A counterexample to Wall's finite D(2) conjecture at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	donuts, coffee cups	Levels:	1
Category:	Topology	Lean version:	not yet
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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.

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LEVEL 1 A Counterexample to Wall's D(2) Problem

released 2026-10-06 | 2 theorems · 3 lemmas · 7 proofs · 4,498 words | PLAY LEVEL 1 » (pdf)

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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