On topological complexity of Eilenberg-MacLane spaces (Q2800384)

This very short paper is dedicated to a first step in investigating a question of M. Farber who asked about computing \(\mathrm{TC}(K(\pi,1))\), where \(\mathrm{TC}\) denotes the topological complexity of the Eilenberg-MacLane space. The author proves thatNEWLINENEWLINETheorem 1. For every natural number \(k\) and every natural \(\ell\) with \(k\leq \ell\leq 2k\), there exists a discrete group \(\pi\) such that \(\mathrm{cat}K(\pi,n)=k\) and \(\mathrm{TC}(K(\pi,1))=\ell\).NEWLINENEWLINEThe proof of this fact is constructive in the sense that \(\pi:= \mathbb{Z}^k* \mathbb{Z}^{\ell-k}\). Combining Theorem 1 with the well known fact that \(\mathrm{cat}(X)\leq \mathrm{TC}(X)\leq \mathrm{cat}(X\times X)\) and the fact that \(\mathrm{cat}(K(\pi,1))\) has a known purely group theoretic construction, the author concludes that Farber's question then becomes a question of searching for a purely group-theoretic construction of \(\mathrm{TC}(K(\pi,1))\).