Finite Generation of Graph-Configuration Homology Functors

Definitions and mathematical statement

The supplied question concerns finite generation of graph-configuration homology as a functor. Its precise formulation requires a category $\mathcal C$ of graphs, a coefficient ring $R$, a particle number $k$, a convention for configuration spaces, and compatible maps on homology. These data are not specified in the supplied record.

For a topological realization of a graph $G$ and a nonnegative integer $k$, the usual ordered configuration space is \[ \{(x_1,\ldots,x_k)\in G^k : x_i\ne x_j\text{ whenever }i\ne j\}. \] The usual unordered configuration space is its quotient by the symmetric group permuting coordinates. The notation $\operatorname{Conf}_k(G)$ in the source does not specify which convention is intended. The allowed graphs and graph morphisms must also be specified so that the displayed assignment actually defines a functor.

If the intended meaning is finite generation of a functor $F:\mathcal C\to R\text{-}\mathrm{Mod}$, its usual module interpretation is the existence of finitely many objects $G_1,\ldots,G_m$ and elements $v_i\in F(G_i)$ such that, for every object $G$, \[ F(G)=\operatorname{span}_R\{F(f)(v_i):1\le i\le m,\ f\in\operatorname{Hom}_{\mathcal C}(G_i,G)\}. \] This definition is conditional: the record does not establish that this is the precise finite-generation notion used in the cited result.

Subject to recovering the hypotheses and conventions of that result, the supplied extension question is: for the same fixed particle number $k$, graph category, coefficients, and functorial structure, is \[ F_{q,k}:G\longmapsto H_q(\operatorname{Conf}_k(G);R) \] finitely generated for every integer $q\ge0$? Here $R$ records an unresolved coefficient choice, rather than an additional assertion about arbitrary coefficients. The record reports an existing finite-generation result in degrees $q=0,1$ and asks whether it extends to all nonnegative degrees. It does not specify the quantification over $k$, or whether any uniformity in $k$ or $q$ is intended.

The supplied dated assessment also reports finite generation for certain FI-graph families, but does not identify those families or their hypotheses. Those qualifications must be recovered before a fully determined statement can be made.

Short English statement

The question asks whether the reported finite-generation theorem for graph-configuration homology in degrees zero and one extends to every nonnegative homological degree, with the original hypotheses retained. The graph category, morphisms, coefficients, configuration-space convention, particle-number quantifiers, and relevant FI-graph families are missing, so this is a conditional statement requiring mathematical review.

Sources

• Supplied UnsolvedMath local registry record, permanent ID $30003704$, source problem code catalogue entry, entitled Finite Generation of Graph-Configuration Homology Functors. The record lists proposed year 2018 and identifies no individual proposer.
• Supplied report citation: Topology of Arrangements and Representation Stability (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/2, pp. 43--123. https://doi.org/10.4171/owr/2018/2. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no exact passage or theorem specifying the omitted hypotheses is supplied.
• Supplied literature reference: Ramos, FI-algebras and graph configuration spaces. https://arxiv.org/abs/1805.05316.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports finite generation in homological degrees at most one and for the stated FI-graph families, while reporting the general higher-degree question as open. This is supplied dated curation, not an independently verified literature assessment. Its time-sensitive status and cited mathematical claims require independent verification.