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

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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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The paper states

Theorem 1.1

In A Counterexample to Wall’s D(2) Problem, 6 October 2026

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There is a finite connected three-dimensional CW complex X such that

H_i(\widetilde X;\Z)=0\quad(i>2),\qquad H^3(X;M)=0\quad\text{for every }\Z[\pi_1(X)]\text{-module }M,

but X is not homotopy equivalent to any finite CW complex of dimension at most two.

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A Counterexample to Wall’s D(2) Problem

6 October 2026, current edition.

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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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14 references in A Counterexample to Wall’s D(2) Problem

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An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Author not stated. Source

Matthew Sparkes, OpenAI announces 722 mathematical discoveries in one go, 7 October 2026. Source
Johnson, who spent 25 years on Wall’s D(2) problem, says AI settled it far faster than they expected.

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