  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.
