Relative topological complexity of a pair

TL;DR

This paper introduces a new variant of relative topological complexity for pairs of spaces, providing estimates, computations for examples, and connections to other topological invariants and axial maps.

Contribution

It defines the relative topological complexity of a pair of spaces, explores its properties, and computes it for various examples, linking it to Lusternik-Schnirelmann category and axial maps.

Findings

Computed for wedges of spheres, topological groups, and polygon spaces.

Established bounds and relationships with existing invariants.

Connected the invariant to the existence of certain axial maps.

Abstract

For a pair of spaces $X$ and $Y$ such that $Y \subseteq X$, we define the relative topological complexity of the pair $(X,Y)$ as a new variant of relative topological complexity. Intuitively, this corresponds to counting the smallest number of motion planning rules needed for a continuous motion planner from $X$ to $Y$. We give basic estimates on the invariant, and we connect it to both Lusternik-Schnirelmann category and topological complexity. In the process, we compute this invariant for several example spaces including wedges of spheres, topological groups, and spatial polygon spaces. In addition, we connect the invariant to the existence of certain types of axial maps.