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
- Positive gallery category, cancellation and length
- Path presentations, congruences and formal inverses
- Complexified complements in an open convex cone
- Eilenberg--Mac Lane space
- Fundamental groupoid of a space
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- Lifts of maps, paths, and homotopies through a covering map
- Existence and uniqueness of path lifts through a covering map
- Existence and uniqueness of homotopy lifts through a covering map
- The endpoint of a lifted path depends only on its endpoint-fixed homotopy class
- Nullhomotopic maps and contractible spaces
- Every nonempty contractible space is path-connected
- A contractible space has trivial fundamental group
- Equivalence, quasi-inverse, and adjoint equivalence of categories
- Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · Introduction theorem, printed p. 273; Proposition 1.27, pp. 283–284; (2.10)–(2.12), pp. 289–290; text after (3.3), (3.4)(i), (3.5), (3.6), Theorem 3.7 and part A, pp. 292–294
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Divisibility and atomicity in a category
- Categorical Garside structures and simple arrows
- Positive degree and atoms of the presented Artin monoid
- Identifying the presented monoid with its positive image
- Reduced-word lifts and the prefix-order convention
- Reduced Coxeter lifts in the presented positive monoid
- Both-sided divisors of the longest positive lift
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
- Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · §2.3, Definition 2.1 (printed p. 5), Example 2.3 (printed p. 6), and Definition 2.5 (printed p. 7)
- Luis Paris, Artin monoids inject in their groups, Commentarii Mathematici Helvetici 77 (2002), 609–637 · §1, Theorem 1.1, printed p. 610; defining presentations on printed pp. 609–610
- 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
- Left group actions, transitive actions, and faithful actions
- A free group action has no nonidentity element fixing a point
- Monoid homomorphism and group homomorphism
- Group isomorphisms, automorphisms and the set Aut(G)
- Ronald Brown, Topology and Groupoids, third edition, corrected electronic text · 10.3.1–10.3.3 and Corollary 1, printed pp. 371–374; 10.4.2–10.4.3 and Corollary 1, pp. 376–377; 10.6.3–10.6.4, pp. 393–394
- Sign facets of a cone arrangement
- Facet incidence and localized chamber signs
- Partial order and partially ordered set
- Luis Paris, K(π, 1) conjecture for Artin groups (2014) · §3.1, printed pp.385–386 (PDF pages 26–27), definitions of facet incidence, localized chambers and the displayed Sal₀(A) order after Figure 3.1; arrangement hypotheses on printed p.364 (PDF page 5).
- Bounded prefix ideals and rank-two closure
- Normal subgroup: invariance under conjugation
- Cell attachment by a characteristic map
- An abstract simplicial complex
- The geometric realization of an abstract simplicial complex
- CW complex with closure finiteness and weak topology
- Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · §1, (1.3), (1.12)–(1.13), (1.26)(i),(ii); printed pp. 275, 277–278, 283
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
Gallery fractions, fundamental groupoid and asphericity
Lemma
Transport of a Garside structure to the presented monoid
Theorem
Independent spherical Artin Garside input
Lemma
Normal coverings and the convention-correct deck quotient
Definition
The incident facet and chamber poset
Lemma
Both gallery divisibility orders are lattices
Lemma
Universal groupoid covering from the trivial subgroup
Lemma
The Salvetti order complex and its finite CW case
Lemma