  This paper is concerned with a covering problem of smooth manifolds of
dimension $n-1$ using a specific arrangement, called Q-arrangement, of $2^{n}$
$n$-simplices which could be degenerate and are allowed to overlap (preventing
therefore the immediate use of simplicial homology). We leverage the underlying
inductive nature of the problem to give a (non-constructive) topological
characterization. We show that for low dimensions such characterization reduces
to studying the local geometry around the specific points serving to form the
simplices, solving thereby the problem for $n \leq 3$. The motivation of such
work comes from a relatively old, yet unsolved, problem that originated in the
optimization community: under which conditions on the $n\times n$ matrix $M$,
does the so called linear complementarity problem given by $w - M z = q$,
$w,z\geq 0$, and $w.z = 0$, have a solution $(w,z)$ for all vectors $q\in
\mathbb{R}^n$. If the latter property holds, the matrix $M$ is said to be a
Q-matrix.
