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Simple Connectivity of Spheres in the Curve Complex

Richard Cao, Rishibh Prakash

TL;DR

This work studies spheres in the curve complex CΣ of a high-complexity surface and proves that, for ξ(Σ)≥10, every loop in a sphere S_r bounds an embedded disk contained in a thin annulus B_{r+M}\setminus B_{r-3}, i.e., the spheres are almost simply connected. The authors integrate Teichmüller-theoretic tools with combinatorial maneuvering in CΣ, notably the triangle, diamond, and cone moves, to push triangulations outward while maintaining good relations to subsurfaces and applying the Bounded Geodesic Image Theorem. The result builds on Wright’s connectivity work and boundary theory (Klarreich, Gabai) and advances toward higher connectivity and potential linear simple connectivity of the Gromov boundary ∂CΣ for large ξ(Σ). The approach provides a robust inductive framework that could extend to higher-dimensional connectivity and offers new insight into how the curve complex's large-scale geometry governs the topology of its spheres and boundary.

Abstract

For a fixed radius $r$ and a point $o$ in the curve complex of a surface, we define the sphere of radius $r$ to be the induced subgraph on the set of vertices of distance $r$ from $o$. We show that these spheres are almost simply connected for surfaces of high enough complexity, in the sense that loops in the sphere bound an embedded disk contained in a small neighborhood of the sphere.

Simple Connectivity of Spheres in the Curve Complex

TL;DR

This work studies spheres in the curve complex CΣ of a high-complexity surface and proves that, for ξ(Σ)≥10, every loop in a sphere S_r bounds an embedded disk contained in a thin annulus B_{r+M}\setminus B_{r-3}, i.e., the spheres are almost simply connected. The authors integrate Teichmüller-theoretic tools with combinatorial maneuvering in CΣ, notably the triangle, diamond, and cone moves, to push triangulations outward while maintaining good relations to subsurfaces and applying the Bounded Geodesic Image Theorem. The result builds on Wright’s connectivity work and boundary theory (Klarreich, Gabai) and advances toward higher connectivity and potential linear simple connectivity of the Gromov boundary ∂CΣ for large ξ(Σ). The approach provides a robust inductive framework that could extend to higher-dimensional connectivity and offers new insight into how the curve complex's large-scale geometry governs the topology of its spheres and boundary.

Abstract

For a fixed radius and a point in the curve complex of a surface, we define the sphere of radius to be the induced subgraph on the set of vertices of distance from . We show that these spheres are almost simply connected for surfaces of high enough complexity, in the sense that loops in the sphere bound an embedded disk contained in a small neighborhood of the sphere.

Paper Structure

This paper contains 11 sections, 38 theorems, 23 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Context and Previous Results
  3. Main Results
  4. Acknowledgments
  5. Sufficient Conditions for Connectivity
  6. Key Lemmas
  7. Triangle
  8. Diamond
  9. Cone
  10. Almost simply connectedness
  11. Base Case

Key Result

Theorem 1.1

Let $\xi(\Sigma)\geq 10$. There is some $M$ depending only on $\Sigma$ such that for any embedded loop $\gamma \subset S_r$ there exists an embedded disk $\Delta \subset B_{r+M}{\smallsetminus} B_{r-3}$ with boundary $\gamma$.

Figures (12)

  • Figure 1: a) An eventually non-separating multicurve, b) a pants curve, c) an essentially non-separating multicurve
  • Figure 2: Pushing simplices away from the origin
  • Figure 3: Picking a non-separating or pants curve on $S{\smallsetminus} (x\cup y\cup z)$
  • Figure 4: replacing bad triangles
  • Figure 5: Plan for diamond case
  • ...and 7 more figures

Theorems & Definitions (85)

  • Theorem 1.1
  • Remark 1.2
  • Conjecture 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • ...and 75 more