In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have nontrivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space, relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs.
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PathConnectivity of Fréchet Spaces of Graphs
We examine topological properties of spaces of paths and graphs
mapped to $\R^d$ under the Fr\'echet distance. We show that these spaces
are pathconnected if the map is
either continuous or an immersion. If the map is an embedding, we show
that the space of paths is pathconnected, while the space of graphs only
maintains this property in dimensions four or higher.
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 NSFPAR ID:
 10351571
 Date Published:
 Journal Name:
 Young Researcher's Forum (CG Week)
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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