The Singer conjecture in dimension four: L2-Betti numbers of closed aspherical four-manifolds vanish outside degree two
For a closed connected aspherical -manifold with fundamental group , the -Betti numbers are von Neumann dimensions of the -homology of the universal cover. Singer proposed obtaining the sign of the Euler characteristic of nonpositively curved manifolds from vanishing of -harmonic forms outside the middle degree (recorded by Dodziuk, 1979). The Singer conjecture in its aspherical form (Davis-Okun 2001, Conjecture 0.3; Kirstein-Kremer-Lück 2025, Conjecture 4.1) asserts for , which in dimension gives . In dimension four Poincaré duality reduces it to ; known cases needed extra structure (right-angled Coxeter groups, hierarchies, Kähler surfaces). Does every closed connected aspherical four-manifold satisfy for all ?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- L2-invariants; aspherical manifolds
- Posed by
- I. M. Singer (as recorded by Dodziuk); aspherical formulation by Davis and Okun and by Lück
- Year posed
- 1979
- Years open
- 47y
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if is a finite connected aspherical integral Poincaré complex of formal dimension four, with any orientation character, then for . Corollary 1.2: the same holds for every closed connected aspherical topological four-manifold, so the Singer conjecture holds in dimension four. Stated consequences: , so closed aspherical four-manifolds have nonnegative Euler characteristic; for oriented ; limits of normalized Betti numbers along residual towers of covers. Not shown: the Singer conjecture in any other dimension, or any computation of .
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'The Singer conjecture in dimension four' (September 25, 2026). It cites a companion release manuscript constructing a PD4 group without an aspherical manifold model, to which its theorem applies.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the Singer conjecture as the manuscript states it. Corollary 1.2 is exactly the four-dimensional case for closed connected aspherical topological four-manifolds, orientable or not, smoothable or not; Theorem 1.1 extends it to finite aspherical integral Poincaré complexes of formal dimension four, a formulation the manuscript attributes to Lück (Remark 11.3 of his approximation survey). There is no Lean formalization for this family. The proof, a long argument on boundaries, annuli, pretrees and path pushing, was not refereed. The conjecture in other dimensions is untouched, so the entry is the dimension-four special case.