On the higher topological complexity of manifolds with abelian fundamental group

We study the higher (or sequential) topological complexity ${\mathrm{T}\mathrm{C}}_s$ of manifolds with abelian fundamental group. We give sufficient conditions for ${\mathrm{T}\mathrm{C}}_s$ to be non-maximal in both the orientable and non-orientable cases. In combination with cohomological lower bounds, we also obtain some exact values for certain families of manifolds.