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Math-1993-A1 · revision r1 · on Math-1993

OpenAI claim summary for Math-1993

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The main theorem provides exactly the requested existence example; finite generation and residual finiteness are additional properties. Within these assumptions, this is a claimed affirmative answer to the stated question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Questions in Geometric Group Theory — Q 2.3
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Questions in Geometric Group Theory — Q 2.3

OpenAI's claim

OpenAI presents a proof in the selected manuscript. A finitely generated residually finite group is claimed to have integral cohomological dimension two but no two-dimensional classifying space.

Mathematical statement of the claimed result

Result 1.
Write $\operatorname{cd}$ and $\operatorname{gd}$ for integral cohomological dimension and the minimum dimension of a classifying space. There is a group $G$, residually finite and finitely generated, for which \[ \operatorname{cd}G=2,\qquad\operatorname{gd}G=3. \] It takes a height kernel in a right-angled Artin group built from an acyclic flag triangulation of the presentation complex \[ \langle x,y\mid x^2=y^5,\ x^2=(xy^{-1})^3\rangle. \] Its nonexistence conclusion applies to every two-dimensional $K(G,1)$, including those with infinitely many cells.

Outline of the argument

Degree estimates on conjugation-invariant parts of word fibers obstruct arbitrary aspherical presentations. Exact boundary-chain counts and quadratic bounds compare any candidate low-dimensional model with the constructed group, forcing geometric dimension three despite the shorter cohomological resolution.

Formal verification

For family 249, the documented example is finitely generated and residually finite, has integral cohomological dimension two, and has geometric dimension three. The listed Comparator files are EilenbergGanea.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources