Let \(\bar M_{n,r}\) denote the space of isometry classes of \(n\)-gons in the plane with one side of length \(r\) and all the others of length 1, and assume that \(1\leq r<n-3\) and \(n-r\) is not an odd integer. The question is what is the topological complexity of \(\bar M_{n,r}\), denoted by \(TC(\bar M_{n,r})\). For some variation of the hypothesis on the integers \(n\) and \(r\) results about this question for the space \(\bar M_{n,r}\) are known.
The paper under review shows a very good estimation for \(TC(\bar M_{n,r})\). The author shows:
Theorem 1.1. If \(r\) is a real number such that \(1\leq r<2n-3\) and \(n-r\) is not an odd integer, then \(TC(\bar M_{n,r})\geq 2n-6\).
The above result together with some known results leads to the inequality \(2n-5\geq TC(\bar M_{n,r})\geq 2n-6\), so the result is within \(1\) of being optimal. The proof of Theorem 1.1 is obtained using a very careful and non trivial analysis of the cohomology ring of \(H ^*(\bar M_{n,r};\mathbb{Z}_2)\), using some previous works. In fact even properties of the ring \(H ^*(\bar M_{n,r};\mathbb{Z}_2)\otimes H ^*(\bar M_{n,r};\mathbb{Z}_2)\) are analyzed. The results about the cohomology ring \(H ^*(\bar M_{n,r};\mathbb{Z}_2)\) which are proved in order to show Theorem 1.1 lead to the result:
Theorem 1.2. If \(1\leq r < n-3\) is not an odd integer, the zero-divisors-cup-length of \(H ^*(\bar M_{n,r},\mathbb{Z}_2)\) equals \(2n-7\).