Higher analogues of discrete topological complexity (Q6577491)

Let \(\text{TC}_n^{\text{R}}(-)\) be the higher topological complexity introduced by \textit{Y. B. Rudyak} in [Topology Appl. 157, No. 5, 916--920 (2010; Zbl 1187.55001)]. The authors use the reduced version, i.e., \(\text{TC}_n^{\text{R}}(X)=0\) if and only if \(X\) is contractible. Let \(\text{scat}(-)\) be the simplicial Lusternik-Schnirelmann category introduced in [\textit{D. Fernández-Ternero} et al., Topology Appl. 194, 37--50 (2015; Zbl 1327.55004)].\N\NIn the present paper, the authors present the notion of \(n\)th discrete topological complexity (Definition 2.2). Namely, let \(K\) be a simplicial complex and \(n\geq 2\). \(K^n\) denotes the \(n\)th cartesian product of \(K\) joint with the categorical product of simplicial complexes. Let \(\Delta:K\to K^n\) be the diagonal map. The \textit{\(n\)th discrete topological complexity} of \(K\), denoted by \(\text{TC}_n(K)\), is the least non-negative integer \(k\) such that \(K^n\) can be covered by subcomplexes \(\Omega_0, \Omega_1,\ldots, \Omega_k\) and for each \(0\leq i\leq k\) there is a simplicial map \(\sigma_i:\Omega_i\to K\) so that \(\Delta\circ\sigma_i\) and the inclusion map \(\iota_i:\Omega\hookrightarrow K^n\) are in the same contiguity class.\N\NThe authors show the following properties: For \(n\geq 2\), \N\begin{enumerate}\N\item \(\text{TC}_n(K)\leq \text{TC}_{n+1}(K)\) (Theorem 2.2).\N\item \(\text{TC}_n(K)=\text{TC}_n(L)\) whenever \(K\) and \(L\) have the same strong homotopy type (Theorem 2.3).\N\item \(\text{scat}(K^{n-1})\leq\text{TC}_n(K)\leq \text{scat}(K^{n})\) provided that \(K\) is an edge-path connected simplicial complex \(K\) (Theorem 2.4).\N\item \(\text{TC}_n^{\text{R}}(\mid K\mid)\leq\text{TC}_n(K)\) (Theorem 2.6).\N\end{enumerate}