Graphs of 2-torus actions

Details Graphs of 2-torus actions Graphs of 2-torus actions

2007-11-23

arxiv.org/abs/0711.3778v2

Contemp. Math. 460 (2008), 261-272.

Mathematics

Combinatorics

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

Scientific paper

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).

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