Explicit motion planning in digital projective product spaces

TL;DR

This paper introduces digital projective product spaces, establishing bounds for their digital LS-category and topological complexity, and relates these complexities to those of digital spheres through explicit motion planning methods.

Contribution

It defines digital projective product spaces and derives bounds for their digital LS-category and topological complexity, linking these to digital spheres and validating previous theoretical results.

Findings

Upper bounds for digital LS-category and topological complexity are established.

A relation between the complexities of digital projective product spaces and digital spheres is proven.

Explicit motion planning constructions demonstrate the theoretical results.

Abstract

We introduce the digital projective product spaces based on Davis' projective product spaces. We determine an upper bound for the digital LS-category of the digital projective product spaces. In addition, we obtain an upper bound for the digital topological complexity of these spaces. We prove the relation between the digital topological complexity of the digital projective product spaces and sum of the digital topological complexity of the digital projective space by associating with the first digital sphere and the digital topological complexity of the remaining digital spheres through an explicit motion planning construction, which shows digital perspective validity of the results given by S. Fi\c{s}ekci and L. Vandembroucq. We apply our outcomes on specific spaces in order to be more clear.