Criteria for balance in abelian gain graphs, with applications to piecewise-linear geometry

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 non-trivial 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.