View source for Classifying spaces and geometric obstructions for Artin groups

'''Classifying spaces and geometric obstructions for Artin groups''' is the subject of three manuscripts dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). They concern the Artin ''K''(π,1) conjecture, intersections of parabolic subgroups, and the question whether every Artin group acts geometrically on a CAT(0) space. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== The original result ==
* '''The ''K''(π,1) conjecture.''' The standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. Source: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026 Harmonic heights and the Artin K(pi,1) conjecture], dated 23 September 2026.
* '''Parabolic intersections.''' Arbitrary intersections of parabolic subgroups of a finite-rank Artin group are parabolic. Source: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026 Parabolic intersections in Artin groups], dated 23 September 2026.
* '''No geometric CAT(0) action.''' An Artin group on 116 generators admits no proper, cocompact isometric action on a nonempty proper CAT(0) space; the manuscript states that this refutes the CAT(0) conjecture for Artin groups. Source: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026 An Artin group with no geometric CAT(0) action], dated 23 September 2026.
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/254.md lean/docs/254.md] states that the Lean formalization proves the contractibility of the universal cover of the Salvetti complex for every finite-rank Artin group, the 116-generator counterexample, and the parabolic intersection and closure results with some structural consequences; later corollaries of the manuscripts are not included.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-09 || frogscooper, [https://github.com/frogscooper/artin-cat0-29 frogscooper/artin-cat0-29] (GitHub) || Extension (formal) || An edited version of the CAT(0) manuscript with a shorter argument for its braid-angle estimate, showing that 11-strand braid blocks suffice; this gives an Artin group on 29 generators with no geometric CAT(0) action. A second version uses a single braid chain to give an Artin group on 11 generators. || The README states that the 29-generator theorem is checked in Lean in a cut-down copy of OpenAI's Lean project, with only the three standard axioms, and that the 11-generator version is checked modulo one stated hypothesis, a special case of van der Lek's theorem on parabolic subgroups.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]