Connectivity of spaces of directed paths in geometric models for concurrent computation

Higher Dimensional Automata (HDA) are higher dimensional relatives to transition systems in concurrency theory taking into account which degree various actions commute. Mathematically, they take the form of labelled cubical complexes. It is important know, and challenging from a geometric/topological perspective, whether space directed paths (executions model) between two vertices (states) connected; more generally, estimate connectedness these path spaces. This paper presents an approach for such estimation particularly simple HDA modelling access number processors resources with given limited capacity each. defines spare concurrent program prescribed periods resources. shows that spaces can be estimated (from above) by capacities. Moreover, capacities also used detect deadlocks critical states HDA. The key theoretical ingredient calculation local bounds (of upper links HDA) global ones applying version nerve lemma due Anders Bj\"orner.