Genus two trisections are standard

TL;DR

This paper classifies all closed 4-manifolds with genus two trisections, showing they are limited to specific known manifolds, and proves the uniqueness of such trisections, using combinatorial methods related to Heegaard diagrams.

Contribution

It establishes a complete classification of closed 4-manifolds with genus two trisections and proves their uniqueness up to diffeomorphism, a significant advancement in 4-manifold topology.

Findings

Only specific manifolds admit genus two trisections.

Each admits a unique genus two trisection up to diffeomorphism.

Classifies certain links in genus two Heegaard surfaces with cosmetic Dehn surgery.

Abstract

We show that the only closed 4-manifolds admitting genus two trisections are $S^2 \times S^2$ and connected sums of $S^1 \times S^3$, $\mathbb{CP}^2$, and $\overline{\mathbb{CP}}^2$ 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 two-component links contained in a genus two Heegaard surface for $S^3$ with a surface-sloped cosmetic Dehn surgery.