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

Authors:
Konstantin Rybnikov;Thomas Zaslavsky
Affiliations:
Department of Mathematical Sciences, University of Massachusetts at Lowell, Lowell, MA 01854, USA;Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA
Venue:
Discrete & Computational Geometry
Year:
2005

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Abstract

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.