Generalization of the Menger's Theorem to Simplicial Complexes and Certain Invariants of the Underlying Topological Spaces

Abstract

We extend the edge version of the classical Menger's Theorem for undirected graphs to $n$-dimensional simplicial complexes with chains over the field $\mathbb{F}_2$. The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by $k$ pairwise edge-disjoint paths if, and only if, after a deletion of any $k-1$ edges from the graph, there will still will exist a path connecting these two vertices. We introduce the notion of $k$-boundance of $(n-1)$-dimensional cycles in an $n$-dimensional simplicial complex over $\mathbb{F}_2$, which is a generalization of the classical notion of $k$-edge-connectivity in an undirected graph. For the case $n=1$, $k$-boundance of $0$-dimensional cycles in an undirected graph is just an extension of the classical notion of $k$-edge-connectivity of pairs of vertices, stated in the language of cycles and boundaries. Using the notion of $k$-boundance, we prove that a non-trivial $(n-1)$-dimensional cycle in an $n$-dimensional simplicial complex over $\mathbb{F}_2$ is a boundary of $k$ pairwise disjoint $n$-dimensional chains if, and only if, after a deletion of any $k-1$ $n$-dimensional simplices from that complex, there still remains some $n$-dimensional chain in it, for which this $(n-1)$-dimensional cycle is a boundary. In our last section we restate both the original Menger's Theorem and our generalization to $k$-boundance in $n$ dimensions, in terms of the underlying topological space. Thus, $k$-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, $k$-boundance of an $(n-1)$-dimensional cycle is a topological property of the topological subspace, underlying that $(n-1)$-dimensional cycle, in the topological space, underlying the $n$-dimensional simplicial complex.