The Configuration Space of Two Particles Moving on a Graph

Abstract

In this thesis we study the configuration space, F (Γ, 2), of two particles moving without collisions on a graph Γ with a view to calculating the Betti numbers of this space. We develop an intersection theory for cycles in graphs inspired by the classical intersection theory for cycles in manifolds and we use this to develop an algorithm to calculate the second Betti number of F (Γ,2) for any graph Γ. We also use this intersection theory to provide a complete description of the cohomology algebra H ^*(F (Γ, 2), Q) for any planar graph Γ and to calculate explicit formulae for the Betti numbers of F (Γ, 2) when Γ is a complete graph or a complete bipartite graph. We also investigate the generators of group H_2 (F (Γ, 2), Z) and show that for any planar graph this group is entirely generated by tori induced by disjoint cycles in the graph. For non-planar graphs the situation is more complicated and we show that there can exist a generator of H_2 (F (Γ, 2), Z) which is not the fundamental class of a surface embedded in the space F (Γ, 2)