TOPOLOGICAL COMPLEXITY OF KHALIMSKY CIRCLES

Abstract

We determine the topological complexity of a Khalimsky circle 𝕊¹_𝑛, a finite space of 2𝑛 points for all 𝑛 ≥ 2, which is weakly homotopy-equivalent to a circle 𝑆¹. This answers two conjectures on topological complexity for finite spaces raised by K. Tanaka (Algebr. Geom. Topol. 18(2)(2018), 779–796), namely, we obtain (1) tc(Sd^𝑘𝕊¹₂) = 1 for all 𝑘 ≥ 2, and (2) tc(𝕊¹_𝑛) < cat(𝕊¹_𝑛 ×𝕊¹_𝑛) for all 𝑛 ≥ 5.