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A Counterexample to Wall's D(2) Problem
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Abstract
We give a negative answer to Wall's finite $D(2)$ problem. We construct a finite connected three-dimensional CW complex satisfying the $D(2)$ finiteness condition but not homotopy equivalent to any finite CW complex of dimension at most two. The example has infinite fundamental group.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- Pyronis-Inc, WallD2: Lean formalization and proof structure visualization of counterexample to Wall's D(2) conjecture (GitHub, Lean 4/Mathlib, README states work since 2026-10-14). The README calls this a conditional formalization of family 321: it proves perfectness of the normal subgroup (Lemma 4.1), the Fox-row independence behind Lemma 4.2, and that the image of x1 has order 2. Mannan's finite plus construction (
plus_assembly) and the Howie-type tower obstruction (howie_tower) are left as named axioms.
Possibly relevant literature (found by the agent, not checked)
- W. H. Mannan, Realizing algebraic 2-complexes by cell complexes (arXiv:2310.07722; Math. Proc. Cambridge Philos. Soc. 146 (2009) 671–673). The abstract says the D(2) property and the realization property are equivalent for all finitely presented groups, which bears on the realization viewpoint the introduction attributes to Johnson.
- X. Jin, Y. Su, L. Yu, On Homology Roses and the D(2)-problem (arXiv:1311.1051; Sci. China Math. 58 (2015) 1753–1770). It discusses looking for D(2) counterexamples among F-homology roses and F-acyclic spaces, a search related to the paper's plus-construction approach along a perfect normal subgroup.
Digests and expositions
- Claude Opus 5.5 / Open Mathematics Courses, A counterexample to Wall's D(2) problem (Open Mathematics Courses, October 2026). Four lessons expand the paper's twisted homology, cyclic-tower obstruction, finite plus construction and explicit counterexample, with exercises and solutions; the source states that independent review is incomplete. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 321
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Counterexample-to-Walls-D2-Problem-October-6-2026,
author = {{OpenAI}},
title = {{A Counterexample to Wall's $D(2)$ Problem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf}{OAI:A-Counterexample-to-Walls-D2-Problem-October-6-2026}},
year = {2026}
}Family 321: A counterexample to Wall's finite D(2) problem
Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.
This is the only paper in this family.