On the connectedness of subcomplexes of a disk complex

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 $\mathrm{(torus)} \times I$.

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