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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 Fr(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 Fr(Γ,2) varies. Given Γ and r, we provide an algorithm for computing the number of path components of Fr(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 Fr(Γ,2) changes. We study the transition across a critical value R (a,b) by computing the ranks of the relative homology groups of (Fa(Γ,2),Fb(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