Criteria for Balance in Abelian Gain Graphs, with Applications to   Piecewise-Linear Geometry
  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. 
Konstantin Rybnikov, Thomas Zaslavsky