Overview
The research focuses on the introduction and subsequent analysis of discrete versions for two specific measures of topological complexity of maps: those proposed by Scott and by Murillo-Wu. The study establishes core theoretical properties and computational considerations for these discrete analogs, particularly in the context of simplicial maps.
Research Context
Topological complexity of maps, as defined by Scott and Murillo-Wu, serves as a foundational concept within topological studies. This work investigates discrete counterparts to these established continuous measures. The aim is to translate these concepts into a discrete framework, enabling their study and application within discrete mathematical structures.
Approach
The methodology involved introducing discrete analogs of the specified topological complexity measures. Following their introduction, the research proceeded to investigate their behavior and intrinsic properties. This included examining their invariance under contiguity and their equivalency.
Findings
- The study successfully introduced discrete analogs for Scott's topological complexity of maps.
- Discrete analogs for Murillo-Wu's topological complexity of maps were also introduced.
- These newly introduced discrete analogs were found to be contiguity invariants.
- A key finding is the proof that these discrete analogs are equivalent to one another.
- The research established fundamental theoretical properties of the discrete topological complexity of simplicial maps.
- Computational aspects related to the discrete topological complexity of simplicial maps were also established.