Equivariant and Invariant Parametrized Topological Complexity
Ramandeep Singh Arora, Navnath Daundkar
TL;DR
The paper introduces the invariant parametrized topological complexity TC^G[p: E → B] for G-equivariant fibrations and proves it generalizes both the parametrized TC and the invariant TC. It establishes an invariance theorem equating TC^G[p] with the orbit-space TC under free group actions and develops a robust framework via equivariant sectional category and equivariant LS-category. It also defines and analyzes TC_G[p], the equivariant parametrized TC, and proves a main invariance result for TC^G[p], along with several bounds and cohomological criteria. The authors compute TC^{Σ_s} for equivariant Fadell–Neuwirth fibrations, showing explicit dependence on the number of obstacles and ambient dimension, with results aligning with orbit-space reductions. Collectively, the work advances motion-planning complexity in symmetric and parametrized settings, offering tools and exact computations for symmetry-aware configurational problems.
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.