#### Volume 16, issue 3 (2016)

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

### Jungsoo Kim

Algebraic & Geometric Topology 16 (2016) 1427–1451
##### Abstract

Let $\left(\mathsc{V},\mathsc{W};F\right)$ be a weakly reducible unstabilized genus three Heegaard splitting in an orientable, irreducible $3$–manifold $M\phantom{\rule{0.3em}{0ex}}$. In this article, we prove that either the disk complex $\mathsc{D}\left(F\right)$ is contractible or $F$ is critical. Hence, the topological index of $F$ is $2$ if $F$ is topologically minimal.

##### Keywords
$3$–manifold, Heegaard splitting, disk complex, topologically minimal surface
Primary: 57M50