Equivariant trisections for group actions on four-manifolds

TL;DR

This work introduces $G$-equivariant trisections for smooth 4-manifolds with finite group actions and establishes that such trisections, together with $G$-equivariant bridge positions for invariant surfaces, are determined by their spines via shadow diagrams. The authors prove existence (via equivariant triangulations) and show quotient results that encode $X/G$ and the induced diagrams diagrammatically; they develop a rich suite of examples including branched coverings, hyperelliptic involutions, and linear actions from subgroups of $PU(3)$ acting on $P^2$. They classify low-genus equivariant trisections (genus 0–2), showing genus-zero and many genus-one cases are geometric, with explicit models for $D_4$ and $D_6$ actions at genus two. The framework connects 4-manifold equivariant topology to 2-dimensional diagrammatic data, enabling a tractable approach to understanding quotients, branched covers, and symmetry phenomena in familiar manifolds such as $S^4$, $S^2 imes S^2$, and $P^2$, and opens avenues for orbifold trisections and broader group-action classifications.

Abstract

Let $G$ be a finite group, and let $X$ be a smooth, orientable, connected, closed 4-dimensional $G$-manifold. Let $\mathcal{S}$ be a smooth, embedded, $G$-invariant surface in $X$. We introduce the concept of a $G$-equivariant trisection of $X$ and the notion of $G$-equivariant bridge trisected position for $\mathcal{S}$ and establish that any such $X$ admits a $G$-equivariant trisection such that $\mathcal{S}$ is in equivariant bridge trisected position. Our definitions are designed so that $G$-equivariant (bridge) trisections are determined by their spines; hence, the 4-dimensional equivariant topology of a $G$-manifold pair $(X,\mathcal{S})$ can be reduced to the 2-dimensional data of a $G$-equivariant shadow diagram. As an application, we discuss how equivariant trisections can be used to study quotients of $G$-manifolds. We also describe many examples of equivariant trisections, paying special attention to branched covering actions, hyperelliptic involutions, and linear actions on familiar manifolds such as $S^4$, $S^2\times S^2$, and $\mathbb{CP}^2$. We show that equivariant trisections of genus at most one are geometric, and we give a partial classification for genus-two.

Figures (13)

• Figure 1: Decomposition of tetrahedron into star neighborhoods using the double barycentric subdivision. These figures are derived from the wonderful Wolfram Demonstration by Aleksandr Berdnikov Ber_24_Star-Neighborhoods-in-Double.
• Figure 2: A schematic illustrating how the $X_i$ are decomposed based on the fundamental pieces $\mathcal{V}$, $\mathcal{E}$, $\mathcal{F}$, and $\mathcal{T}$ at each depth level. The purple segments indicate the location of the linearly parting ball-systems $\mathcal{B}_i$.
• Figure 3: A schematic illustrating how the $H_i$ (represented as black segments) arise as the intersections of the sectors $X_i$ and $X_{i+1}$, which are decomposed based on the fundamental pieces $\mathcal{V}$, $\mathcal{E}$, $\mathcal{F}$, and $\mathcal{T}$ at each depth level.
• Figure 4: Diagrams for the link $\mathcal{L}$ and some of its branched covers
• Figure 5: The branched coverings of the link $\mathcal{L}\subset S^4$ of projective planes corresponding to the representations $\rho\colon\pi_1(S^4\setminus\nu(\mathcal{L}))\to Q_8$

Theorems & Definitions (104)

• Remark 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Definition 3.7
• Definition 3.8
• Lemma 3.9
• proof