q-Analog Singular Homology of Convex Spaces

In this article we study some interesting properties of the $q$-Analog singular homology, which is a generalization of the usual singular homology, suitably adapted to the context of $N$-complex and amplitude homology \cite{kapranov}. We calculate the $q$-Analog singular homology of a convex space. Although it is a local matter; this is an important step in order to understand the presheaf of $q$-chains and its algebraic properties. Our result is consistent with those of Dubois-Viol\`ette & Henneaux \cite{dubois3}. Some of these results were presented for the XVIII Congreso Colombiano de Matem\'aticas in Bucaramanga, 2011.