Connectivity of Cluster-Tilting Exchange Graphs

Definitions and mathematical statement

A Frobenius category is an exact category $\mathcal C$ with enough projective objects and enough injective objects, in which the projective and injective objects coincide. Its stable category $\underline{\mathcal C}$ is the quotient by morphisms factoring through projective-injective objects; it is triangulated, with suspension denoted by $\Sigma$. Under the usual additional convention that this stable category is linear over a field $k$ and has finite-dimensional morphism spaces, the stably $2$-Calabi--Yau condition means that there are bifunctorial isomorphisms \[ \operatorname{Hom}_k\!\left(\operatorname{Hom}_{\underline{\mathcal C}}(X,Y),k\right) \cong \operatorname{Hom}_{\underline{\mathcal C}}(Y,\Sigma^2X) \qquad (X,Y\in\underline{\mathcal C}). \] The supplied record does not specify the field, finiteness hypotheses, or precise convention for this condition; the displayed interpretation is conditional on that usual setting.

For an object $T$, let $\operatorname{add}(T)$ consist of the objects isomorphic to direct summands of finite direct sums of copies of $T$. A standard cluster-tilting convention requires \[ \operatorname{add}(T) =\{X\in\mathcal C:\operatorname{Ext}^1_{\mathcal C}(T,X)=0\} =\{X\in\mathcal C:\operatorname{Ext}^1_{\mathcal C}(X,T)=0\}, \] with $\operatorname{add}(T)$ functorially finite: every object admits both a right and a left $\operatorname{add}(T)$-approximation. A right approximation $A\to X$, with $A\in\operatorname{add}(T)$, is a morphism through which every morphism from an object of $\operatorname{add}(T)$ to $X$ factors; a left approximation is defined dually. The exact cluster-tilting convention used in the report is not included in the supplied excerpt.

In a Krull--Schmidt setting, a basic object is a finite direct sum of pairwise nonisomorphic indecomposable objects. Under the usual basic-object interpretation, the full exchange graph $\mathcal E(\mathcal C)$ has as vertices the isomorphism classes of all basic cluster-tilting objects. An undirected edge represents one admissible cluster-tilting mutation, replacing a single indecomposable summand while retaining the other summands. The supplied record does not give the exchange-sequence or approximation conditions defining admissible mutation, so this description does not assert that every replacement or every summand admits mutation.

Question (under these explicitly stated conventions). For every stably $2$-Calabi--Yau Frobenius category $\mathcal C$ in the intended class, is the full graph $\mathcal E(\mathcal C)$ connected? Equivalently, for every pair of its vertices represented by $T,T'$, do there exist $m\in{\mathbb{Z}}_{\geq 0}$ and cluster-tilting objects \[ T=T_0,T_1,\ldots,T_m\cong T' \] such that each consecutive pair is related by one mutation?

Here the full graph includes all cluster-tilting objects under the chosen vertex convention, without restricting to objects already reachable from a fixed object. The intended ambient hypotheses, vertex convention, and mutation convention require confirmation from the report. Existence of a cluster-tilting object is also not explicitly assumed in the supplied statement: the pairwise-path formulation is vacuous for an empty vertex set, whereas a convention requiring a connected graph to be nonempty would require a separate existence hypothesis.

Short English statement

The question asks whether any two cluster-tilting objects in a stably $2$-Calabi--Yau Frobenius category can be joined by finitely many mutations in the full exchange graph. The supplied record labels this problem open. Its precise ambient assumptions, basic-object convention, mutation rules, and treatment of categories without cluster-tilting objects remain to be confirmed.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006600$, title Connectivity of Cluster-Tilting Exchange Graphs, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2026.
• Representation Theory of Quivers and Finite-Dimensional Algebras (2026), Oberwolfach Reports, pp. 345--432, DOI 10.4171/owr/2026/6: https://doi.org/10.4171/owr/2026/6. The supplied original excerpt identifies p. 413 and describes the vertices as cluster-tilting objects and the edges as mutations.
• Supplied literature source: Oberwolfach Report 6/2026, https://doi.org/10.4171/owr/2026/6.
• Supplied literature source: Bruestle--Yang, Ordered exchange graphs, https://arxiv.org/abs/1302.6045.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that the source explicitly treats connectivity of the full exchange graph as open, while important subclasses are known to be connected. This is supplied dated curation, not an independently verified literature assessment; the subclasses and their precise hypotheses are not identified in the record.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not independent verification performed for this draft.