Manifolds with weakly reducible genus-three trisections are standard

TL;DR

This work proves Meier's genus‑three trisection conjecture in the case of weakly reducible trisections by importing robust ideas from 3‑manifold topology, notably weak reducibility and generalized Heegaard splittings, and introducing five‑chains to relate genus‑three trisections to genus‑two data. The authors develop a diagrammatic framework for surgery on loops and spheres within trisections, enabling a classification of 4‑manifolds arising from weakly reducible genus‑three trisections: they must be spun lens spaces $S_p$ or $S_p'$, $S^4$, or connected sums of copies of $inom{ ext{CP}^2}{- ext{CP}^2}$, $S^1 imes S^3$, and $S^2 imes S^2$. The paper further connects five‑chain surgery to loop surgery, leverages results on Heegaard triples, and extends the analysis to dependent triples, yielding corollaries about trisection‑genus additivity and the structure of potential exotic manifolds. Overall, the work strengthens the bridge between 3‑manifold topology and 4‑manifold trisection theory and provides a concrete, diagrammatic toolkit for understanding genus‑three trisections. It also opens avenues for future exploration of five‑chain creation, non‑standard trisections of $S^4$, and related questions on manifold decompositions.

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.

Figures (19)

• Figure 1: Two examples of standard Heegaard diagrams for a genus-three Heegaard splitting of $S^1 \times S^2$
• Figure 2: Schematic diagrams for the two different untelescopings of a weakly reducible genus-three Heegaard splitting.
• Figure 3: A genus-three trisection diagram $(\Sigma;\alpha,\beta,\gamma)$ for the manifold $S_2$, the spin of $\mathbb{RP}^3$, in which curves in $\alpha$ are red, curves in $\beta$ are blue, and curves in $\gamma$ are green. Note that the teal curve belongs to both $\beta$ and $\gamma$ and is disjoint from a curve in $\alpha$, so this trisection is weakly reducible.
• Figure 4: At top left, curves $c^*$ and $c_2$ are homotopic to disjoint curves in $\Sigma_{c_1}$ (shown with scars). At top right, slides of $c^*$ over $c_1$ in $\Sigma$ (homotopies over scars in $\Sigma_{c_1}$) yield $c^{**}$ disjoint from $c_2$ in $\Sigma_{c_1}$. At bottom, further slides of $c^{**}$ over $c_1$ yield a reducing curve $c^{***}$ for $\Sigma$.
• Figure 5: At left, a non-standard genus-two diagram for $S^3$ and a wave, surgery on which yields the standard genus-two diagram at right.

Theorems & Definitions (61)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Remark 2.1
• Lemma 2.2
• Remark 2.3
• Proposition 2.4