Generalized Heegaard splittings and the disk complex (Q2176720)

Let \((\mathcal{V}, \mathcal{W}; F)\) be a weakly reducible, unstabilized Heegaard splitting of genus at least three in an orientable, irreducible 3-manifold \(M\). The author defines an equivalence relation on the set of generalized Heegaard splittings obtained by weak reductions (this notion is introduced in [\textit{D. Bachman}, Geom. Topol. 12, No. 4, 2327--2378 (2008; Zbl 1152.57020)]), and later shows that there are special subsets of the disk complex \(D(F)\) (named ``equivalent clusters'') and there is a canonical function from the set of the equivalent clusters to the set of equivalence classes for the relation. When the surface \(F\) has genus three, the author proves that the canonical function is bijective.