Topological complexity of spatial polygon spaces (Q2809220)

The topological complexity \(\mathrm{TC}(X)\) 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.NEWLINENEWLINEIf \(\overline{\ell}=(\ell _1,\dots,\ell_n)\) is an \(n\)-tuple of positive real numbers, the author of the paper under review deals with the computation of the topological complexity of \(N(\overline{\ell})\), the space of equivalence classes of oriented \(n\)-gons in \(\mathbb R^3\) with consecutive sides of length \(\ell_1,\dots,\ell_n\), identified under translation and rotation of \(\mathbb R^3\). Namely, as far as \(N(\overline{\ell})\) is nonempty and contains no straight-line polygons, the author proves by using cohomological arguments that \(\mathrm{TC}(N(\overline{\ell}))=2n-5\).