Equivariant and invariant parametrized topological complexity

TL;DR

This paper introduces and studies the invariant parametrized topological complexity for G-equivariant fibrations, generalizing previous concepts and providing computations for specific fibrations relevant to motion planning with obstacles.

Contribution

It defines the invariant parametrized topological complexity, relates it to orbit space fibrations, and computes it for equivariant Fadell-Neuwirth fibrations, advancing understanding in equivariant motion planning.

Findings

Invariant parametrized topological complexity coincides with the complexity of the induced orbit space fibration when G acts freely.

Computed the invariant parametrized topological complexity for equivariant Fadell-Neuwirth fibrations.

Studied equivariant sectional category and parametrized topological complexity as tools for these analyses.

Abstract

For a $G$-equivariant fibration $p \colon E\to B$, we introduce and study the invariant analogue of Cohen, Farber and Weinberger's parametrized topological complexity, called the invariant parametrized topological complexity. This notion generalizes the invariant topological complexity introduced by Lubawski and Marzantowicz. When $G$ is a compact Lie group acting freely on $E$, we show that the invariant parametrized topological complexity of the $G$-fibration $p \colon E\to B$ coincides with the parametrized topological complexity of the induced fibration $\overline{p} \colon \overline{E} \to \overline{B}$ between the orbit spaces. Furthermore, we compute the invariant parametrized topological complexity of equivariant Fadell-Neuwirth fibrations, which measures the complexity of motion planning in the presence of obstacles with unknown positions, where the order of their placement is irrelevant. In addition, we study the equivariant sectional category and the equivariant parametrized topological complexity, which serve as essential tools for obtaining several results in this paper.