Wall's D(2) problem
A finite connected CW complex with fundamental group satisfies the condition if for and for every finitely generated -module . Wall showed such has a finite three-dimensional model and asked whether the three-cells can be removed. Stably the answer is yes (Cohen), and it is yes for many finite groups (Johnson, Hambleton, Hofmann-Nicholson); candidate counterexamples (Cohen-Dyer, Bridson-Tweedale) stayed unresolved or conditional on relation-gap assertions. Wall's finite problem: is every finite complex homotopy equivalent to a finite CW complex of dimension at most two?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Algebraic topology; homotopy types of 2-complexes
- Posed by
- C. T. C. Wall
- Year posed
- 1965
- Years open
- 61y
- Solved
- 2026-10-06
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 46 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is a finite connected three-dimensional CW complex with for and for every -module that is not homotopy equivalent to any finite complex of dimension at most two. The obstruction: on a finite 2-complex, a rank-one local system with vanishing twisted is trivial on every finite-order element of . is infinite. It does not settle the D(2) problem for finite fundamental groups, the relation gap problem, or the Eilenberg-Ganea problem.
What the AI did
The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, 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, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Wall's question in the formulation the manuscript takes from Wall (1965, 1979) and Hofmann-Nicholson: finiteness of both complexes, no asphericity hypothesis. The construction is a Quillen plus construction on a five-generator four-relator presentation complex (finite D(2) by Mannan's theorem), with an obstruction from rank-one twisted second homology proved by a Howie-style cyclic tower. The example has infinite fundamental group, so the problem for finite groups, where most work concentrated, remains open. The proof was not refereed. No Lean formalization exists for this family.