The Baum-Connes Conjecture (coefficient-free, reduced)
For a countable discrete group , the Baum-Connes assembly map sends equivariant -homology of the classifying space for proper actions to the -theory of the reduced group -algebra. Baum and Connes conjectured that it is an isomorphism. It was proved for a-T-menable groups (Higson-Kasparov) and hyperbolic groups (Mineyev-Yu, Lafforgue); Higson-Lafforgue-Skandalis disproved the version with coefficients, but the coefficient-free group conjecture stayed open. Is an isomorphism for every countable discrete group ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Operator algebras; noncommutative geometry; K-theory
- Posed by
- Paul Baum and Alain Connes (preprint circulated 1982, published 2000); reformulated with Nigel Higson (1994)
- Year posed
- 1982
- Years open
- 44y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 60 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Three counterexamples. (1) A finitely generated group with a projection in of irrational canonical trace, so is not surjective. (2) A finitely generated torsion-free group whose degree-zero reduced assembly map kills a class of infinite order, so rational injectivity fails (the reduced form of the strong Novikov conjecture). (3) The Kadison-Kaplansky counterexample also gives a torsion-free group with a class outside the assembly image. Not shown: failure of maximal assembly, a finitely presented counterexample, or failure of the classical Novikov conjecture.
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). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the main theorems of the three family manuscripts were read against the coefficient-free reduced conjecture as stated by Baum-Connes-Higson. The constructions were not refereed. No Lean formalization is listed. The surjectivity failure rests on Luck's theorem that traces of assembly classes are rational. The groups are finitely generated; finite presentation is not claimed. Maximal assembly and the classical Novikov conjecture are not disproved.