
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 -action on a smooth connected closed manifold fixing a finite set can be associated to a -colored regular graph. In this paper, we consider abstract graphs of -actions, called abstract 1-skeletons. We study when an abstract 1-skeleton is a colored graph of some -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 -action).
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Taxonomy
TopicsGeometric and Algebraic Topology · Topological and Geometric Data Analysis · Advanced Graph Theory Research
