Alphabeta MathArticle items · 12 — Universal Brink–Howlett covering and its deck group
Presents 27 items from the article and its immediate prerequisites, covering dependency levels 17–20.
1 · Prerequisites
- Coxeter-system, roots, length and weak-order conventions
- Groupoid coverings, universal and normal covers
- Normaliser action on labels, orbits and adjacency
- The universal wall groupoid is generated by adjacent crossings
- Quotient groupoids give normal covers
- Independent groupoid covering and universal-cover input
- Brink–Howlett groupoid
- Universal wall groupoid on relative chambers
- Tits cone intersections, relative roots and arrangements
- Labelling theorem for relative chambers
- Standard expressions and their lengths
- Adjacent chambers and inverse wall-crossing elements
- Algebraic chamber-label set
- Root-permuting normaliser complement
- Coxeter normaliser semidirect product
- Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink–Howlett groupoids · §§1.3–1.4, PDF pp.3–4; §1.7, pp.4–5; Theorem 2.9, PDF p.22; Proposition 2.10(b)–(d) and Proposition 2.11(c), PDF p.23
- Real Coxeter representation and contragredient notation
- Relative complexified hyperplane complement
- Chamber open sets used for the topological equivalence
- Contractibility and intersection components of chamber open sets
- Faithful normaliser-quotient action on the relative cone
- Positive arrangement category and its groupoid completion
- Fundamental groupoid of a space
- Paths, path-connected spaces and path components
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- Nullhomotopic maps and contractible spaces
- A3 relative arrangement and its six labelled chambers
- Simple wall-crossing elements and maps
- Independent finite local wall-crossing formula
- The root-length criterion and faithfulness of the canonical reflection representation
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- D4 chamber labels and complete adjacency graph
- D4 diagram and generator conventions
- Correspondence between chamber labels and Brink–Howlett morphisms
- Path presentations, congruences and formal inverses
- The seven D4 planes give a finite spherical cellulation
- CW complex with closure finiteness and weak topology
- Standard parabolic subgroups and parabolic roots
- Brink–Howlett generators and invariance of standard length
- Finite relative longest element without spherical ambient rank
- Elementary inversions detect the two initial prefixes
- Simultaneous elementary prefixes force the relative longest prefix
- Exactly two standard words in the finite corank-two residue
- Every standard expression counts the same parallel classes
- Chamber paths, concatenation and wall words
- Geodesic chamber paths
- Length additivity and the base-chamber half-space
- Translation between relative arrangements in the geodesic argument
- Finite galleries and separating hyperplanes
- Convexity, stratification and faces of the relative cone
- Local finiteness of the interior relative arrangement
- Set-sized groupoids, functors and composition conversion
- Normalisers and their quotients
- Tits cone geometry with its exact interior boundary
- Complete-local smooth threefold flops and their slice data
- Generic slicing identifies flop stability walls with restricted ADE walls
- The Axiom of Choice
- The flop restriction complement is the finite ADE relative complement
- Chambers label smooth minimal models and their NCCR categories
- Minimal flop paths and the arrangement-complement action
- Chambers, walls and adjacency
- Empty-subset wall crossings are simple reflections
- Reflecting hyperplanes, fundamental chamber and Tits cone
- Labelled directed graphs, their underlying simple graphs, and label-preserving isomorphisms
- Categorical Garside structures and simple arrows
- Independent relative chamber labelling input
- Coordinate faces span their annihilator subspaces
- Independent finite simplicial arrangement theorem
- Gallery fractions, fundamental groupoid and asphericity
- Associates of a parabolic subset
- Reduced ribbon groupoid
- Ribbon generators lift wall-crossing morphisms
- Free actions and quotient groupoids
- Simplified wall crossing for a finite parabolic
- Finite wall-crossing connectivity
- Free object actions give orbit coverings
- Minimal representatives identify core cosets with root-transport arrows
- Rotation parameters and the two switchback completions
- Finite parabolic longest elements and disconnected factors
- Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · §2.2, Definition 2.1 and (2.4), PDF p. 9; Remark 4.8 (4.8), PDF p. 15
- Relative free action, normal cover and exact sequence
- Covering-space actions and fundamental-groupoid quotients
- Equivalence, quasi-inverse, and adjoint equivalence of categories
- Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
- Linear hyperplane
- A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces
- 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
- Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point
- Two ribbon-lcm expressions and presentation
- Divisibility of positive lifts and weak order
- Reduced-word lifts and the prefix-order convention
- Positive degree and atoms of the presented Artin monoid
- A finite connected component gives the opposite relative chamber
- Deg:π₁(ℝ/ℤ,[0])→(ℤ,+) is an isomorphism
- [t]↦(cos 2π t,sin 2π t) is a homeomorphism from ℝ/ℤ to the unit circle
- The degree of a based circle loop
- Root sign coherence and the action of simple reflections on positive roots
- Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
- Eilenberg--Mac Lane space
- Complexification as ℂ⊗_ℝV with its canonical real-linear embedding
- Open subsets of Euclidean space have the standard smooth structure
- Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word
- Based loops and the fundamental group
- Loop classes form the group π₁(X,x₀) under concatenation
- Higher homotopy group by based cubes
- The geometric simplex spanned by affinely independent vertices
- Cell attachment by a characteristic map
- Every path-connected space is connected, and every path component lies inside a component
- Jacob Lurie, Whitehead Triangulations (Lecture 3) · PDF p. 1, Definition 1 and Whitehead assertion (1)
- Jacob Lurie, Existence of Triangulations (Lecture 4) · PDF p. 1, paragraph before Definition 3; p. 2, Variant 8
- José F. Fernando and Riccardo Ghiloni, Smooth Approximations in PL Geometry, American Journal of Mathematics 144 (2022), 967–1007 · §3.C, proof of Theorem 1.6, Step I, printed p. 993 (PDF p. 27), first two paragraphs; topology convention in §1.A, printed p. 969 (PDF p. 3)
- Atomic reduced compositions and standard products use the same order
- Finite roots and finite Coxeter groups at finite rank
- Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · Section 4 opening and Notation 4.1, PDF p. 13; Proposition 4.9, equation (4.9), and Definition 4.10, PDF p. 15
- Groupoid completion and its universal property
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
TheoremDependency level 17
Universal Brink–Howlett covering and its deck group
DefinitionDependency level 18
Equivariant chamber-point and path assignment
ExampleDependency level 18
A3 doubled adjacency graph
ExampleDependency level 18
Four colour orbits of eight D4 chambers
ExampleDependency level 18
Eight-element D4 normaliser quotient vertex group
ExampleDependency level 18
Universal groupoid interpretation of the D4 graph
LemmaDependency level 18
Flag triangles construct the dual sphere
LemmaDependency level 18
Two-generator minimal common left multiple
PropositionDependency level 18
Geodesics are exactly length-additive wall words
RemarkDependency level 18
Natural functors to the universal wall groupoid
RemarkDependency level 18
Normaliser-action wording in the D4 exposition
RemarkDependency level 18
Induced normaliser action on arrangement groupoids
RemarkDependency level 18
Threefold-flop realization of relative arrangements and derived actions
RemarkDependency level 18
Universal wall graph as a groupoid Cayley graph
TheoremDependency level 18
Spherical relative arrangements and Deligne topology
DefinitionDependency level 19
Atomic braid relations
DefinitionDependency level 19
Assignment from wall crossings to ribbon generators
ExampleDependency level 19
D4 Brink–Howlett generating graph
LemmaDependency level 19
The corank-two atomic coset has relative longest minimum
LemmaDependency level 19
Equivariant equivalence with fundamental groupoids
LemmaDependency level 19
Ribbon lcms lift two-generator Brink–Howlett elements
LemmaDependency level 19
Standard expressions correspond to geodesics
PropositionDependency level 19
Path-class independence and equivariance in the topological construction
RemarkDependency level 19
Root characterization of the inverse two-generator element
TheoremDependency level 19
Deligne asphericity used for the relative complement
LemmaDependency level 20
Ko atomic braid relations are exactly the finite BH rank-two relations
LemmaDependency level 20