Alphabeta MathArticle items · 08 — Artin–Tits group, pure subgroup and positive lifts
Presents 27 items from the article and its immediate prerequisites, covering dependency levels 6–9.
1 · Prerequisites
- Coxeter-system, roots, length and weak-order conventions
- Classical complexified hyperplane complement
- Classical free action and covering exact sequence
- Reduced-word lifts and the prefix-order convention
- Independent Artin fundamental-group identification
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Fundamental groupoid of a space
- Sign facets of a cone arrangement
- Facet incidence and localized chamber signs
- The incident facet and chamber poset
- The Salvetti order complex and its finite CW case
- Complexified complements in an open convex cone
- Left group actions, transitive actions, and faithful actions
- A simplicial map and its geometric realization
- Antipodal transformation and finite-type Garside structure
- Gallery fractions, fundamental groupoid and asphericity
- Categorical Garside structures and simple arrows
- Path presentations, congruences and formal inverses
- Eilenberg--Mac Lane space
- Positive gallery category, cancellation and length
- Independent relative chamber labelling input
- Finite parabolic longest elements and disconnected factors
- The full-cone and interior arrangements have identical chambers
- Tits cone geometry with its exact interior boundary
- Locally finite arrangements and relative face terminology
- Tits Cone Intersections and Applications · Definitions 1.8, 1.16 and 1.19 and Theorem 1.20(1)(a),(c),(d) and (2)(a), printed pp. 6, 8–9 (PDF pp. 30, 32–33)
- Set-sized groupoids, functors and composition conversion
- The quotient group G/N and coset product (gN)(hN)=ghN
- Unique lifts, endpoint fibres and characteristic subgroups
- Explicit connected cover associated with a vertex subgroup
- Pointed classification of connected groupoid coverings
- Normal coverings and the convention-correct deck quotient
- Universal groupoid covering from the trivial subgroup
- Normal subgroup: invariance under conjugation
- Group isomorphisms, automorphisms and the set Aut(G)
- Connected small groupoids and vertex groups
- Group presentation by generators and relations
- The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups
- A free group action has no nonidentity element fixing a point
- Equivalence relation, equivalence class, and the quotient set A/∼
- Monoid homomorphism and group homomorphism
- The kernel and image of a group homomorphism
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- Equivariant maps and isomorphisms of group actions
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- Luis Paris, K(pi,1) conjecture for Artin groups, Annales de la Faculté des sciences de Toulouse 23 (2014), 361–415 · §1, arrangement convention p.364, Vinberg system and its cone/arrangement pp.365–366, complement p.368; §3.1, construction pp.385–386 and Theorem 3.1 p.386 (printed pagination).
- Covariant functor, identity functor, composite functor, and contravariant functor
- Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
- Normalizer quotients, kernels and conditional surjectivity
- The intersection of a nonempty family of subgroups of G is a subgroup of G
- The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
- Semigroup and monoid
- Parabolic coordinate subspaces and fundamental faces
- Reflecting hyperplanes, fundamental chamber and Tits cone
- Annihilators, relative hyperplanes and complexification
- Parabolic root subsystems and unique coset minima
- Root signs, inversions and descent translation
- Connected component notation [I,a]
- The root-length criterion and faithfulness of the canonical reflection representation
- Basic Tits-cone properties
- Coverings preserve the higher-homotopy obstruction to K(pi,1)
- Universal covering spaces
- Higher homotopy group by based cubes
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications · §§1.1–1.3, printed pp.3–5 / PDF pp.27–29
- Luis Paris, K(pi,1) conjecture for Artin groups · Introduction, printed pp.361–362; §1, pp.363–365; §3.1 Theorem 3.1, p.386; §3.2 Theorem 3.3 and Corollary 3.4, pp.393–394; §3.3 quotient cell construction, p.398, and Theorem 3.10, p.400; §4.1 Proposition 4.2, p.401 (PDF pp.2–6,27,34–35,39,41–42)
