Relative topological complexity of a pair

Robert Short

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arXiv · 1710.06201

Relative topological complexity of a pair

Robert Short

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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.

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Robert Short. 2017-10-17. Relative topological complexity of a pair. https://arxiv.org/abs/1710.06201

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