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The primitive curve complex for a handlebody

Sangbum Cho, Jung Hoon Lee

TL;DR

The paper studies the primitive curve complex $\mathcal{PC}(V)$ for a genus $g$ handlebody $V$, a subcomplex of the curve complex $\mathcal{C}(\Sigma)$ consisting of primitive curves on the boundary $\Sigma=\partial V$. It proves that $\mathcal{PC}(V)$ is connected for every $g\ge 2$ by constructing a sequence of primitive curves between any two given primitive curves, with consecutive curves satisfying a separation condition via dual disks. The authors first establish a weaker result (Theorem 2.4) where two primitive curves with a common dual disk can be connected through a chain of curves with disjoint dual disks, using arc-surgery techniques. They then treat the cases $g=2$ and $g\ge 3$ separately, employing a detailed analysis of how curves intersect a fixed dual disk and, for $g\ge 3$, a dual-pair framework that yields $p$-connectedness of dual pairs. These results illuminate the combinatorial structure of primitive curves and have potential implications for understanding the handlebody group and related 3-manifold decompositions.

Abstract

A simple closed curve in the boundary surface of a handlebody is called primitive if there exists an essential disk in the handlebody whose boundary circle intersects the curve transversely in a single point. The primitive curve complex is then defined to be the full subcomplex of the curve complex for the boundary surface, spanned by the vertices of primitive curves. Given any two primitive curves, we construct a sequence of primitive curves from one to the other one satisfying a certain property. As a consequence, we prove that the primitive curve complex for the handlebody is connected.

The primitive curve complex for a handlebody

TL;DR

The paper studies the primitive curve complex for a genus handlebody , a subcomplex of the curve complex consisting of primitive curves on the boundary . It proves that is connected for every by constructing a sequence of primitive curves between any two given primitive curves, with consecutive curves satisfying a separation condition via dual disks. The authors first establish a weaker result (Theorem 2.4) where two primitive curves with a common dual disk can be connected through a chain of curves with disjoint dual disks, using arc-surgery techniques. They then treat the cases and separately, employing a detailed analysis of how curves intersect a fixed dual disk and, for , a dual-pair framework that yields -connectedness of dual pairs. These results illuminate the combinatorial structure of primitive curves and have potential implications for understanding the handlebody group and related 3-manifold decompositions.

Abstract

A simple closed curve in the boundary surface of a handlebody is called primitive if there exists an essential disk in the handlebody whose boundary circle intersects the curve transversely in a single point. The primitive curve complex is then defined to be the full subcomplex of the curve complex for the boundary surface, spanned by the vertices of primitive curves. Given any two primitive curves, we construct a sequence of primitive curves from one to the other one satisfying a certain property. As a consequence, we prove that the primitive curve complex for the handlebody is connected.
Paper Structure (4 sections, 11 theorems, 22 figures)

This paper contains 4 sections, 11 theorems, 22 figures.

Table of Contents

  1. Introduction
  2. Primitive curves having a common dual disk
  3. Proof of the main theorem in the case of $g = 2$
  4. Proof of the main theorem in the case of $g \ge 3$

Key Result

Theorem 1.1

Let $C$ and $C'$ be primitive curves in $\Sigma$, the boundary of a genus-$g$ handlebody $V$ for $g \ge 2$. Then there exists a sequence $C = C_1, C_2, \ldots, C_n = C'$ of primitive curves in $\Sigma$ such that $C_i$ and $C_{i+1}$ are separated for each $i \in \{ 1, 2, \ldots, n-1 \}$.

Figures (22)

  • Figure 1: (a) $\Delta_{+} \cap \Delta_{-} = p$ and (b) $\Delta_{+} \subset \Delta_{-}$.
  • Figure 2: The arc $\delta_{+}$ and $\delta_{-}$.
  • Figure 3: The arc $\epsilon$.
  • Figure 4: The arc $\delta_{+}$ and $\delta_{-}$.
  • Figure 5: The arc $\delta = \delta_{+} \cup \delta_{-}$.
  • ...and 17 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1: Intersecting in a single point
  • proof
  • Lemma 2.2: A stronger version of Lemma \ref{['lem2.1']}
  • proof
  • Lemma 2.3: Arc surgery
  • proof
  • Theorem 2.4
  • proof
  • ...and 9 more