Configuration spaces of thick particles on a metric graph
From the abstract: ``We study the topology of configuration spaces \(F_r(\Gamma,2)\) of two thick particles (robots) of radius \(r>0\) moving on a metric graph \(\Gamma\). As the size of robots increases, the topology of \(F_r(\Gamma,2)\) varies. Given \(\Gamma\) and \(r\), we provide an algorithm for computing the number of path-connected components of \(F_r(\Gamma,2)\). Using our main tool of \(\mathrm{PL}\) Morse-Bott theory, we show that there are finitely many critical values of \(r\) where the homotopy type of \(F_r(\Gamma,2)\) changes. We study the transition across a critical value \(R\in (a,b)\) by computing the ranks of the relative homology group of \((F_a(\Gamma,2), F_b(\Gamma,2))\).'' Some definitions: Let \(X\) be a space. The \(n\)th ordered configuration space \(F(X,n)\) is defined by setting \[ F(X,n)=\{(x_1,\ldots,x_n) \in X^n \;| \;x_i\not=x_j \text{ for } i\not=j\} \] with the subspace topology of the Cartesian product \(X^n\). In this article, the space \(X\) is a metric graph \(\Gamma\); the thick configuration space of radius \(r>0\) is defined by setting \[ F_r(\Gamma,2)=\{(x,y)\in\Gamma \;| \;d(x,y)\geq 2r\} \] with the subspace topology of \(F(\Gamma,2)\).
scientific article; zbMATH DE number 5911869
scientific article