Alphabeta MathArticle items · 10 — Local finiteness of the interior relative arrangement
Presents 27 items from the article and its immediate prerequisites, covering dependency levels 10–12.
1 · Prerequisites
- Tits cone intersections, relative roots and arrangements
- Linear hyperplane arrangements and local finiteness
- Finite pair separation yields a finite relative ball
- The finite-negativity criterion, the reduction step, and convexity of the Tits cone
- D4 diagram and generator conventions
- Seven relative D4 hyperplanes and thirty-two chambers
- Core and atomic double cosets
- Complete-local smooth threefold flops and their slice data
- Yuki Hirano and Michael Wemyss, Faithful actions from hyperplane arrangements · Example 2.2, preprint PDF p. 4
- Michael Wemyss, Flops and clusters in the homological minimal model programme · Example 3.4, printed p.20; §7.1 opening, p.47; Example 7.4, pp.48–49 and Figure 6; Remark 7.5 and Example 7.6, pp.49–50, through Step 1
- Hankyung Ko, An atomic Coxeter presentation · §8.2 opening, PDF p. 28; Example 1.9, p. 6; Example 4.14, p. 16
- Michael Wemyss, A lockdown survey on cDV singularities · Definition 1.1, printed p.1; §2.1, p.9; §2.3, pp.12–13; §3.1, pp.13–14; §3.3, p.15
- Associates of a parabolic subset
- Simple wall-crossing elements and maps
- Connected component notation [I,a]
- Longest parabolic elements and opposition permutation
- Coxeter-system, roots, length and weak-order conventions
- Standard parabolic subgroups and parabolic roots
- Algebraic chamber-label set
- Brink–Howlett groupoid
- Set-sized groupoids, functors and composition conversion
- Opposite category Cᵒᵖ
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · PDF/printed p. 5, §§1.8–1.9; arbitrary-rank convention p. 4
- Artin–Tits group, pure subgroup and positive lifts
- Positive Artin and parabolic monoids
- Positive elementary ribbon elements
- Categorical Garside structures and simple arrows
- Independent spherical Artin Garside input
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.4, PDF p.4; §1.24–§1.27, PDF pp.9–10, especially Lemma 1.25 and Remark 1.26
- Wall-crossing elements are Brink–Howlett morphisms
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.24 (component convention) and Lemma 1.25, especially (a),(c), PDF/printed p. 10; arbitrary-rank convention §1.4, p. 4
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §§1.2–1.4, PDF/printed pp. 3–4 (reflections, root signs and positive-root bijection); §1.28, PDF/printed p. 11 (reflection-subgroup root subsystem and standard parabolics)
- Shrink the finite relative ball to a central arrangement
- A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces
- Invariant choices of facet points for finite linear symmetry groups
- Natural finite Salvetti good covers
- Disjoint translates of invariant Salvetti cover members
- The Salvetti order complex and its finite CW case
- CW complex with closure finiteness and weak topology
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- Equivariant maps and isomorphisms of group actions
- Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word
- Refinements, locally finite families, point-finite families, and star refinements
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- Cell attachment by a characteristic map
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- A free group action has no nonidentity element fixing a point
- Free and properly discontinuous group actions
- Luis Paris, K(pi,1) conjecture for Artin groups (2014) · §2.1, Proposition 2.2, printed pp.376–377
- Finite outside roots give a finite complementary parabolic
- Finite roots and finite Coxeter groups at finite rank
- Component roots intersect the complementary parabolic exactly
- Ribbon groupoid in spherical type
- Simplified wall crossing for a finite parabolic
- Finite parabolic longest elements and disconnected factors
- Reduced-word lifts and the prefix-order convention
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- The root-length criterion and faithfulness of the canonical reflection representation
- Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order
- Coxeter diagrams: edges, labels, components and finite type
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- The canonical reflection homomorphism, roots, reflections, and the positive cone
- Descent of the reflection representation, unit root norms, and conjugation of reflections
- Chamber open sets used for the topological equivalence
- Sign cells and local finite-arrangement reductions
- Complexified complements in an open convex cone
- Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints
