Generalization of the Menger's Theorem to Simplicial Complexes and Certain Invariants of the Underlying Topological Spaces
arXiv:1208.5439
Abstract
We extend the edge version of the classical Menger's Theorem for undirected graphs to -dimensional simplicial complexes with chains over the field . The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by pairwise edge-disjoint paths if, and only if, after a deletion of any edges from the graph, there will still will exist a path connecting these two vertices. We introduce the notion of -boundance of -dimensional cycles in an -dimensional simplicial complex over , which is a generalization of the classical notion of -edge-connectivity in an undirected graph. For the case , -boundance of -dimensional cycles in an undirected graph is just an extension of the classical notion of -edge-connectivity of pairs of vertices, stated in the language of cycles and boundaries. Using the notion of -boundance, we prove that a non-trivial -dimensional cycle in an -dimensional simplicial complex over is a boundary of pairwise disjoint -dimensional chains if, and only if, after a deletion of any -dimensional simplices from that complex, there still remains some -dimensional chain in it, for which this -dimensional cycle is a boundary. In our last section we restate both the original Menger's Theorem and our generalization to -boundance in dimensions, in terms of the underlying topological space. Thus, -edge-connectivity of a pair of points in an undirected graph is really a topological property of the corresponding pair of points in the topological space, underlying that graph. Similarly, -boundance of an -dimensional cycle is a topological property of the topological subspace, underlying that -dimensional cycle, in the topological space, underlying the -dimensional simplicial complex.
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