Topological complexity of some planar polygon spaces

Abstract

Let ${\overline{M}}_{n,r}$ denote the space of isometry classes of n-gons in the plane with one side of length r and all others of length 1, and assume that $1\le r<n-3$ and $n-r$ is not an odd integer. Using known results about the mod-2 cohomology ring, we prove that its topological complexity satisfies ${\text {TC}}({\overline{M}}_{n,r})\ge 2n-6$. Since ${\overline{M}}_{n,r}$ is an $(n-3)$-manifold, ${\text {TC}}({\overline{M}}_{n,r})\le 2n-5$. So our result is within 1 of being optimal.