Wall's manifold-realization question for finitely presented Poincare duality groups

Theorem 1.1: there is a finitely presented group $G$ with a finite oriented four-dimensional aspherical Poincare complex as classifying space, $H^i(G;\mathbb ZG)=0$ for $i\ne4$ and $H^4(G;\mathbb ZG)\cong\mathbb Z$ with trivial action, such that no closed aspherical topological 4-manifold has fundamental group $G$ . This answers Wall's question negatively for finitely presented integral $PD$ groups with finite classifying space. Not shown: a counterexample in dimension 3 or in dimensions at least 5, or anything about smooth or homology-manifold realizations beyond the topological category.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 24, 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.

Canonical aliases: Wall's manifold-realization question for finitely presented Poincare duality groups · Wall's question on PD groups (dimension 4)

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VibeMathed record: Wall's manifold-realization question for finitely presented Poincare duality groups

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'A PD4 group without an aspherical manifold model' and its introduction were read against Wall's Problem G2 in the finitely presented form the paper cites. The proof (a marked chamber with a filling obstruction, Davis reflection, Farrell-Jones for CAT(0) groups, Kasprowski-Land 4-dimensional surgery and a direction-control argument) was not refereed. No Lean formalization. The result is conditional on the unreviewed marked tensor obstruction companion in the same release. Scope the paper itself states: a single dimension-four counterexample, which refutes the question quantified over all dimensions; it says nothing about dimension three (the PD3 conjecture) or about dimensions five and up.

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Input (family 305): A PD4 group without an aspherical manifold model

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Companion: A marked tensor obstruction to four-dimensional disk embedding

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paper link supplied by the VibeMathed registry.

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Davis, Poincare duality groups (survey, Question 3.4), author's version

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problem-record link supplied by the VibeMathed registry.

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Input (family 305): A PD4 group without an aspherical manifold model evidence for this event

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Davis, Poincare duality groups (survey, Question 3.4), author's version evidence for this event

VibeMathed record: Wall's manifold-realization question for finitely presented Poincare duality groups evidence for this event

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