Alphabeta MathArticle items · 13 — Two-geodesic wall-crossing lemma
Presents 27 items from the article and its immediate prerequisites, covering dependency levels 20–27.
1 · Prerequisites
- Coxeter-system, roots, length and weak-order conventions
- Standard parabolic subgroups and parabolic roots
- Brink–Howlett groupoid
- Standard expressions and their lengths
- Simple wall-crossing elements and maps
- Geodesic chamber paths
- Correspondence between chamber labels and Brink–Howlett morphisms
- Two-generator minimal common left multiple
- Standard expressions correspond to geodesics
- Simultaneous elementary prefixes force the relative longest prefix
- Tits cone intersections, relative roots and arrangements
- Chambers, walls and adjacency
- D4 chamber labels and complete adjacency graph
- Spherical relative arrangements and Deligne topology
- Four colour orbits of eight D4 chambers
- Finite galleries and separating hyperplanes
- Categorical Garside structures and simple arrows
- Partial order and partially ordered set
- Independent finite simplicial arrangement theorem
- Seven relative D4 hyperplanes and thirty-two chambers
- Hankyung Ko, An atomic Coxeter presentation · §8.2, PDF pp. 28–29, especially diagram immediately before Remark 8.6 on p. 29
- Atomic braid relations
- Brink–Howlett generators and invariance of standard length
- Concatenation of length-additive standard expressions
- Every standard expression counts the same parallel classes
- Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws
- Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · Lemma 1.25(a), Remark 1.26, and Lemma 3.12(a) and its proof
- Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · §1.4, rank convention and equation (1.4.4); §1.5, first displayed equivalence defining weak right order
- Algebraic chamber-label set
- Labelling theorem for relative chambers
- Chamber paths, concatenation and wall words
- Well-defined Deligne-to-ribbon functor and geodesic images
- Groupoid coverings, universal and normal covers
- Two ribbon-lcm expressions and presentation
- Ribbon lcms lift two-generator Brink–Howlett elements
- Geodesics are exactly length-additive wall words
- Positive arrangement category and its groupoid completion
- Independent finite local wall-crossing formula
- Adjacent chambers and inverse wall-crossing elements
- Assignment from wall crossings to ribbon generators
- Reduced ribbon groupoid
- Parabolic root subsystems and unique coset minima
- Finite wall-crossing connectivity
- Core and atomic double cosets
- Simplified wall crossing for a finite parabolic
- Minimal representatives identify core cosets with root-transport arrows
- Atomic reduced compositions and standard products use the same order
- Rotation parameters and the two switchback completions
- The corank-two atomic coset has relative longest minimum
- Ko atomic braid relations are exactly the finite BH rank-two relations
- Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · Section 1.2, equation (1.3) and following sentence, PDF p. 3; Theorem 1.8 (= Theorem 5.1), PDF pp. 6 and 18; Definition 4.10, pp. 15–16
- D4 Brink–Howlett generating graph
- D4 diagram and generator conventions
- The twelve D4 positive roots
- Finite relative longest element without spherical ambient rank
- Translation between relative arrangements in the geodesic argument
- Parabolic coordinate subspaces and fundamental faces
- Longest parabolic elements and opposition permutation
- Normaliser action on labels, orbits and adjacency
- Induced normaliser action on arrangement groupoids
- Free object actions give orbit coverings
- Quotient groupoids give normal covers
- Path presentations, congruences and formal inverses
- Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
- Componentwise crossings are inverse elements
- Standard expressions are connected by atomic braid moves
- Every finite generator path reduces to a standard expression
- Flag triangles construct the dual sphere
- The seven D4 planes give a finite spherical cellulation
- Independent relative chamber labelling input
- Reduced-word lifts and the prefix-order convention
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares
- CW complex with closure finiteness and weak topology
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- The disjoint union (coproduct) ⨆ᵢ Xᵢ with the final topology of the canonical injections: a set is open exactly when each of its traces is
- 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
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)
- Cell attachment by a characteristic map
- Oriented smooth manifolds and oriented charts
- Boundary orientation of the unit sphere by the outward normal
- Induced boundary orientation
