A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical

Abstract

Let $\left(\mathcal{V},\mathcal{W};F\right)$ be a weakly reducible unstabilized genus three Heegaard splitting in an orientable, irreducible $3$–manifold $M$. In this article, we prove that either the disk complex $\mathcal{D}\left(F\right)$ is contractible or $F$ is critical. Hence, the topological index of $F$ is $2$ if $F$ is topologically minimal.