#### Volume 21, issue 3 (2017)

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Genus-two trisections are standard

### Jeffrey Meier and Alexander Zupan

Geometry & Topology 21 (2017) 1583–1630
##### Abstract

We show that the only closed $4$–manifolds admitting genus-two trisections are ${S}^{2}×{S}^{2}$ and connected sums of ${S}^{1}×{S}^{3}$, ${ℂℙ}^{2}$ and ${\overline{ℂℙ}}^{2}$ with two summands. Moreover, each of these manifolds admits a unique genus-two trisection up to diffeomorphism. The proof relies heavily on the combinatorics of genus-two Heegaard diagrams of ${S}^{3}$. As a corollary, we classify tunnel number one links with an integral cosmetic Dehn surgery.

##### Keywords
trisections, Heegaard splittings, waves
##### Mathematical Subject Classification 2010
Primary: 57N12, 57R65
Secondary: 57M25