On unstabilzed genus three critical Heegaard surfaces

Abstract

Let $M$ be a compact orientable irreducible 3-manifold and $H$ be an unstabilized genus three Heegaard splitting of $M$. In this article, we will define a simplicial complex of weak reducing pairs for $H$ and find several properties of this complex. Using this method, we will prove that an unstabilized Heegaard splitting of genus three is critical in a certain condition. In addition, we will show that the standard genus three Heegaard splitting for $T^3$ and the induced Heegaard splitting of the three component chain exterior by a certain tunnel system are critical as examples of the main theorem.