Higher dimensional automata (HDA) are highly expressive models for<br/>concurrency in Computer Science, cf van Glabbeek (Theor Comput Sci 368(1–2):<br/>168–194, 2006). For a topologist, they are attractive since they can be modeled as cubical complexes—with an inbuilt restriction for directions of allowable (d-)paths.<br/>In Raussen (Algebr Geom Topol 10:1683–1714, 2010), we developed a new method describing, for a certain subclass of HDA, the homotopy type of the space of execution paths (d-paths) as a finite simplicial complex. Several restrictions that were made to ease the presentation in that latter paper will be removed in the present article in order to make the results applicable in greater generality. Furthermore, we take a close look at semaphore models with semaphores all of arity one. It turns out that execution spaces for these are always homotopy discrete with components representing sets of “compatible” permutations. Finally, we describe a model for the complement of the execution space seen as a subspace of a product of spheres—with the aim to make the calculation of topological invariants easier and faster.