Topological complexity of the Klein bottle (Q1659254)

The topological complexity \(\text{TC}(X)\) of a space \(X\) is a numerical homotopy invariant introduced by \textit{M. Farber} in [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)] in order to study the motion planning problem in robotics from a topological perspective. In that foundational paper Farber computed, among other examples, the topological complexity of all orientable surfaces by mainly using the bound given by the zero-divisors cup-length with coefficients in a field. However the used techniques were not applicable to non-orientable surfaces and this case had been left as an open problem. Only the case of the real projective plane \(\mathbb{RP}^2\) was solved (its normalized topological complexity being 3) in the paper [\textit{M. Farber, S. Tabachnikov} and \textit{S. Yuzvinsky}, Int. Math. Res. Not. 2003, No. 34, 1853--1870 (2003; Zbl 1030.68089)]. In [Proc. Am. Math. Soc. 144, No. 11, 4999--5014 (2016; Zbl 1352.55002)], \textit{A. Dranishnikov} made a big step towards the solution of this problem by proving that the (normalized) topological complexity of \(N_g\), the non-orientable surface of genus \(g\), is 4 when \(g\geq 5.\) In a subsequent paper [Topology Appl. 232, 61--69 (2017; Zbl 1378.55001)] he also showed that \(\text{TC}(N_4)=4\) and that his methods do not extend to the cases \(g=2\) (i.e. the Klein bottle) and \(g=3\). Since then only these two cases were still open. In the paper under review the authors prove that the (normalized) topological complexity of the Klein bottle is 4. Their result is based on computations of zero-divisors cup-length with local coefficients as in \textit{A. Costa} and \textit{M. Farber}'s paper [Commun. Contemp. Math. 12, No. 1, 107--119 (2010; Zbl 1215.55001)] and also by using an explicit bar resolution for the Klein bottle as a \(K(G,1)\)-space. Actually, using an inductive argument starting at \(g=2,\) they are capable to prove that \(\text{TC}(N_g)=4\), for all \(g\geq 2.\) This way the program of computing the topological complexity of all non-orientable surfaces is fulfilled. As noted by the authors, \textit{D. M. Davis} also found an independent computation of the topological complexity of the Klein bottle based on certain \(\Delta \)-complex structures [``An approach to the topological complexity of the Klein bottle'', Preprint, \url{arXiv:1612.02747}].