An upper bound for higher topological complexity and higher strongly equivariant complexity

Abstract

We prove an upper bound of higher topological complexity $T{C}_n(X)$ using higher $\mathcal{D}$-topological complexity $T{C}_n^{\mathcal{D}}(X)$ of a space $X$. An intermediate invariant ${\widetilde{TC}}_n(X)$ is used in the proof. We interpret this invariant ${\widetilde{TC}}_n(X)$ as higher analogue of strongly equivariant topological complexity of the universal cover of $\tilde{X}$ with the action of the fundamental group of $X$.