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This paper is concerned with a covering problem of smooth manifolds ofdimension $n-1$ using a specific arrangement, called Q-arrangement, of $2^{n}$$n$-simplices which could be degenerate and are allowed to overlap (preventingtherefore the immediate use of simplicial homology). We leverage the underlyinginductive nature of the problem to give a (non-constructive) topologicalcharacterization. We show that for low dimensions such characterization reducesto studying the local geometry around the specific points serving to form thesimplices, solving thereby the problem for $n \leq 3$. The motivation of suchwork comes from a relatively old, yet unsolved, problem that originated in theoptimization 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 aQ-matrix.

On Covering Smooth Manifolds with Overlapping Simplicies: An inductive Characterization of Q-matrices