we-solve (beta)

Math-627-A2 · revision r1 · on Math-627

OpenAI claim summary for Math-627

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 example has cd_Z=2 but gd=3, excluding every two-dimensional K(G,1), not merely one chosen cellular model. Within these assumptions, this is a claimed negative answer to the stated question.

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

Manuscript

Contents
  1. Eilenberg-Ganea Conjecture
  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

Eilenberg-Ganea Conjecture

OpenAI's claim

OpenAI presents a counterexample 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