Generalized Heegaard Splittings and the Disk Complex

Abstract

Let $M$ be an orientable, irreducible $3$-manifold and $(\mathcal{V},\mathcal{W};F)$ a weakly reducible, unstabilized Heegaard splitting of $M$ of genus at least three. In this article, we define an equivalence relation $\sim$ on the set of the generalized Heegaard splittings obtained by weak reductions and find special subsets of the disk complex $\mathcal{D}(F)$ named by the ``$equivalent$ $clusters$'', where we can find a canonical function $\Phi$ from the set of equivalent clusters to the set of the equivalent classes for the relation $\sim$. These equivalent classes are more detailed than the isotopy classes of the generalized Heegaard splittings obtained by weak reductions from $F$. In the last section, we prove $\Phi$ is a bijection if the genus of $F$ is three.