Alphabeta MathGroupoid covers, arrangement topology and ribbons
Connect path groupoids and covering quotients with arrangement complements, Salvetti and Deligne models, spherical Garside structures and ribbons, preserving each finite or spherical hypothesis.
1 · Prerequisites
- Path presentations, congruences and formal inverses
- Set-sized groupoids, functors and composition conversion
- Independent groupoid covering and universal-cover input
- An indexed family (Aᵢ)_i ∈ I is a function with domain I; {Aᵢ : i ∈ I} is its range
- Injection, surjection, bijection
- Complexified complements in an open convex cone
- The Salvetti order complex and its finite CW case
- Linear symmetries act on the Salvetti model
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- Independent Artin fundamental-group identification
- 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).
- Independent spherical Artin Garside input
- Rigidity of two-letter words below braid length
- Exactly two words for a two-generator braid element
- Elementary ribbons have explicitly determined endpoints
- Opposition as conjugation by a longest Artin lift
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Definition 2.7 (printed p. 7); Example 3.3 and Definition 3.7 (printed p. 12); Definition 3.8 and following positive-ribbon convention (printed p. 13); opening paragraph of §3.3 (printed p. 14)
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · §4.2 opening notation after Corollary 4.10 (printed p. 20); defining paragraph immediately before Lemma 4.17 and the generating convention and vertex-set clause of Corollary 4.18 (printed p. 23)
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Example 3.14, printed/PDF p. 15, opening paragraph and displayed-in-text formula for s outside X
- Unique conjugator factorization restricts to a normalizer splitting
- Finite parabolic longest elements and disconnected factors
- 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)
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Example 3.14 (printed/PDF p. 15); §4.2 opening notation (p. 20); paragraph immediately before Lemma 4.17 (p. 23)
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · §2.2, natural transformations for groups (printed/PDF p. 4); Definition 2.7 (p. 7); Definitions 3.7–3.8 (pp. 12–13); monoidal description immediately preceding Definition 3.12 (p. 14)
- Braid congruence preserves word length and support
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Example 3.14 (printed p. 15); §4.2 opening notation and Theorem 4.14 relation (3) (printed pp. 20–22); Corollary 4.18 (printed pp. 23–24); Proposition 4.19 (printed p. 24)
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
Proposition
A criterion for lifting a presented groupoid
Theorem
Independent Salvetti homotopy-model input
Lemma
The two-generator braid element is the lcm
Lemma
The reduced ribbon subgroupoid uses exactly the exterior atoms
Theorem
Independent spherical ribbon splitting input
Lemma
The empty-object ribbon component and its braid relations
Theorem