$q$-Analog Singular Homology of Convex Spaces

TL;DR

This paper explores the properties of $q$-Analog singular homology, a generalization of classical homology, and computes it for convex spaces, advancing understanding of its algebraic structure.

Contribution

It provides the first calculation of $q$-Analog singular homology for convex spaces, linking it to existing theories and exploring its algebraic properties.

Findings

$q$-Analog singular homology of convex spaces computed.

Results align with previous work by Dubois-Viol ext{ }tte & Henneaux.

Advances understanding of $q$-chains and their algebraic structure.

Abstract

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.