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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

Citations:

Zbl 1206.57020
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