On Lusternik-Schnirelmann category and topological complexity of non-\(k\)-equal manifolds (Q2139566)

The non-\(k\)-equal manifold \(M^{(k)} _d (n\)) is defined as the complement in \((\mathbb{R}^d)^n\) of the diagonal-subspace arrangement formed by the union of subspaces \[A_I =\{(x_1, \dots , x_n) \in (\mathbb{R}^d)^n\colon x_{i_1} = \cdots= x_{i_k}\},\] where \(I = \{i_1, \dots, i_k\}\) runs through all cardinality-\(k\) subsets of the set \( \{1, 2, \dots , n\}\). For \(k = 2\) this yields the classical configuration space of \(n\) distinct ordered points in \(\mathbb{R}^d\). In the paper under review the authors compute the Lusternik-Schni\-rel\-mann category [\textit{O. Cornea} et al., Lusternik-Schnirelmann category. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1032.55001)] and all the Farber-Rudyak higher topological complexities [\textit{Y. B. Rudyak}, Topology Appl. 157, No. 5, 916--920 (2010; Zbl 1187.55001)] of the manifolds \(M^{(k)}_d (n)\) for certain values of \(d\), \(k\) and \(n\). A key ingredient in these computations is the description of the cohomology ring \(H^*(M^{(k)}_d (n))\) by \textit{N. Dobrinskaya} and \textit{V. Turchin} [Homology Homotopy Appl. 17, No. 2, 261--290 (2015; Zbl 1346.18013)] .