A sufficient condition is stated for non-separating weak reducing pairs to be separated by reducing spheres on the genus-4 Heegaard surface in S^3, thereby reducing the connectivity problem in the reducing sphere complex to spheres disjoint from a fixed non-separating compressing disk.
Bachman, Topological index theory for surfaces in 3-manifold s, Geom
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A note on reducing spheres for the genus-4 Heegaard surface in the 3-sphere
A sufficient condition is stated for non-separating weak reducing pairs to be separated by reducing spheres on the genus-4 Heegaard surface in S^3, thereby reducing the connectivity problem in the reducing sphere complex to spheres disjoint from a fixed non-separating compressing disk.