Wall’s manifold-realization question for PD4 groups
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Must a finitely presented integral Poincaré-duality group of dimension four with a finite classifying space be the fundamental group of a closed aspherical topological four-manifold?
Why it matters and what counts as progress
Why it matters: A counterexample would distinguish finite Poincaré-duality complexes from manifold models in Wall’s realization question.
What counts as progress: A group with its integral duality and finite classifying-space properties proved but no manifold model, or a realization theorem covering the entire stated class.
How the problem developed
- 24 Sept 2026 · Reported claim [1]
OpenAI Reported a finitely presented integral PD₄ group with a finite classifying space that is not the fundamental group of any closed aspherical topological four-manifold, giving a negative answer to the finitely presented version of Wall's question in dimension four.
AI contribution
A PD4 group without an aspherical manifold model
An OpenAI manuscript claims a finitely presented integral Poincaré-duality group of dimension four with a finite oriented aspherical Poincaré complex, but no closed aspherical topological four-manifold model.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI reports a PD4 counterexample to Wall’s manifold question