Genus-two trisections are standard

Abstract

We show that the only closed $4$–manifolds admitting genus-two trisections are $S^2\times S^2$ and connected sums of $S^1\times S^3$, ${ℂ\mathbb{P}}^2$ and ${\overline{ℂ\mathbb{P}}}^2$ 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.