Higher topological complexity of Seifert fibred manifolds

In this article, we investigate the higher topological complexity of oriented Seifert fibred manifolds that are Eilenberg–MacLane spaces $K(G,1)$ with infinite fundamental group $G$ . We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the $r^{\text{th}}$ topological complexity of these manifolds lies in $\{3r-1, 3r, 3r+1\}$ , and characterize large families where the value is $3r$ or $3r+1$ . Additionally, we establish a sufficient condition for higher topological complexity to be exactly $3r$ when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical $3$ -manifolds is exactly $3r+1$ .