manifolds relate to intrinsic geometric data. By introducing a triangulated Lusternik–Schnirelmann category and employing symplectic tools such as Lagrangian cobordisms and sectorial covers, the authors derive new bounds tying $\mathrm{Rdim}$ and $\mathrm{Ddim}$ to Morse-theoretic data, action values of Hamiltonians, Reeb chords, and sectorial decompositions. They extend these bounds from cotangent bundles to general Weinstein domains, strengthen prior results via sectorial descent, and connect dimension theory to Lefschetz fibrations and embeddings, while also proving lower bounds arising from cuplength. The work thus provides a cohesive framework linking categorical complexity to geometric and topological invariants, with conjectures and questions aimed at refining these connections and exploring polarized/arboreal structures and coefficient-dependence. Together, these results offer new avenues for understanding how symplectic topology encodes algebraic complexity and its implications for mirror symmetry and Orlov-type conjectures.

Abstract

We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we relate the Rouquier dimension of the wrapped Fukaya category of either the cotangent bundle of a smooth manifold $M$ or more generally a Weinstein domain $X$ to quantities of geometric interest. These quantities include the minimum number of critical values of a Morse function on $M$, the Lusternik-Schnirelmann category of $M$, the number of distinct action values of a Hamiltonian diffeomorphism of $X$, and the smallest $n$ such that $X$ admits a Weinstein embedding into $\mathbb{R}^{2n+1}$. Along the way, we introduce a notion of the Lusternik-Schnirelmann category for dg-categories and construct exact Lagrangian cobordisms for restriction to a Liouville subdomain.

Relating categorical dimensions in topology and symplectic geometry

TL;DR

The paper investigates how categorical dimensions, notably the Rouquier dimension and diagonal dimension , of wrapped Fukaya categories associated to topology and Weinstein manifolds relate to intrinsic geometric data. By introducing a triangulated Lusternik–Schnirelmann category and employing symplectic tools such as Lagrangian cobordisms and sectorial covers, the authors derive new bounds tying and to Morse-theoretic data, action values of Hamiltonians, Reeb chords, and sectorial decompositions. They extend these bounds from cotangent bundles to general Weinstein domains, strengthen prior results via sectorial descent, and connect dimension theory to Lefschetz fibrations and embeddings, while also proving lower bounds arising from cuplength. The work thus provides a cohesive framework linking categorical complexity to geometric and topological invariants, with conjectures and questions aimed at refining these connections and exploring polarized/arboreal structures and coefficient-dependence. Together, these results offer new avenues for understanding how symplectic topology encodes algebraic complexity and its implications for mirror symmetry and Orlov-type conjectures.

Abstract

We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we relate the Rouquier dimension of the wrapped Fukaya category of either the cotangent bundle of a smooth manifold or more generally a Weinstein domain to quantities of geometric interest. These quantities include the minimum number of critical values of a Morse function on , the Lusternik-Schnirelmann category of , the number of distinct action values of a Hamiltonian diffeomorphism of , and the smallest such that admits a Weinstein embedding into . Along the way, we introduce a notion of the Lusternik-Schnirelmann category for dg-categories and construct exact Lagrangian cobordisms for restriction to a Liouville subdomain.
Paper Structure (34 sections, 28 theorems, 109 equations, 2 figures, 2 tables)

This paper contains 34 sections, 28 theorems, 109 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Further context and questions
  4. Outline
  5. Acknowledgements
  6. Categorical setup
  7. Context: constructing a manifold "efficiently"
  8. Open covers
  9. Handlebody Decompositions
  10. Relations
  11. Rouquier Dimension
  12. Diagonal Dimension
  13. Triangulated LS category of a category
  14. Symplectic tools: resolutions and LS-coverings in Liouville domains
  15. Background: Liouville domains and wrapped Fukaya categories
  16. ...and 19 more sections

Key Result

Theorem B

If $M$ is a smooth manifold and $f$ is any Morse function on $M$, where $\mathop{\mathrm{cVal}}\nolimits(f)$ is the set of critical values of $f$ and $|\mathop{\mathrm{cVal}}\nolimits(f)|$ is the number of critical values, and where $\mathop{\mathrm{LS}}\nolimits(M)$ is the Lusternik-Schnirelmann category of $M$.

Figures (2)

  • Figure 1: Replacing a transverse intersection (between the red Lagrangian and the core represented by the $x$-axis) with one restricting to a cocore (blue Lagrangian) in the polydisk $D(r/2)$.
  • Figure 2:

Theorems & Definitions (80)

  • Theorem B: \ref{['cor:Morsebound']}, \ref{['prop:LSboundLS']}
  • Theorem C: \ref{['prop:ddimgeneral']}, \ref{['eq:lssect']}
  • Theorem D: \ref{['prop:Lefschetzddim']}, \ref{['prop: Lefschetz_covering_bound']}
  • Theorem E: \ref{['cor:trivialembed']}
  • Conjecture G: \ref{['conj:cuplength']}
  • Definition 2.2: Rouquier dimension and variations
  • Example 2.3
  • Remark 2.4
  • Lemma 2.6
  • proof
  • ...and 70 more