Simple Connectivity of Spheres in the Curve Complex

TL;DR

This paper proves that spheres in the curve complex of high-complexity surfaces are nearly simply connected, with loops bounding disks close to the sphere, revealing topological properties of these geometric structures.

Contribution

It introduces the concept of spheres in the curve complex and demonstrates their near simple connectivity for high-complexity surfaces.

Findings

Spheres in the curve complex are almost simply connected.

Loops in these spheres bound disks within a small neighborhood.

The results apply to surfaces of sufficiently high complexity.

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.