Volume 11, issue 4 (2011)

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Configuration spaces of thick particles on a metric graph

Kenneth Deeley

Algebraic & Geometric Topology 11 (2011) 1861–1892
Abstract

We study the topology of configuration spaces ${F}_{r}\left(\Gamma ,2\right)$ of two thick particles (robots) of radius $r>0$ moving on a metric graph $\Gamma$. As the size of the robots increases, the topology of ${F}_{r}\left(\Gamma ,2\right)$ varies. Given $\Gamma$ and $r$, we provide an algorithm for computing the number of path components of ${F}_{r}\left(\Gamma ,2\right)$. Using our main tool of PL Morse–Bott theory, we show that there are finitely many critical values of $r$ where the homotopy type of ${F}_{r}\left(\Gamma ,2\right)$ changes. We study the transition across a critical value $R\in \left(a,b\right)$ by computing the ranks of the relative homology groups of $\left({F}_{a}\left(\Gamma ,2\right),{F}_{b}\left(\Gamma ,2\right)\right)$.

Keywords
topology of configuration spaces, metric graph, PL topology, topological robotics
Mathematical Subject Classification 2010
Primary: 55R80, 57Q05
Secondary: 57M15
Publication
Received: 23 October 2010
Revised: 15 March 2011
Accepted: 26 April 2011
Published: 14 June 2011
Authors
 Kenneth Deeley Department of Mathematical Sciences Durham University Durham DH1 3LE UK