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.