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Torus decomposition and foliation detected slopes

Qingfeng Lyu

TL;DR

The paper proves that a closed 3-manifold $M=M_1\cup_f M_2$ admits a co-oriented taut foliation if and only if the gluing map $f$ identifies a rational CTF-detected slope on each side of the gluing torus. The authors treat rational-homology-sphere cases via torus decomposition and positive first Betti number cases via Gabai’s sutured-manifold techniques, while irrational slopes are analyzed with Li’s laminar-branched-surface theory to show such slopes lie in the interior of the detected set and have nearby rational slopes that are strongly CTF-detected. They further extend the framework to multislope settings, proving a multislope version of gluing coherence and providing partial converses to existing multislope gluing theorems. The work unifies JSJ-analysis, sutured manifold theory, and laminar branched-surface methods to characterize when foliations can be glued across tori, with implications for boundary slope detection and related conjectures in the foliation/L-space landscape.

Abstract

Let $M_1$ and $M_2$ be knot manifolds and $M=M_1\cup_f M_2$ be the closed 3-manifold obtained by gluing up $M_1$ and $M_2$ via $f:\partial M_1\xrightarrow{\cong} \partial M_2$. We show that if $M$ admits a co-oriented taut foliation, then $f$ identifies some CTF-detected rational boundary slopes of $M_1$ and $M_2$, affirming a conjecture proposed by Boyer, Gordon and Hu.

Torus decomposition and foliation detected slopes

TL;DR

The paper proves that a closed 3-manifold admits a co-oriented taut foliation if and only if the gluing map identifies a rational CTF-detected slope on each side of the gluing torus. The authors treat rational-homology-sphere cases via torus decomposition and positive first Betti number cases via Gabai’s sutured-manifold techniques, while irrational slopes are analyzed with Li’s laminar-branched-surface theory to show such slopes lie in the interior of the detected set and have nearby rational slopes that are strongly CTF-detected. They further extend the framework to multislope settings, proving a multislope version of gluing coherence and providing partial converses to existing multislope gluing theorems. The work unifies JSJ-analysis, sutured manifold theory, and laminar branched-surface methods to characterize when foliations can be glued across tori, with implications for boundary slope detection and related conjectures in the foliation/L-space landscape.

Abstract

Let and be knot manifolds and be the closed 3-manifold obtained by gluing up and via . We show that if admits a co-oriented taut foliation, then identifies some CTF-detected rational boundary slopes of and , affirming a conjecture proposed by Boyer, Gordon and Hu.
Paper Structure (10 sections, 18 theorems, 4 figures)

This paper contains 10 sections, 18 theorems, 4 figures.

Table of Contents

  1. introduction
  2. Torus decomposition and rational homology spheres
  3. Observations and preparing lemmas
  4. Main proof
  5. Manifolds with positive first betti number
  6. Laminar branched surface and irrational detected slopes
  7. Laminar branched surface
  8. Knot manifolds
  9. General case
  10. Multislope detections

Key Result

Theorem 1.2

Suppose that $M_1$ and $M_2$ are knot manifolds and $M=M_1\cup_fM_2$, where $f:\partial M_1\rightarrow \partial M_2$ is the gluing homeomorphism. If $f$ identifies CTF-detected rational slopes $[\alpha_1]\in \mathcal{D}_{CTF}(M_1)$ and $[\alpha_2]\in \mathcal{D}_{CTF}(M_2)$, then $M$ admits a co-ori

Figures (4)

  • Figure 1: Persistent foliar decoration
  • Figure 2: Local picture around the graph manifold piece $G$
  • Figure 3: Branched Surfaces
  • Figure 4: Sink disk and half sink disk

Theorems & Definitions (38)

  • Definition 1.1: boyer2021slope, Definition 5.1
  • Theorem 1.2: boyer2021slope, Theorem 5.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 28 more