Configuration spaces of thick particles on graphs

Abstract

In this thesis, we study the topology of configuration spaces of particles of variable radius r>0 moving on a metric graph. Our main tool is a piecewise linear (PL) Morse-Bott theory for affine polytope complexes, which extends the Morse theory for such complexes introduced by M. Bestvina and N. Brady.
As the size parameter r increases, the topological properties of the corresponding configuration spaces vary. We show that there are finitely many critical values where the homotopy type of these spaces changes, and describe these critical values in terms of metric properties of the graph. This provides an upper bound on the number of critical values in terms of the metric data. Moreover, we apply PL Morse-Bott theory to analyse the change in homotopy type of configuration spaces of thick particles when the radius transits a critical value.
Provided that r is sufficiently small, we show that the thick particle configuration space is homotopy equivalent to the familiar configuration space of zero-size points on the graph. We also investigate discrete models for configuration spaces of two thick particles. Moreover, given a metric graph and the size parameter r, we provide an algorithm for computing the number of path-components of the configuration space of two thick particles.