The Eilenberg-Ganea conjecture

Theorem 1.1: there is a finitely generated residually finite group $G$ with $\mathrm{cd}\,G=2$ and $\mathrm{gd}\,G=3$ ; no two-dimensional $K(G,1)$ exists regardless of the number of cells. $G$ is a Bestvina-Brady group (the kernel of the height map of a right-angled Artin group); it is of type FL and FP $_\infty$ but not finitely presented. The obstruction compares an acyclic two-dimensional level complex with any hypothetical aspherical presentation, using small SU(2) labelings and a degree argument. It does not give a finitely presented counterexample, and by itself it does not decide Whitehead's asphericity conjecture.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Eilenberg-Ganea conjecture · Eilenberg-Ganea conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The Eilenberg-Ganea conjecture

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture as the paper states it (with the 1957 reference); it claims a finitely generated, residually finite group $G$ , the height kernel of a right-angled Artin group over an acyclic flag triangulation of the presentation complex of $\langle x,y\mid x^2=y^5,x^2=(xy^{-1})^3\rangle$ , with $\mathrm{cd}\,G=2$ and $\mathrm{gd}\,G=3$ . The proof was not refereed. The challenge is not in the formalization catalogue (lean/formalization.yaml); it is found through lean/docs/249.md, and its solution module OAI.Topology.EilenbergGanea.Main exists at the pinned commit. The statement ComparatorChallenges/EilenbergGanea.lean was read here: it defines the group as the kernel of the height map on the Artin group of an explicit seed graph and asserts it is finitely generated, residually finite, has cohomological dimension 2 (projective dimension of the trivial module), has a 3-dimensional classifying CW space, and has no 2-dimensional one in any universe, with no bound on cells. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.