Combinatorial Conditions for Directed Collapsing

Abstract

The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In this paper, we give combinatorially equivalent conditions for preserving the topology of the links, allowing for the implementation of an algorithm for collapsing a directed Euclidean cubical complex. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of directed path spaces, as well as examples when link-preserving directed collapses do not preserve the number of connected components of the path space between the minimum and a given vertex.

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

“…Section 3 focusses on the local behaviour of the state space. As already explained by Ziemiański [20] and Belton etal [1,2], the key information is the topology (in particular, the connectivity) of the future links (or past links) of vertices in the state space. It turns out that these future links are joins (aka convex combinations) of skeleta of simplices.…”