- Equivariant equivalence with fundamental groupoids
- Spherical Artin normaliser splitting
- Vertex-group exact sequence of a normal groupoid cover
- Normaliser quotient exact sequence
- Relative free action, normal cover and exact sequence
- Ribbon generators lift wall-crossing morphisms
- Morphisms of group extensions
- Equivalence of group extensions with fixed kernel and fixed quotient
- Artin–Tits group, pure subgroup and positive lifts
- The Axiom of Choice
- Universal wall groupoid on relative chambers
- Relative complexified hyperplane complement
- Normalisers and their quotients
- Root-permuting normaliser complement
- Natural functors to the universal wall groupoid
- Universal Brink–Howlett covering and its deck group
- Coxeter normaliser semidirect product
- Classification of finite Coxeter systems, including the H and dihedral families
- Fundamental groupoid of a space
- Subcategory and full subcategory
- Seifert–van Kampen identifies the fundamental group with a group pushout
- Stereographic projection identifies the Riemann sphere with the unit two-sphere
- Empty-subset wall crossings are simple reflections
- The rank-two half-space alternative and the chamber-length induction (Pₙ), (Qₙ)
- The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness
- Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects
- Faithful normaliser-quotient action on the relative cone
- Existence and uniqueness of path lifts through a covering map
- A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces
- Deligne asphericity used for the relative complement
- Coverings preserve the higher-homotopy obstruction to K(pi,1)
- Paths, path-connected spaces and path components
- Based loops and the fundamental group
- Higher homotopy group by based cubes
- Continuity of a map of topological spaces at a point and globally
- Universal groupoid interpretation of the D4 graph
- Sⁿ is simply connected for every n≥2
- Equivalence, quasi-inverse, and adjoint equivalence of categories
- Covariant functor, identity functor, composite functor, and contravariant functor
- A functor is an isomorphism of categories exactly when its object and morphism maps are bijective
- Natural transformation and its components
- Identity natural transformation and vertical composition
- Natural isomorphism
- Finite equivariant Salvetti and orbit homotopy models
- Relative Salvetti models and the finiteness qualification
- The Salvetti order complex and its finite CW case
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- Higher homotopy groups are functorial and based homotopy invariant
- Higher homotopy basepoint transport and moving homotopies
- Reflecting hyperplanes, fundamental chamber and Tits cone
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
LemmaDependency level 20
Two-geodesic wall-crossing lemma
RemarkDependency level 20
D4 grading, region poset and simple-morphism divisibility
TheoremDependency level 20
Atomic Matsumoto theorem
DefinitionDependency level 21
Codimension-two relations
LemmaDependency level 21
Deligne-to-ribbon functor is a covering
RemarkDependency level 21
Translation to Ko’s atomic cosets and the scope extension
CorollaryDependency level 22
Geodesics differ by codimension-two relations
ExampleDependency level 22
Explicit D4 atomic braid relations and lifted paths
PropositionDependency level 22
Reduced ribbon groupoid is the Deligne orbit quotient
CorollaryDependency level 23
Normal Deligne–ribbon cover and its deck group
LemmaDependency level 23
The atomic-quadratic path quotient is faithful
LemmaDependency level 23
The dual face cycles are the D4 atomic polygons
RemarkDependency level 23
Reported uniformly weakly Dynkin extension
ExampleDependency level 24
D4 Deligne and ribbon graphs without quadratic relations
LemmaDependency level 24
Doubling edges with bigons preserves the spherical space
LemmaDependency level 24
Intermediate exact sequences and deck morphism in Theorem 6.7
RemarkDependency level 24
Commutative square of arrangement, ribbon and Brink–Howlett groupoids
TheoremDependency level 24
Brink–Howlett presentation by atomic and quadratic relations
ExampleDependency level 25
Explicit D4 comparison functors
LemmaDependency level 25
The vertex fundamental groupoid of a sphere is thin
RemarkDependency level 25
The ordinary Coxeter presentation is the empty-J case
RemarkDependency level 25
Explicit equivalence chain for the middle isomorphism
TheoremDependency level 25
Isomorphism of algebraic and topological normaliser sequences
CorollaryDependency level 26
Spherical normaliser quotient classifying spaces
ExampleDependency level 26
The D4 groupoid and the spherical cell complex
RemarkDependency level 27
Finite CW-models for spherical normaliser quotients
TheoremDependency level 27