Homology generators and relations for the ordered configuration space of a star graph

We study the ordered configuration spaces of star graphs. Inspired by the representation stability results of Church–Ellenberg–Farb for the ordered configuration space of a manifold and the edge stability results of An–Drummond-Cole–Knudsen for the unordered configuration space of a graph, we determine how the ordered configuration space of a star graph with k k leaves behaves as we add particles at the leaves. We show that, as a module over the combinatorial category FI k , o k, o , the first homology of this ordered configuration space is finitely generated by 4 4 particles for k = 3 k=3 , by 3 3 particles for k = 4 k=4 , and by 2 2 particles for k ≥ 5 k≥ 5 . Additionally, we prove that every relation among homology classes can be described by relations on at most 6 6 particles for k = 4 k=4 , at most 5 5 particles when k = 5 k=5 , at most 4 4 particles when k = 6 k=6 , and at most 3 3 particles for k ≥ 7 k≥ 7 , while proving that adding particles always introduces new relations when k = 3 k=3 . This proves that there is no finite universal presentation for the homology of ordered configuration spaces of graphs.