The Euler characteristic of the configuration space of planar spidery linkages
type: Item
instance of: scholarly article
title: The Euler characteristic of the configuration space of planar spidery linkages
zbMATH Open document ID: 1321.55011
author: Yasuhiko Kamiyama
author: Shuichi Tsukuda
published in: Algebraic \& Geometric Topology
publication date: 2015-01-26T00:00:00Z
review text: 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…
reviewed by: Gwénaël Massuyeau
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 70B15
zbMATH DE Number: 6390562
zbMATH Keywords: configuration space
zbMATH Keywords: planar linkage
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Morse-Bott function
MaRDI profile type: Publication
full work available at URL: agt.2014.14.3659
OpenAlex ID: W2058826275
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DOI: 10.2140/AGT.2014.14.3659
description: scientific article; zbMATH DE number 6390562