Graphs of 2-torus actions

Abstract

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

12 pages with 3 figures. To appear in Contemporary Mathematics of AMS (Proceedings of the International Conference on Toric Topology)