Foundational Challenges in Proximal and Digital Topological Complexity: A Critical Review of Recent Developments

The theory of topological complexity is an important theory in the study of motion planning problems. Over the last decade, several attempts have been made to extend the theory of complexity (and its higher analogues) to proximal and digital topological settings. As a result, there are now digital versions of concepts like path spaces, motion planning algorithms, the Schwarz genus, the homotopic distance, digital fibrations, digital topological groups, and generalized topological complexity. Despite the number of publications on the topic of digital and proximal topological complexity, several mathematical issues within the area have yet to be resolved. For example, the theory of classical topological complexity suggests the construction of path spaces, product adjacencies, continuous maps, homotopy structures, lifting structures, and fibrations. Each of these concepts inherently relates to the notion of topology, which is a concept that applies to digital spaces only in a restricted manner. As a result, each of these concepts must be defined within digital spaces, and their mathematical consistency within those spaces must be established. In this paper, a collection of existing publications on the topic of proximal and digital topological complexity will be reviewed. Each of these publications has encountered issues related to adjacency structures, digital continuity, path spaces, digital fibrations, and generalized complexity. A focus will be placed upon the role that product adjacencies, digital coverings, the unique path lifting property, and the homotopy lifting property play in the definition of digital analogues of concepts from classical topology. Overall, while the current state of the known results related to digital and proximal topological complexity can be summarized, several challenges must be overcome in developing future results on the topic. Suggestions for future research in the field will be made and suggested challenges will be discussed. Specifically, it is clear from the published results that a theory based upon adjacencies in digital spaces is needed, as well as established results regarding digital continuity, digital covering maps, and lifting structures.