Topological complexity of spatial polygon spaces

Abstract:

Let $\overline {\ell }=(\ell _1,\ldots ,\ell _n)$ be an $n$-tuple of positive real numbers, and let $N(\overline {\ell })$ denote the space of equivalence classes of oriented $n$-gons in $\mathbb {R}^3$ with consecutive sides of lengths $\ell _1,\ldots ,\ell _n$, identified under translation and rotation of $\mathbb {R}^3$. Using known results about the integral cohomology ring, we prove that its topological complexity satisfies $\operatorname {TC}(N(\overline {\ell }))= 2n-5$, provided that $N(\overline {\ell })$ is nonempty and contains no straight-line polygons.