## A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical.(English)Zbl 1362.57027

For a surface of genus $$\geq 2$$ in a compact orientable 3-manifold, the disc complex $$\mathcal{D}(F)$$ is the complex such that (i) vertices are isotopy classes of compressing disks of $$F$$, (ii) $$m+1$$ vertices span an $$m$$-simplex if they have pairwise disjoint representatives. D. Bachman [Geom. Topol. 14, No. 1, 585–609 (2010; Zbl 1206.57020)] defined a separating surface $$F$$ with no torus components to be topologically minimal if $$\mathcal{D}(F)$$ is either empty or non-contractible. In this case the topological index of $$F$$ is the smallest $$n$$ such that $$\pi_{n-1}(\mathcal{D}(F))$$ is nontrivial; it is $$0$$ if $$\mathcal{D}(F)=\emptyset$$. If $$F$$ is topologically minimal with topological index $$2$$, then $$F$$ is called a critical surface.
Recent results by several authors suggest that it is more common for a weakly reducible Heegaard surface to be topologically minimal than not. The main result of the paper is that for a weakly reducible unstabilized genus $$3$$ Heegaard splitting of an orientable irreducible $$3$$-manifold, either the disk complex $$\mathcal{D}(F)$$ of the Heegaard surface $$F$$ is contractible or $$F$$ is critical. In particular, if $$F$$ is topologically minimal, then its topological index is $$2$$.

### MSC:

 57M50 General geometric structures on low-dimensional manifolds

### Keywords:

3-manifolds; weakly reducible; Heegaard splittings

Zbl 1206.57020
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