On topological complexity of twisted products (Q465849)

The topological complexity \(\text{TC}(X)\) of a space \(X\) is the sectional category (or Schwarz genus) of the end-points evaluation fibration NEWLINE\[NEWLINE\pi :X^{[0,1]}\rightarrow X\times X,\quad\pi (\alpha )=(\alpha (0),\alpha (1)) NEWLINE\]NEWLINE It was introduced by \textit{M. Farber} [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)] motivated by the motion planning problem in Robotics. The topological complexity is a homotopy invariant so given any discrete group \(\pi \) its topological complexity can be defined as \(\text{TC}(\pi ):=\text{TC}(K(\pi ,1)),\) where \(K(\pi ,1)\) denotes the Eilenberg-MacLane space of type \((\pi ,1)\).NEWLINENEWLINEIn the paper under review the author proves the inequality NEWLINE\[NEWLINE\text{TC}(X)\leq \text{TC}(\pi )+\text{dim}(X)NEWLINE\]NEWLINE \noindent for any CW-complex \(X\) with fundamental group \(\pi .\) In order to prove this inequality the author first shows that NEWLINE\[NEWLINE\text{TC}(X)\leq \text{TC}(B)+\text{TC}^*_G(F)NEWLINE\]NEWLINE \noindent in the case of a twisted product \(X=B\tilde{\times}F\) over \(B\) with fiber \(F\) and structure group \(G.\) Here \(\text{TC}^*_G(F)\) is a certain equivariant version of topological complexity, which is also defined in the paper.