Homology Generators and Relations for the Ordered Configuration Space of a Star Graph

TL;DR

The paper addresses the homology of ordered configuration spaces on star graphs Γ_k as the number of particles grows, using FI_{k,o}-modules and a Mayer–Vietoris framework to capture how new particles at the leaves generate and relate homology classes. By modeling F_n(Γ_k) with Lütgehetmann’s cube complex and analyzing a tailored open cover, the authors prove that H_i(F_•(Γ_k)) forms a finitely generated FI_{k,o}-module for k≥3, with explicit generation degrees for i=1 depending on k, and they establish finite presentability for k≥4 (with explicit degrees) while showing k=3 fails to be finitely presentable. The work also demonstrates that no finite universal presentation exists for the homology of ordered graph configuration spaces, highlighting intrinsic complexity in the ordered setting compared to unordered graphs and manifolds. Overall, the study extends representation stability phenomena to ordered graph configurations, introduces FI_{d,o} as a robust framework, and provides concrete generation/presentation data via a detailed Mayer–Vietoris analysis.

Abstract

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.

Figures (21)

• Figure 1: The star graph $\Gamma_{3}$.
• Figure 2: The action of the inclusion map on a point in $F_{3}(T^{\circ})$, where $T^{\circ}$ is the once-punctured torus. This map is induced by a map on the torus that retracts the torus away from the puncture; note that in the coimage of this retraction, which is a tubular neighborhood of the puncture, one can embed a copy of $\mathbb{R}^{2}$ (see the dotted circle). To get a point in $F_{4}(T^{\circ})$ from a point in $F_{3}(T^{\circ})$ first apply this retraction to the three particles constituting this point; then add a new particle centered at the image of the origin of the embedded $\mathbb{R}^{2}$.
• Figure 3: The composition of two FI$_{2,o}$-morphisms. Here the ordering is denoted by the subscripts.
• Figure 4: A decomposition of the FI$_{2,o}$-morphism depicted on the right of Figure \ref{['compFIO2morph']} into FI$_{2,o}$-morphisms of the form $\sigma\in S_{5}$ and $(\iota_{l-1}, c_{j_{l}}, o_{j_{l}})$.
• Figure 5: The action of $\iota_{4,1}$ on a point in $F_{4}(\Gamma_{3})$. From now on we will suppress the naming of the particles by natural numbers in our figures, and instead use colors to identify them. These are not the colors associated to the FI$_{k,o}$-modules, which are the edges of $\Gamma_{k}$.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Example 2.1
• Theorem 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Lemma 2.5
• proof