A finitely generated counterexample to the Eilenberg-Ganea conjecture

A finitely generated residually finite group with cohomological dimension 2 and geometric dimension 3: the Eilenberg-Ganea conjecture fails. The group is the kernel of the height homomorphism of the right-angled Artin group of an acyclic flag complex whose fundamental group maps nontrivially into the unit quaternions; a degree argument on finite quotients and compactness show that a two-dimensional classifying space would force small quaternion labellings that are trivial around triangles to be trivial around every closed path, and a slowly varying map of the level complex into the flag complex produces a labelling that is not. The course is an exposition of OpenAI's proof (2026).

Eleven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Cohomological and geometric dimension
2. Word maps on unit quaternions and degree
3. Aligned presentations of the cycle module
4. Exact counts and a quadratic estimate
5. Finite quotients and the comparison theorem
6. Right-angled Artin groups and residual finiteness
7. The cube complex and its universal cover
8. The level complex and its acyclicity
9. An acyclic seed with quaternion monodromy
10. Traces and direction vectors
11. Small transport and the counterexample

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Explicit CW models and the first Hurewicz map (Complex threefolds from marked torus families)
• Homotopy fibres and the Serre spectral sequence · Characteristic classes
• Regular values and degree modulo two (Borsuk's conjecture fails in dimension nine)
• Cell complexes and homotopy equivalences (Cell complexes, fundamental groups and covering spaces)
• Fundamental groups of graphs and two-complexes (Cell complexes, fundamental groups and covering spaces)
• Universal covers and cellular chains (Cell complexes, fundamental groups and covering spaces)
• Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian · Foundations of von Neumann algebras
• Cone pictures and asphericity (A torsion-free group algebra with zero divisors)
• Group algebras and zero divisors (A torsion-free group algebra with zero divisors)
• Torsion-freeness and the zero divisors (A torsion-free group algebra with zero divisors)
• Modules over group rings, projective resolutions and group cohomology (developed in the first lesson).
• Fundamental groups, covering spaces and cellular homology (core course Algebraic Topology), the Hurewicz and Whitehead theorems from earlier programme lessons.
• Unit quaternions and the mapping degree (developed in the second lesson).