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The notion of a homotopy flow on a directed space was introduced in Raussen (Appl Categ Struct 15(4):355–386, 2007) as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all directed maps along such a 1-parameter deformation preserve the homotopy types of path spaces, such a flow and the parameter maps are called inessential. For a directed space, one may consider various categories whose objects are pairs of reachable points to which a functor associates the space of directed paths between them. The monoid of all inessential maps acts on such a category by endofunctors leaving the associated path spaces invariant up to homotopy. We construct a pair component category as quotient category: it has as objects pair components along which the homotopy type is invariant—for a coherent and transparent reason. This paper follows up Fajstrup et al. (J Homotopy Relat Struct 12(1):81–108, 2004); Goubault and Haucourt (Appl Categ Struct 15(4):387–414, 2007); Raussen (Appl Categ Struct 15(4):355–386, 2007) and removes some of the restrictions for their applicability. At least in several examples, it gives reasonable results for spaces with non-trivial directed loops. If one uses homology equivalence instead of homotopy equivalence as the basic relation, it yields an alternative to computable versions of “natural homology” introduced in Dubut et al (in: Automata, languages, and programming. 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015. Proceedings. Part II, 171–183, Springer, Berlin 2015) and elaborated in Dubut (Directed homotopy and homology theories for geometric models of true concurrency. École normale supérieure Paris-Saclay 2017). It refines, for good and for evil, the stable components introduced and investigated in Ziemiański (Appl Categ Struct 27:217–244, 2019).

We compute the homology of the spaces of directed paths on a certain class of cubical subcomplexes of the directed Euclidean space $$\mathbb{R }^n$$Rnby a recursive process. We apply this result to calculate the homology and cohomology of the space of directed loops on the $$(n-1)$$(n-1)-skeleton of the directed torus $$\vec {T}^n$$T→n.

This paper focusses on the determination of the dihomotopy classes of dipaths in cubical complexes, representing the essentially different computations in a given concurrent program due to different schedules. Several new notions have to be developed, for example, the domains of dependence (Definition 2.4), which are borrowed from relativity theory (Penrose 1972). However, it turns out that the algorithm determining deadlocks and unsafe regions described in Fajstrup et al. (1998a; 1998b) can be modified and applied to do the essential calculational work.

We investigate the rational homotopy classification problem for the components of some function spaces with $ {S^n}$ as target space.

The aim of this paper is to follow up the program set in [LR85, Rau92], i.e., to show the existence of nontrivial group actions ("symmetries") on certain classes of manifolds. More specifically, given a manifold $ X$, I would like to construct nontrivial actions of cyclic groups on $ X$ as fixed point set. Of course, this is not always possible, and a list of necessary conditions for the existence of an action of the circle group $ T = {S^1}$ with fixed point set $ F$ and $ X$-action on the complement $ X\backslash F$-bundle action on the normal bundle of $ F$. In a subsequent paper [Rau94], I plan to show how to realize this $ T$ rationally homotopy equivalent to $ X$ and how to "propagate" all but finitely many of the restricted cyclic group actions to itself.