The Euler characteristic of the configuration space of planar spidery linkages
This paper studies the topology of the configuration space of ``planar spidery linkages''. Let \(m,n\geq 2\) be integers and let \(\ell >0\) be a real number. The authors consider the set \(M(n,m,\ell)\) of ``polygonal spiders'' in \(\mathbb{C}\) obtained by connecting the vertices \(v_1,\dots,v_n\) of the regular \(n\)-gon (the ``claws'' of the spider) to a single point of \(\mathbb{C}\) (the ``body'' of the spider) by means of \(n\) polygonal lines (the \(n\) ``legs'' of the spider -- possibly \(n\neq 8\)), each of these lines consisting of \(m\) segments of length \(\ell\). To be more specific, the authors define the topological space \[ M(n,m,\ell) :=\left\{ (a_1,\dots,a_n) \in (\mathbb{T}^m)^n\, :\, v_i+ \ell \sum_{k=1}^m a_{i,k}\in \mathbb{C} \text{ does not depend on \(i\)} \right\} \] where \(\mathbb{T}^m=\mathbb{S}^1 \times \cdots \times \mathbb{S}^1\) denotes the \(m\)-dimensional torus. The configuration space \(M(n,m,\ell)\) of ``planar spidery linkages'' has already been considered in the literature in the particular cases \(m=2\) and \(n=2\) (usually under different names). It is proved in the paper that there exist finitely many ``bad'' lengths \[ 1/m=\ell_0< \ell_1 < \cdots < \ell_{\text{max}} < +\infty \] such that, for every \(i\in \{0,\dots, \text{max}\}\), the space \(M(n,m,\ell)\) is an \({(nm-2n+2)}\)-dimensional closed orientable manifold of constant diffeomorphism type for all \(\ell \in (\ell_i,\ell_{i+1})\); moreover, the values of \(\ell_1\) and \(\ell_{\text{max}} \) are determined, and \(M(n,m,\ell)\) is proved to be homeomorphic to a certain moment-angle complex for \(\ell \in (\ell_0,\ell_1)\). Furthermore, for any \(\ell\in (\ell_0,+\infty) \setminus \{\ell_1,\dots, \ell_{\text{max}}\}\) satisfying some mild conditions, the authors show that the map \({f: M(n,m,\ell) \to \mathbb{R}}\) defined by \[ f(a_1,\dots,a_n) := \left| v_1+ \ell \sum_{k=1}^m a_{1,k}\right|^2 \] is a Morse-Bott function, whose critical submanifolds are identified and whose indices are explicitly computed. This is used to show that the Euler characteristic of \(M(n,m,\ell)\) vanishes for \(m\) odd, and it is explicitly calculated for \(m\) even and certain values of \(\ell\). Finally, the authors conclude that \(M(n,m,\ell)\) is a sphere if and only if \(n\in \{2,3\}\) and \(\ell \in (\ell_0,\ell_1)\).
scientific article; zbMATH DE number 6390562