Result 249, Group theory

A finitely generated Eilenberg–Ganea counterexample

Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Algebraic and geometric ways of measuring a group's dimension need not agree, according to this manuscript. Its claimed example exposes a limit on using algebraic information to predict how simply a group can be represented by a space.

What changes?

The unreviewed manuscript reports a finitely generated, residually finite group with integral cohomological dimension two and geometric dimension three. Cohomological dimension measures how high nonzero cohomology, an algebraic invariant, can occur with arbitrary integral coefficient systems. Geometric dimension is the smallest dimension of a classifying space, a space with this fundamental group whose universal cover can be shrunk to a point. The claimed group has no two-dimensional classifying space, even with infinitely many cells.

What does that help mathematicians do?

The claimed example would disprove the Eilenberg–Ganea conjecture under notable restrictions. Finitely generated means that finitely many elements generate the entire group. Residually finite means that every nonidentity element remains nonidentity in some finite quotient. Thus, the dimension gap would not depend on having infinitely many generators or elements invisible in all finite quotients. Neither restriction suffices to force a two-dimensional classifying space.

Are there practical applications?

Its immediate value is foundational: it provides a test case for proposed criteria connecting algebraic dimension to topological models. Researchers seeking two-dimensional classifying spaces would need additional hypotheses beyond those satisfied here. Allowing infinitely many cells does not remove the reported obstruction, so simply enlarging a two-dimensional construction cannot overcome it.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A finitely generated counterexample to the Eilenberg–Ganea conjecture

September 23, 2026 25 pages

We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A finitely generated counterexample to the Eilenberg--Ganea conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf}{OAI:A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/249.md.

A finitely generated Eilenberg–Ganea counterexample

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Eilenberg–Ganea conjecture predicts that a group of integral cohomological dimension two has a two-dimensional classifying space. The formalization constructs a finitely generated residually finite group of cohomological dimension two that has a three-dimensional classifying space but no two-dimensional one. Thus its geometric dimension is three, contradicting the conjecture.

Comparator links

Result Comparator statement
Counterexample to the Eilenberg–Ganea conjecture EilenbergGanea.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.