  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.
