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Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry | | Konstantin Rybnikov
; Thomas Zaslavsky
; | | Date: |
3 Oct 2002 | | Subject: | Combinatorics; Algebraic Topology; Computational Geometry; Data Structures and Algorithms; Discrete Mathematics MSC-class: Primary: 05C22, 52C25; Secondary: 05C38, 52C25, 52C22 | math.CO cs.CG cs.DM cs.DS math.AT | | Abstract: | 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. | | Source: | arXiv, math.CO/0210052 | | Services: | Forum | Review | PDF | Favorites | |
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