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

TL;DR

This paper establishes an upper bound for higher topological complexity using a new invariant related to equivariant topological complexity, with implications for understanding the complexity of spaces via their universal covers.

Contribution

It introduces an intermediate invariant and relates higher topological complexity to strongly equivariant complexity, providing new bounds and interpretations.

Findings

Proves an upper bound for $TC_n(X)$ using $TC_n^{\ ext{\mathcal{D}}}(X)$.

Defines and interprets a new invariant $\ ilde{TC}_n(X)$ as a higher analogue of strongly equivariant complexity.

Connects the complexity of a space with the complexity of its universal cover under fundamental group action.

Abstract

We prove an upper bound of higher topological complexity $TC_n(X)$ using higher $\mathcal{D}$-topological complexity $TC_n^{\mathcal{D}}(X)$ 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 $\widetilde{X}$ with the action of the fundamental group of $X$.