Wall's manifold-realization question for finitely presented Poincare duality groups
A group is an integral Poincare duality group of dimension if has a finite projective -resolution and vanishes for and is for . Fundamental groups of closed aspherical -manifolds are groups. Wall asked whether the converse holds (Problem G2, 1979). Davis built groups that are not finitely presented for every , which a closed manifold cannot realize, so the question is asked for finitely presented groups (Davis's survey, Question 3.4; Luck's survey, Conjecture 7.29). Is every finitely presented integral Poincare duality group the fundamental group of a closed aspherical manifold?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Geometric group theory; aspherical manifolds and Poincare duality groups
- Posed by
- C. T. C. Wall, List of problems, Problem G2, in Homological Group Theory (LMS Lecture Notes 36, 1979); finitely presented form in Davis's survey (Question 3.4) and Luck's (Conjecture 7.29)
- Year posed
- 1979
- Years open
- 47y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 42 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is a finitely presented group with a finite oriented four-dimensional aspherical Poincare complex as classifying space, for and with trivial action, such that no closed aspherical topological 4-manifold has fundamental group . This answers Wall's question negatively for finitely presented integral groups with finite classifying space. Not shown: a counterexample in dimension 3 or in dimensions at least 5, or anything about smooth or homology-manifold realizations beyond the topological category.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (September 24, 2026) imports its geometric and algebraic obstruction package from the family's companion 'A marked tensor obstruction to four-dimensional disk embedding' and proves the transfer to topological fillings itself.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'A PD4 group without an aspherical manifold model' and its introduction were read against Wall's Problem G2 in the finitely presented form the paper cites. The proof (a marked chamber with a filling obstruction, Davis reflection, Farrell-Jones for CAT(0) groups, Kasprowski-Land 4-dimensional surgery and a direction-control argument) was not refereed. No Lean formalization. The result is conditional on the unreviewed marked tensor obstruction companion in the same release. Scope the paper itself states: a single dimension-four counterexample, which refutes the question quantified over all dimensions; it says nothing about dimension three (the PD3 conjecture) or about dimensions five and up.