Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

Details Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

2002-10-03

arxiv.org/abs/math/0210052v5

Discrete and Computational Geometry, 34 (2005), no. 2, 251-268.

Mathematics

Combinatorics

Changes(28 Dec. 2004): revised title and abstract; shortened, mainly by omitting inessentials; minor errors fixed. Changes (16

Scientific paper

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.

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