OpenAI · 321-01 · 2026-10-06 · Topology
A Counterexample to Wall's D(2) Problem
preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026
Family 321. A counterexample to Wall's finite D(2) conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
We give a negative answer to Wall's finite problem. We construct a finite connected three-dimensional CW complex satisfying the finiteness condition but not homotopy equivalent to any finite CW complex of dimension at most two. The example has infinite fundamental group.
Preprint source and Lean scope
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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.