Equivariant and Invariant Parametrized Topological Complexity

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. We establish the equivariant fibrewise homotopy invariance of this notion and derive several bounds, including a cohomological lower bound and a dimensional upper bound. Additionally, we compare invariant parametrized topological complexity with other well-known invariants. 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. Finally, we compute the invariant parametrized topological complexity of equivariant Fadell-Neuwirth fibrations, which measures the complexity of motion planning in presence of obstacles having unknown positions such that the order in which they are placed is irrelevant. Apart from this, we establish several bounds, including a cohomological lower bound, an equivariant homotopy dimension-connectivity upper bound and various product inequalities for the equivariant sectional category. Applying them, we obtain some interesting results for equivariant and invariant parametrized topological complexity of a $G$-fibration.