Graphs of 2-torus actions

TL;DR

This paper investigates the conditions under which abstract graphs, called 1-skeletons, can be realized as colored graphs arising from effective smooth actions of $(\mathbb{Z}_2)^k$ on manifolds, exploring their geometric significance.

Contribution

It characterizes when abstract 1-skeletons correspond to actual $(\mathbb{Z}_2)^k$-actions and examines the existence of faces with geometric implications.

Findings

Criteria for realizing abstract 1-skeletons as $(\mathbb{Z}_2)^k$-action graphs

Conditions for the existence of faces in abstract 1-skeletons

Connections between abstract graphs and geometric structures

Abstract

It has been known that an effective smooth $({\Bbb Z}_2)^k$-action on a smooth connected closed manifold $M^n$ fixing a finite set can be associated to a $({\Bbb Z}_2)^k$-colored regular graph. In this paper, we consider abstract graphs $(\Gamma,\alpha)$ of $({\Bbb Z}_2)^k$-actions, called abstract 1-skeletons. We study when an abstract 1-skeleton is a colored graph of some $({\Bbb Z}_2)^k$-action. We also study the existence of faces of an abstract 1-skeleton (note that faces often have certain geometric meanings if an abstract 1-skeleton is a colored graph of some $({\Bbb Z}_2)^k$-action).