Towards Directed Collapsibility (Research)

Abstract

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Combinatorial Conditions for Directed Collapsing

“…The topology of the past links is intrinsically related to the one of the spaces of directed paths. Specifically, in [1] we prove that the contractability and/or connectedness of past links of vertices in directed Euclidean cubical complexes with a minimum vertex 3 implies that all spaces of directed paths with w as initial point are also contractible and/or connected.…”

“…We refer to (K, − → P (K)) as a directed Euclidean cubical complex. 1 The connected components of − → P q p (K) are exactly the equivalence classes of directed paths, up to dihomotopy. If two dipaths, f and g are homotopic through a continuous family of dipaths, then f and g are called dihomotopic.…”

“…Since we are interested in preserving the directed path spaces through collapses, the results from Section 2.3 motivate us to study a type of directed collapse (DC) via past links, introduced in [1]. However, we will call it link-preserving directed collapse (LPDC) (as opposed to directed collapse) since we show in the last sections of this paper that when the spaces of directed paths starting from the minimum vertex are not connected, the following definition of collapse does not preserve the number of components.…”

“…This paper builds on our prior work [1], as well as work by others [6,8,9,14,18]. In this section, we recall the definitions of directed Euclidean cubical complexes, which are the objects that we study in this paper.…”

“…They were introduced in [18] as a means to show that any finite homotopy type can be realized as a connected component of the space of execution paths for some P V -model. In [1], we found conditions for when the local information of past links preserve the global information on the homotopy type of spaces of directed paths. Because of these relationships between past links and directed path spaces, we define collapsing in terms of past links.…”