Trisections of PL 4-manifolds arising from colored triangulations

TL;DR

This paper advances trisection theory by extending gem-induced trisections to non-orientable compact 4-manifolds and clarifying how boundary structure (specifically, boundaries that are connected sums of $\mathbb{S}^2$-bundles over $\mathbb{S}^1$) yields trisections of the associated closed manifolds with the same central surface. It formalizes the construction of gem-induced trisections from colored triangulations (gems) in both orientable and non-orientable cases, providing explicit genus formulas for the constituent handlebodies and linking the central surface genus to the regular genus $\rho_{\hat{4}}(\Gamma_{\hat{4}})$. The authors then show that a gem-induced trisection of $M$ induces a trisection of the closed manifold $\bar{M}=M\cup \mathbb{Y}^4_m$ (or $\tilde{\mathbb{Y}}^4_m$), yielding upper bounds on $g_T(\bar{M})$ in terms of combinatorial data from colored triangulations and Kirby diagrams, and they establish additivity results under (boundary) connected sums. The work also demonstrates how to obtain trisections for many closed 4-manifolds via Kirby diagrams, and discusses the compatibility between gem-induced trisections and classical trisections, with explicit minimal-genus instances for standard manifolds. Overall, it provides a robust combinatorial framework to bound and realize trisection genera for a wide class of orientable and non-orientable 4-manifolds using colored triangulations and graph encodings.

Abstract

The purpose of the present paper is twofold: firstly to extend to non-orientable compact 4-manifolds the notion of gem-induced trisection, directly obtained from colored triangulations (or, equivalently, from colored graphs encoding them, called gems); secondly to prove that, both in the orientable and non-orientable case, if the boundary is homeomorphic to a connected sum of sphere bundles over $\mathbb S^1$, gem-induced trisections naturally give rise to trisections of the corresponding closed 4-manifold. As a consequence, an estimation of the trisection genus of any closed orientable 4-manifold in terms of surgery description is obtained via colored triangulations.

Figures (4)

• Figure 1: the intersections of a 4-simplex of $K(\Gamma)$ with $H_{01}$, $H_{02}$ and $H_{12}$ respectively
• Figure 2: a gem of the genus one non-orientable handlebody $\tilde{\mathbb Y}^4_1$
• Figure 3: Factorization of a $\rho_3$-pair switching into an edge insertion and a 3-dipole cancellation
• Figure 4: minimal crystallizations of $\mathbb S^3 \times \mathbb S^1$, $\mathbb S^3 \tilde{\times} \mathbb S^1$ and $\mathbb R \mathbb P^4$, with $\rho_1$-pairs in red

Theorems & Definitions (24)

• Theorem 1
• Remark 1
• Definition 1
• Theorem 2
• Proposition 3
• Definition 2
• Remark 2
• Remark 3
• Proposition 4
• Proposition 5