Biequivariant maps on spheres and topological complexity of lens spaces (Q2853985)

The topological complexity of a space \(X,\) denoted in this paper as \(\overline{\text{TC}}(X),\) is the (normalized) Schwarz genus of the fibration \(\pi _X:PX\rightarrow X\times X\) given by \(\pi _X(\gamma )=(\gamma (0),\gamma (1)),\) where \(PX\) denotes the function space of all continuous paths \(\gamma :[0,1]\rightarrow X.\) This numerical homotopy invariant was introduced by \textit{M. Farber} [Discrete Comput. Geom. 29 No. 2, 211--221 (2003; Zbl 1038.68130)] in order to study the motion planning problem in robotics from a purely topological point of view.NEWLINENEWLINEIn the paper under review the authors deal with the computation of the topological complexity of lens spaces. If \(L^{2n+1}(t)\) denotes the standard \(t\)-torsion lens space of dimension \(2n+1,\) then it is known from [\textit{J. González}, Math. Proc. Camb. Philos. Soc. 139, No. 3, 469--485 (2005; Zbl 1087.55003)] that \(\overline{\text{TC}}(L^{2n+1}(t))=2b_{n,t}+\varepsilon _{n,t},\) where \(\varepsilon _{n,t}\geq 0\) (\(\varepsilon _{n,t}\in \{0,1\}\) when \(t\) is even) and \(b_{n,t}\) is the smallest positive integer \(m\) for which there exists a \(\mathbb{Z}_t\)-biequivariant map \(S^{2n+1}\times S^{2n+1}\rightarrow S^{2n+1}.\) If \(b(n,e)\) stands for \(b_{n,2^e}\), then J. González proved that \(b(n,e)=2n\) when \(e> \alpha (n)\), and \(b(n,e)=2n-1\) when \(e=\alpha (n);\) here \(\alpha (n)\) denotes the number of ones in the dyadic expansion of \(n.\) Now the authors give a continuation of this previous work and study the unsolved case \(e=\alpha (n)-1.\) Using mainly connective complex \(K\)-theory they are able to prove the following principal theoremNEWLINENEWLINETheorem. \(b(m+1,\alpha (m)-1)\geq 2m-2,\) provided \(\alpha (m)\geq 2.\)NEWLINENEWLINEThis result gives, as a consequenceNEWLINENEWLINECorollary. \(\overline{\text{TC}}(L^{2m+3}(2^{\alpha (m)-1}))\geq 4m-4,\) provided \(\alpha (m)\geq 2\)NEWLINENEWLINEThis is an improvement by arbitrarily large amounts on Farber-Grant's general lower bound for the topological complexity of lens spaces [\textit{M. Farber} and \textit{M. Grant}, Proc. Am. Math. Soc. 136, No. 9, 3339--3349 (2008; Zbl 1151.55004)].