Homology Generators and Relations for the Ordered Configuration Space of a Star Graph
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| Format: | Preprint |
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2024
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| _version_ | 1866911467838636032 |
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| author | Wawrykow, Nicholas |
| author_facet | Wawrykow, Nicholas |
| contents | 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 $k$ leaves behaves as we add particles at the leaves. We show that, as a module over the combinatorial category FI$_{k, o}$, the first homology of this ordered configuration space is finitely generated by $4$ particles for $k=3$, by $3$ particles for $k=4$, and by $2$ particles for $k\ge 5$. Additionally, we prove that every relation among homology classes can be described by relations on at most $6$ particles for $k=4$, at most $5$ particles when $k=5$, at most $4$ particles when $k=6$, and at most $3$ particles for $k\ge 7$, while proving that adding particles always introduces new relations when $k=3$. This proves that there is no finite universal presentation for the homology of ordered configuration spaces of graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homology Generators and Relations for the Ordered Configuration Space of a Star Graph Wawrykow, Nicholas Algebraic Topology Category Theory 55R80 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 $k$ leaves behaves as we add particles at the leaves. We show that, as a module over the combinatorial category FI$_{k, o}$, the first homology of this ordered configuration space is finitely generated by $4$ particles for $k=3$, by $3$ particles for $k=4$, and by $2$ particles for $k\ge 5$. Additionally, we prove that every relation among homology classes can be described by relations on at most $6$ particles for $k=4$, at most $5$ particles when $k=5$, at most $4$ particles when $k=6$, and at most $3$ particles for $k\ge 7$, while proving that adding particles always introduces new relations when $k=3$. This proves that there is no finite universal presentation for the homology of ordered configuration spaces of graphs. |
| title | Homology Generators and Relations for the Ordered Configuration Space of a Star Graph |
| topic | Algebraic Topology Category Theory 55R80 |
| url | https://arxiv.org/abs/2401.13821 |