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Math-1811-A1 · revision r1 · on Math-1811

OpenAI claim summary for Math-1811

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 manuscript constructs a finite three-complex satisfying the local-coefficient cohomology condition and excludes every finite two-complex homotopy model. This page allows a two-complex with infinitely many cells, whose nonexistence is not claimed by the paper. Consequently the constructed example is not asserted to meet the target's stronger nonexistence requirement. It is a related result on the finite-target formulation.

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

Manuscript

Contents
  1. Questions in Geometric Group Theory — Q 10.2
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Questions in Geometric Group Theory — Q 10.2

OpenAI's claim

Wall's finite D$2$ question is claimed to have a negative example: a finite three-dimensional complex with infinite fundamental group satisfies D$2$ but has no finite two-dimensional homotopy model. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims a finite connected CW complex $X$ of dimension three such that, for its universal cover $\widetilde X$, \[ H_i(\widetilde X;\mathbb Z)=0\quad(i>2),\qquad H^3(X;M)=0\quad\text{for every }\mathbb Z[\pi_1X]\text{-module }M, \] yet $X$ is without a homotopy equivalent finite CW model in dimension two or below. The cohomology condition ranges over all local coefficient modules, and no asphericity assumption is made.

Outline of the argument

Manuscript 1 (family 321, 2026-10-06): A rank-one twisted-homology obstruction is proved by repeatedly lifting a fixed finite source through cyclic-cover image towers, whose simplex growth cannot continue. A five-generator presentation has a finitely normally generated perfect subgroup and a quotient involution detected by a complex character. Its explicit boundary matrix gives vanishing twisted second homology; finite plus construction preserves this while contradicting the necessary two-dimensional character condition.

Formal verification

Family 321 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources