Sequential parametrized topological complexity of group epimorphisms
Navnath Daundkar
TL;DR
The paper addresses the problem of extending sequential topological complexity to the parametrized setting for group epimorphisms and their associated fibrations. It defines $\mathrm{TC}_n[\alpha:G\to H]$ as $\mathrm{secat}(\Delta_n:G\to G^n_H)$, establishes pullback and fibrewise homotopy invariance results, and derives sharp bounds by relating $G^n_H$ to a semidirect product $(\ker\alpha)^{n-1}\rtimes G$ and by exploiting centralizer data via $k(\alpha)$. It provides a sequential analogue of Grant's lower and upper bounds and extends them to type-F groups and central extensions; it also yields an alternate proof for the sequential parametrized TC of planar Fadell–Neuwirth fibrations, showing $\mathrm{TC}_n= nt+s-2$ in the planar setting. The work connects algebraic invariants such as cohomological dimension with motion-planning complexity in parametrized and group-theoretic contexts, offering tools for concrete computations in configuration-space problems.
Abstract
We introduce and study the sequential analogue of Grant's parametrized topological complexity of group epimorphisms, which generalizes the sequential topological complexity of groups. We derive bounds for sequential parametrized topological complexity based on the cohomological dimension of certain subgroups, thereby extending the corresponding bounds for sequential topological complexity of groups. We also obtain sequential analogs of (new) lower bounds on parametrized topological complexity of epimorphisms which are recently obtained by Espinosa Baro, Farber, Mescher and Oprea. Finally, we utilize these results to provide alternative computations for the sequential parametrized topological complexity of planar Fadell-Neuwirth fibrations.