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
- Complexified complements in an open convex cone
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Fundamental groupoid of a space
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- Regular coverings
- Deck transformations and the deck-transformation group of a covering
- Lifts of maps, paths, and homotopies through a covering map
- K(π, 1) conjecture for Artin groups · §1, definitions on pp. 363–365, Theorem 1.1 and the arrangement on pp. 365–366, complement and action on p. 368, Theorem 1.2 on p. 369
- Categorical Garside structures and simple arrows
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Isomorphism, groupoid, and connected category
- Finite directed paths form the free category on a quiver
- Equivalence relation, equivalence class, and the quotient set A/∼
- Semigroup and monoid
- Small, locally small, and large categories
- Group and abelian group
- A functor is an isomorphism of categories exactly when its object and morphism maps are bijective
- Set-sized groupoids, functors and composition conversion
- Unique lifts, endpoint fibres and characteristic subgroups
- Locally finite arrangements and relative face terminology
- Linear functionals and the algebraic dual V^*=L(V,F)
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
- Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected
- Sign cells and local finite-arrangement reductions
- Kernel and image of a linear map
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Linear hyperplane
- The quotient vector space V/W and its canonical projection
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Rank-nullity: dim_F V=nullityT+rankT
- If dim_F V = n and U is a linear subspace of V, then U is finite-dimensional, dim_F U ≤ n, and dim_F U = n if and only if U = V
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Luis Paris, K(π, 1) conjecture for Artin groups (2014) · §2.3, printed p.382 (PDF page 23), paragraphs defining chambers, induced arrangements and codimensional facets; arrangement hypotheses on printed p.364 (PDF page 5).
- Path presentations, congruences and formal inverses
- Divisibility and atomicity in a category
- Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · §1, (1.1)–(1.3), (1.8), (1.10), (1.12)–(1.13), (1.19)(i),(ii); printed pp. 274–275, 277–278, 280
- Positive degree and atoms of the presented Artin monoid
- Reduced Coxeter lifts in the presented positive monoid
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
- Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · §1, (1.5)–(1.6), (1.8), (1.9.2), (1.13)–(1.14), and Corollary (1.20); printed pp. 275–278, 282
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
Theorem
Independent Artin fundamental-group identification
Definition
Groupoid completion and Garside groupoids
Lemma
Explicit connected cover associated with a vertex subgroup
Definition
Sign facets of a cone arrangement
Lemma
Positive gallery category, cancellation and length
Lemma
Both-sided divisors of the longest positive lift
Lemma
Identifying the presented monoid with its positive image
Lemma
Pointed classification of connected groupoid coverings
Lemma
Facet incidence and localized chamber signs
Lemma