Wall's D(2) problem

Theorem 1.1: there is a finite connected three-dimensional CW complex $X$ with $H_i(\widetilde X;\mathbb Z)=0$ for $i>2$ and $H^3(X;M)=0$ for every $\mathbb Z[\pi_1X]$ -module $M$ that is not homotopy equivalent to any finite complex of dimension at most two. The obstruction: on a finite 2-complex, a rank-one local system with vanishing twisted $H_2$ is trivial on every finite-order element of $\pi_1$ . $\pi_1(X)$ is infinite. It does not settle the D(2) problem for finite fundamental groups, the relation gap problem, or the Eilenberg-Ganea problem.

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Prior state unknown→disproved

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Occurred: Oct 6, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Wall's question in the formulation the manuscript takes from Wall (1965, 1979) and Hofmann-Nicholson: finiteness of both complexes, no asphericity hypothesis. The construction is a Quillen plus construction on a five-generator four-relator presentation complex (finite D(2) by Mannan's theorem), with an obstruction from rank-one twisted second homology proved by a Howie-style cyclic tower. The example has infinite fundamental group, so the problem for finite groups, where most work concentrated, remains open. The proof was not refereed. No Lean formalization exists for this family.

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Wall's D(2) problem

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