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

TL;DR

This paper generalizes Menger's Theorem from graphs to higher-dimensional simplicial complexes, introducing the concept of $k$-boundance to relate combinatorial connectivity with topological properties.

Contribution

It extends the classical Menger's Theorem to $n$-dimensional simplicial complexes using $k$-boundance, linking combinatorial and topological connectivity.

Findings

Introduces $k$-boundance as a generalization of $k$-edge-connectivity.

Proves a higher-dimensional Menger's Theorem relating $k$-boundance to chain disjointness.

Restates the theorem in topological terms, connecting graph theory and topology.

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.