Configuration spaces of thick particles on a metric graph

Deeley, Kenneth

doi:10.2140/agt.2011.11.1861

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Configuration spaces of thick particles on a metric graph

Kenneth Deeley

Algebraic & Geometric Topology 11 (2011) 1861–1892

DOI: 10.2140/agt.2011.11.1861

Abstract

We study the topology of configuration spaces $F_{r} (Γ, 2)$ of two thick particles (robots) of radius $r > 0$ moving on a metric graph $Γ$ . As the size of the robots increases, the topology of $F_{r} (Γ, 2)$ varies. Given $Γ$ and $r$ , we provide an algorithm for computing the number of path components of $F_{r} (Γ, 2)$ . 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 $F_{r} (Γ, 2)$ changes. We study the transition across a critical value $R \in (a, b)$ by computing the ranks of the relative homology groups of $(F_{a} (Γ, 2), F_{b} (Γ, 2))$ .

Keywords

topology of configuration spaces, metric graph, PL topology, topological robotics

Mathematical Subject Classification 2010

Primary: 55R80, 57Q05

Secondary: 57M15

References

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

Volume 11, issue 4 (2011)