Bounds for the higher topological complexity of configuration spaces of trees

TL;DR

This paper establishes that for many positive integers n and s ≥ 2, the higher topological complexity of unordered configuration spaces of trees reaches its maximum, equaling s times the homotopy dimension.

Contribution

It provides bounds showing that the higher topological complexity of configuration spaces of trees is maximal for many cases, linking it directly to the homotopy dimension.

Findings

Higher topological complexity equals s times the homotopy dimension for many n and s.

Configuration spaces of trees have maximal topological complexity in these cases.

The results give a clear formula for the topological complexity of these spaces.

Abstract

For a tree $T$, we show that for many positive integer values of $n$, and an integer $s \geq 2$, the higher topological complexity $TC_s$ of the unordered configuration spaces of trees $U\mathcal{C}^nT$, is maximal. In other words, we prove that, $TC_s(U \mathcal{C}^nT) = s(hdim (U \mathcal{C}^nT))$ where $hdim$ stands for the homotopy dimension.