Manifolds with weakly reducible genus-three trisections are standard

Abstract

Heegaard splittings stratify 3-manifolds by complexity; only $S^3$ admits a genus-zero splitting, and only $S^3$, $S^1 \times S^2$, and lens spaces $L(p,q)$ admit genus-one splittings. In dimension four, the second author and Jeffrey Meier proved that only a handful of simply-connected 4-manifolds have trisection genus two or less, while Meier conjectured that if $X$ admits a genus-three trisection, then $X$ is diffeomorphic to a spun lens space $S_p$ or its sibling $S_p'$, $S^4$, or a connected sum of copies of $\pm \mathbb{CP}^2$, $S^1 \times S^3$, and $S^2 \times S^2$. We prove Meier's conjecture in the case that $X$ admits a weakly reducible genus-three trisection, where weak reducibility is a new idea adapted from Heegaard theory and is defined in terms of disjoint curves bounding compressing disks in various handlebodies. The tools and techniques used to prove the main theorem borrow heavily from 3-manifold topology. Of independent interest, we give a trisection-diagrammatic description of 4-manifolds obtained by surgery on loops and spheres in other 4-manifolds.