- OpenAI, Harmonic heights and the Artin K(pi,1) conjecture · September 23, 2026; Theorem 1.1, PDF p.2; §§2.1–2.4, PDF pp.6–9; §9.4, PDF p.58
- MathVet, openai-math challenge status and fidelity table · HarmonicArtin row; corresponding family 254 row in fidelity-table.csv at the same commit
- Carlos Toledo, cert-machine openai-math kernel audit · HarmonicArtin entry, run38019899278 at 2026-10-10T14:00:12Z; top-level what and decider
- Coxeter diagrams: edges, labels, components and finite type
- Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice
- The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde
- Classification of affine Coxeter diagrams and their Euclidean simplex realization
- Disconnected diagrams, direct products, and comparison of invariant forms
- Proof of the K(pi,1) conjecture for affine Artin groups · §1, Main Theorem; §2.1, Euclidean reflection realizations, affine-classification paragraph and Figure 1, definitions of Y and Y_W and Conjecture 2.1; §8, Theorem 8.15
- K(pi,1) conjecture for Artin groups · Introduction, printed pp. 361–362; §1, pp. 368–371, Theorem 1.2 and Conjecture 1.3; §3.2, Corollary 3.4, p. 394; §3.3, quotient CW decomposition, p. 398; §5, Corollary 5.8, p. 412
- Allen Hatcher, Algebraic Topology, Chapter 1 · §1.B, definition on printed p. 87; Examples 1B.5–1B.6, p. 88
- Allen Hatcher, Algebraic Topology, Chapter 4 · §4.2, Eilenberg–MacLane Spaces, opening paragraph, printed p. 365
- Independent spherical Artin Garside input
- Elementary ribbons have explicitly determined endpoints
- The reduced ribbon subgroupoid uses exactly the exterior atoms
- Unique conjugator factorization restricts to a normalizer splitting
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Definitions 4.23–4.24 (printed p. 27); Theorem 4.25(ii) and following explanatory paragraph (printed p. 28); standard parabolics in Example 3.3 (printed p. 12)
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · §3.3, Definition 3.12 (printed/PDF p. 14), and Example 3.14 (printed/PDF p. 15)
- The parabolic intersection W_I∩ dW_Jd⁻¹ for d∈^IW^J
- Left and right cosets gH and Hg of a subgroup
- An atomic Coxeter presentation · Definitions 1.1 and 1.6, PDF pp. 1 and 4
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Subset and vertex-complement notation
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
DefinitionDependency level 6
Artin–Tits group, pure subgroup and positive lifts
LemmaDependency level 6
Linear symmetries act on the Salvetti model
TheoremDependency level 6
Independent finite simplicial arrangement theorem
TheoremDependency level 6
Independent finite local wall-crossing formula
TheoremDependency level 6
Independent groupoid covering and universal-cover input
DefinitionDependency level 7
Groupoid coverings, universal and normal covers
DefinitionDependency level 7
Standard parabolic subgroups and parabolic roots
PropositionDependency level 7
Free object actions give orbit coverings
TheoremDependency level 7
Independent Salvetti homotopy-model input
DefinitionDependency level 8
Associates of a parabolic subset
DefinitionDependency level 8
Algebraic chamber-label set
DefinitionDependency level 8
Free actions and quotient groupoids
DefinitionDependency level 8
Longest parabolic elements and opposition permutation
DefinitionDependency level 8
Normalisers and their quotients
DefinitionDependency level 8
Positive Artin and parabolic monoids
DefinitionDependency level 8
Tits cone intersections, relative roots and arrangements
LemmaDependency level 8
Component roots intersect the complementary parabolic exactly
LemmaDependency level 8
Finite roots and finite Coxeter groups at finite rank
Open problemDependency level 8
Classical K(pi,1) conjecture
PropositionDependency level 8
Vertex-group exact sequence of a normal groupoid cover
PropositionDependency level 8
Stabilizer of a fundamental parabolic face
RemarkDependency level 8
Reported proved families for the classical conjecture
TheoremDependency level 8
Independent spherical ribbon splitting input
DefinitionDependency level 9
Brink–Howlett groupoid
DefinitionDependency level 9
Core and atomic double cosets
DefinitionDependency level 9
Relative complexified hyperplane complement
DefinitionDependency level 9