The K(pi,1) conjecture for Artin groups of finite rank
For a Coxeter matrix on a finite set , the Artin group is the fundamental group of the quotient of the complexified hyperplane complement of the Coxeter group by , and of the finite Salvetti complex , which has one cell per spherical subset of . Deligne (1972) proved asphericity when is finite; Charney-Davis proved the FC and dimension-two cases, Paolini-Salvetti the affine case, Huang-Przytycki dimension three. The conjecture, attributed to Arnol'd, Brieskorn, Pham and Thom and listed as Problem 1 in Charney's Artin-group problem list, asserts asphericity in general. Is the Salvetti complex (equivalently the hyperplane-complement quotient) a for every Coxeter matrix?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric group theory, Artin groups
- Posed by
- Attributed to Arnol'd, Brieskorn, Pham and Thom; stated in Brieskorn's Bourbaki seminar (LNM 317, 1973); Problem 1 in R. Charney, Problems related to Artin groups
- Year posed
- 1973
- Years open
- 53y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 60 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every Coxeter matrix on a finite set, with arbitrary finite or infinite labels and connected or disconnected diagrams, the universal cover of the full Salvetti complex is contractible, so for . The proof contracts the poset of spherical cosets by a well-ordered vertex filtration, using a framed zigzag-algebra braid action whose layer multiplicities define a harmonic height. Consequences recorded: torsion-freeness, the center , and Dobrinskaya's monoid equivalence. Infinite rank is not covered.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the conjecture as cited (Brieskorn 1973, Charney-Davis 1995). The categorical and harmonic-height arguments were not refereed. Lean-checked on the release's Comparator challenge HarmonicArtin (declaration OAI.HarmonicArtin.salvetti_cover_contractible, listed in lean/formalization.yaml). Its statement was read here: for every Mathlib CoxeterMatrix on a finite type it asserts ContractibleSpace of the geometric realization of the nerve of a face preorder on pairs (Artin element, spherical subset). That is a combinatorial model of the Salvetti universal cover; the identification of this model with the topological universal cover is classical and is not part of the formal statement. Not rebuilt here.