Equivariant trisections for group actions on four-manifolds

TL;DR

This paper introduces the concept of $G$-equivariant trisections for group actions on 4-manifolds, demonstrating their existence, properties, and applications in understanding quotient spaces and specific examples.

Contribution

It defines $G$-equivariant trisections and bridge positions, showing they can be constructed for any $G$-manifold and reducing complex topology to simpler 2D diagrams.

Findings

Existence of $G$-equivariant trisections for all such manifolds.

Equivariant trisections are determined by their spines, simplifying classification.

Examples include actions on $S^4$, $S^2\times S^2$, and $\mathbb{CP}^2$, with classifications for low genus.

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.