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
- Words in an alphabet with formal inverses, elementary cancellation, and reduced words
- Equivalence relation, equivalence class, and the quotient set A/∼
- Semigroup and monoid
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Independent spherical Artin Garside input
- Reduced-word lifts and the prefix-order convention
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- Finite parabolic longest elements and disconnected factors
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
- 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
- Left group actions, transitive actions, and faithful actions
- Normal subgroup: invariance under conjugation
- Group isomorphisms, automorphisms and the set Aut(G)
- Antipodal transformation and finite-type Garside structure
- Gallery fractions, fundamental groupoid and asphericity
- Categorical Garside structures and simple arrows
- Complexified complements in an open convex cone
- Path presentations, congruences and formal inverses
- Eilenberg--Mac Lane space
- Positive gallery category, cancellation and length
- 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
- A simplicial map and its geometric realization
- Coxeter diagrams: edges, labels, components and finite type
- Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- Internal direct products of finitely many normal subgroups
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Definitions 3.7–3.8 (printed pp. 12–13); §3.3 opening paragraph and Definition 3.12 (printed p. 14); Example 3.14 (printed p. 15)
- A free group action has no nonidentity element fixing a point
- Monoid homomorphism and group homomorphism
- The kernel and image of a group homomorphism
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
Lemma
Braid congruence preserves word length and support
Lemma
Support of positive Artin words and longest lifts
Lemma
Opposition as conjugation by a longest Artin lift
Theorem
Independent groupoid covering and universal-cover input
Theorem
Independent finite simplicial arrangement theorem
Lemma
Linear symmetries act on the Salvetti model
Lemma
Rigidity of two-letter words below braid length
Lemma
Exactly two words for a two-generator braid element
Lemma
Elementary ribbons have explicitly determined endpoints
Proposition