On the Homology of Configuration Spaces of Graphs

In this thesis we deal with unordered configuration spaces of finite graphs, and in particular with their homology. The main tool for the computation is the Świątkowski complex. A class in the n-degree homology of the k-th configuration space can be pictured as k particles on the graph, n of which are moving at the same time without collisions. Performing these moves locally in disjoint subgraphs at the same time gives rise to higher degree classes, said toric classes. In this work, the focus is on the description of the homology in low degree. We describe generators for the first homology as a module on the ring of polynomials in the edge of the graph. We also report an explicit formula given in terms of combinatorial invariants of the graph. Lastly, we prove that some well-known relations the generators are subject to form a complete set and they give a presentation of the first homology. We treat a result about the generators of the second homology in the planar case and we furnish two new classes that need to be considered in general. Studying these “exotic” classes, we found that the homology of the configuration space of a graph enjoy functoriality for the contraction of an edge.