On the connectedness of subcomplexes of a disk complex

TL;DR

This paper investigates the topological properties of disk complexes and reducing sphere complexes in certain 3-manifolds, establishing simple connectivity under specific conditions and analyzing intersections of reducing spheres.

Contribution

It proves the simple connectivity of the disk complex for boundary-reducible 3-manifolds with a genus g+1 Heegaard surface and examines the intersection properties of reducing spheres.

Findings

Disk complex of a genus g+1 Heegaard surface is simply connected.

Connectedness properties of the complex of reducing spheres are analyzed.

Intersections of two reducing spheres are characterized for a genus three splitting of (torus) × I.

Abstract

For a boundary-reducible $3$-manifold $M$ with $\partial M$ a genus $g$ surface, we show that if $M$ admits a genus $g+1$ Heegaard surface $S$, then the disk complex of $S$ is simply connected. Also we consider the connectedness of the complex of reducing spheres. We investigate the intersection of two reducing spheres for a genus three Heegaard splitting of $\mathrm{(torus)} \times I$.