Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry
Abstract
A gain graph is a triple (G,h,H), where G is a connected graph with an arbitrary, but fixed, orientation of edges, H is a group, and h is a homomorphism from the free group on the edges of G to H. A gain graph is called balanced if the h-image of each closed walk on G is the identity. Consider a gain graph with abelian gain group having no odd torsion. If there is a basis of the graph's binary cycle space each of whose members can be lifted to a closed walk whose gain is the identity, then the gain graph is balanced, provided that the graph is finite or the group has no nontrivial infinitely 2-divisible elements. We apply this theorem to deduce a result on the projective geometry of piecewise-linear realizations of cell-decompositions of manifolds.
Keywords
Cite
@article{arxiv.math/0210052,
title = {Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry},
author = {Konstantin Rybnikov and Thomas Zaslavsky},
journal= {arXiv preprint arXiv:math/0210052},
year = {2010}
}
Comments
Changes(28 Dec. 2004): revised title and abstract; shortened, mainly by omitting inessentials; minor errors fixed. Changes (16 Jan. 2005): ADDED--Appendix with detailes on some proofs and another counterexample with picture, a few references. Minor typo and notation fixes. To appear in Discrete & Comput. Geometry (without Appendix and extra references)