Homology of spaces of directed paths on Euclidean cubical complexes (Q2255529)

The authors calculate the homology of the spaces of directed paths of a certain class of cubical subcomplexes of the directed Euclidean space \(\mathbb{R}^n\) by a recursive process. The motivation of this kind of calculation is the geometric approach to concurrency by tools coming from algebraic topology. Indeed, in such a calculation, the directed paths (which form a subspace of the space of continuous paths) represent the execution paths and directed homotopy between directed paths represents model concurrency. It is therefore very important, and in general very complicated, to determine the homotopy type of these path spaces. As an application, the homology and the cohomology of the space of directed loops on the \((n-1)\)-skeleton of the directed \(n\)-dimensional torus are calculated.