- Nullhomotopic maps and contractible spaces
- A nonempty space is contractible if and only if its identity map is nullhomotopic
- Every nonempty contractible space is path-connected
- Every path-connected space is connected, and every path component lies inside a component
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Connected components, quasicomponents, and totally disconnected spaces
- Real Coxeter representation and contragredient notation
- Tits cone geometry with its exact interior boundary
- Relative complexified hyperplane complement
- Independent Salvetti homotopy-model input
- The dihedral product complement admits no finite CW model
- The infinite dihedral Tits cone is an open half-plane with its vertex
- Isolated complex meridian disks inside the product of the dihedral cone
- Independent meridian classes detected by individual winding maps
- Chambers, walls and adjacency
- Locally finite arrangements and relative face terminology
- The full-cone and interior arrangements have identical chambers
- Finite galleries and separating hyperplanes
- Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed
- Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected
- Root-permuting normaliser complement
- Minimum representatives of normaliser cosets preserve the simple roots
- Left and right cosets gH and Hg of a subgroup
- Root-permuting elements are minimum coset representatives
- Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
- Common relative facets have one enlarged standard subset
- Root signs, inversions and descent translation
- Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives
- Convexity, stratification and faces of the relative cone
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §§1.4–1.5, PDF/printed p. 4; §1.24 and Lemma 1.25(a), PDF/printed p. 10; §1.28, PDF/printed p. 11
- Minimal representatives identify core cosets with root-transport arrows
- Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · Section 2.2 and Definition 2.1, PDF p. 9, equation (2.4); Definitions 2.2–2.3 and following equivalence, p. 10; Section 1.3, p. 4; Lemma 3.4, p. 12; Definition 3.9 and Remark 3.11, p. 13; Definition 1.4 and equation (1.4), p. 3
- Annihilators, relative hyperplanes and complexification
- Coordinate vertices inside a relative open ball
- A hyperplane near the centre separates opposite coordinate vertices
- Independent relative chamber labelling input
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
PropositionDependency level 10
Local finiteness of the interior relative arrangement
RemarkDependency level 10
cD4 geometric occurrence and core-coset context
RemarkDependency level 10
Empty-subset wall crossings are simple reflections
RemarkDependency level 10
Opposite-category convention for Brink–Howlett
RemarkDependency level 10
Garside divisibility and lcm assertions for A+
RemarkDependency level 10
Wall-crossing notation and Brink–Howlett convention
RemarkDependency level 10
The printed root characterization of a wall-crossing element
DefinitionDependency level 11
Standard expressions and their lengths
LemmaDependency level 11
Finite relative longest element without spherical ambient rank
LemmaDependency level 11
Antipodal generic points in a central relative ball
LemmaDependency level 11
Finite equivariant Salvetti and orbit homotopy models
LemmaDependency level 11
Finite root difference forces finite connected parabolic
LemmaDependency level 11
Ribbon generators lift wall-crossing morphisms
PropositionDependency level 11
Contractibility and intersection components of chamber open sets
RemarkDependency level 11
Relative cones can avoid the interior of the full cone
RemarkDependency level 11
Relative Salvetti models and the finiteness qualification
RemarkDependency level 11
Interior restriction preserves chambers and adjacency
DefinitionDependency level 12
Reduced ribbon groupoid
LemmaDependency level 12
Every normalising element factors through the root-permuting subgroup
LemmaDependency level 12
The root-permuting subgroup intersects the parabolic trivially
LemmaDependency level 12
Each normaliser coset has exactly one root-permuting representative
LemmaDependency level 12
Adjacent relative chambers force a finite crossing component
LemmaDependency level 12
Elementary inversions detect the two initial prefixes
LemmaDependency level 12
Atomic reduced compositions and standard products use the same order
LemmaDependency level 12
Relative central neighbourhood and antipodal generic points
RemarkDependency level 12
Concatenation of length-additive standard expressions
TheoremDependency level 12