321. A counterexample to Wall's finite D(2) problem

A counterexample to Wall's finite D(2) problem

1 paper

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.

Papers 1

A Counterexample to Wall's D(2) Problem

OpenAI · October 6, 2026

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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@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}
}

A counterexample to Wall's finite D(2) problem

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A Counterexample to Wall's D(2) Problem

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