2024/09/22 by Elahe Hoseinzadeh, Hanıeh Mırebrahımı, Hoseinzadeh, Elahe +5
Mathematics · #Functional Equations Stability Results #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2409.14376
In this paper, we investigate a discrete version of the homotopic distance between two s-Lipschitz maps for s ≥ 0. This distance is defined by specifying a step length r to which some homotopy relation corresponds. In spaces with a significant number of holes, where no continuous homotopy exist and the homotopic distance equals infinite, the discrete homotopic distance provides a meaningful classification by effectively ignoring smaller holes. We show that the discrete homotopic distance Dr generalizes key concepts such as the discrete Lusternik-Schnirelmann category catr and the discrete topological complexity TCr. Furthermore, we prove that Dr is invariant under discrete homotopy relations. This approach offers a flexible framework for classifying s-Lipschitz maps, loops, and paths based on the choice of r.