Alphabeta MathArticle items · 07 — Chambers label smooth minimal models and their NCCR categories
Presents 26 items from the article and its immediate prerequisites, covering dependency levels 2–5.
1 · Prerequisites
- Complete-local smooth threefold flops and their slice data
- Generic slicing identifies flop stability walls with restricted ADE walls
- Derived category of an abelian category
- Exact functor between triangulated categories
- Natural isomorphism
- Hirano–Wemyss, Faithful actions from hyperplane arrangements · §1.3, printed p.3; Setup 4.1, printed pp.8–9; §4.1, Notation 4.3 and Remark 4.4, printed p.9
- Wemyss, A lockdown survey on cDV singularities · §1.3, printed p.3; §4.1, printed p.16; Theorem 4.3 and following paragraph, printed p.17
- Wemyss, Flops and clusters in the homological minimal model programme · §1.6, printed p.9; Setup 2.9, equations (2.A)–(2.B), and Notation 2.10, printed p.12; Lemma 2.12, printed p.13; Theorem 4.2(1)–(2), printed p.21
- Donovan–Wemyss, Noncommutative deformations and flops · §1.4, printed p.5, first paragraph and displayed flop-equivalence pair
- Coxeter-system, roots, length and weak-order conventions
- Root signs, inversions and descent translation
- The root-length criterion and faithfulness of the canonical reflection representation
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
- Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups
- 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
- Subset and vertex-complement notation
- Coxeter diagrams: edges, labels, components and finite type
- Connected graphs and connected components defined by the existence of vertex paths
- Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
- Reflecting hyperplanes, fundamental chamber and Tits cone
- Locally finite arrangements and relative face terminology
- Tits cone geometry with its exact interior boundary
- Linear functionals and the algebraic dual V^*=L(V,F)
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Connected components, quasicomponents, and totally disconnected spaces
- Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected
- For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
- D4 diagram and generator conventions
- Real Coxeter representation and contragredient notation
- The infinite dihedral Tits cone is an open half-plane with its vertex
- The relative root deletion set is the restricted-hyperplane union
- Full-dimensional J-chambers lie in the relative interior
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications, author-hosted draft · Sentence following Definition 1.6, printed p. 6 (PDF p. 30)
- Positive gallery category, cancellation and length
- Path presentations, congruences and formal inverses
- Eilenberg--Mac Lane space
- 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
- Invariant metric, product arrangements and essentiality
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- The dual family (b^*)_b∈ B associated to a Hamel basis B, defined by b^*(c)=δ_bc
- The dual family of a finite basis is a basis of the dual space, with the same dimension
- Finite-dimensional Riesz representation: every functional is uniquely v↦⟨ v,w⟩
- Orthogonality and the orthogonal complement
- Michael W. Davis, The Geometry and Topology of Coxeter Groups (author manuscript, August 2007) · §6.6, Theorem 6.6.3(i)–(iii), printed p.89 (PDF p.105); setup printed pp.88–89; proof printed p.91
- The singular chain complex and singular homology
- The singular boundary operator
- The trigonometric loops give π₁({(x,y):x²+y²=1},(1,0))≅ℤ
- The first Hurewicz map is abelianization
- Allen Hatcher, Algebraic Topology · §2.1, induced-chain-map construction on printed pp.110–111; Proposition 2.9, following functoriality identities (i)–(ii), and Theorem 2.10, p.111
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications (author-hosted draft) · §§1.1, 1.3–1.4; Definitions 1.4–1.6 and the sentence immediately after Definition 1.6; printed pp. 3–6, especially p. 6 (PDF pp. 27–30, especially p. 30).
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Internal direct products of finitely many normal subgroups
- A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces
- Injection, surjection, bijection
- The cardinality | A| of a finite set
- Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
- Categorical Garside structures and simple arrows
- Reduced-word lifts and the prefix-order convention
- Positive degree and atoms of the presented Artin monoid
- Reduced Coxeter lifts in the presented positive monoid
- Both-sided divisors of the longest positive lift
- Identifying the presented monoid with its positive image
- Transport of a Garside structure to the presented monoid
- 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
- Component opposition transports the retained simple roots
- Finite parabolic longest elements and disconnected factors
- Sign facets of a cone arrangement
- Cell attachment by a characteristic map
- The incident facet and chamber poset
- An abstract simplicial complex
- The geometric realization of an abstract simplicial complex
- CW complex with closure finiteness and weak topology
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- Allen Hatcher, Algebraic Topology · §2.1, Corollary 2.11, printed p.111; §2.2, Theorem 2.35, p.139, and consequence (ii), p.140 (PDF pages 120, 148–149)
- Root sign coherence and the action of simple reflections on positive roots
- Descent of the reflection representation, unit root norms, and conjugation of reflections
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- Based loops and the fundamental group
- The homomorphism on fundamental groups induced by a pointed continuous map
- Group extensions, sections, complements, and split extensions
- Linear hyperplane arrangements and local finiteness
- Reflection cardinalities of the three connected finite rank-three types
- Classification of finite Coxeter systems, including the H and dihedral families
- Parabolic root subsystems and unique coset minima
- Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications, author-hosted draft · Definition 1.8 and Theorem 1.12, printed pp. 6–7 (PDF pp. 30–31)
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 2
Chambers label smooth minimal models and their NCCR categories
PropositionDependency level 2
Root-sign and inversion-length facts
TheoremDependency level 2
Classical Matsumoto assertion recalled in the article
TheoremDependency level 2
Independent Artin fundamental-group identification
DefinitionDependency level 3
Connected component notation [I,a]
DefinitionDependency level 3
Separation by a hyperplane
ExampleDependency level 3
The twelve D4 positive roots
LemmaDependency level 3
Isolated complex meridian disks inside the product of the dihedral cone
LemmaDependency level 3
Deleting relative root hyperplanes removes all cone-boundary points
LemmaDependency level 3
Gallery fractions, fundamental groupoid and asphericity
LemmaDependency level 3
Chamber walls of an essential finite reflection group give full Coxeter rank
PropositionDependency level 3
Basic Tits-cone properties
DefinitionDependency level 4
Classical complexified hyperplane complement
LemmaDependency level 4
Independent meridian classes detected by individual winding maps
LemmaDependency level 4
The full-cone and interior arrangements have identical chambers
LemmaDependency level 4
Irreducible reflection arrangements give connected Coxeter diagrams
LemmaDependency level 4
The hyperplanes are in bijection with Coxeter reflections
LemmaDependency level 4
Only finitely many hyperplanes separate two Tits-cone points
LemmaDependency level 4
Segment intersections and the inversion bound
TheoremDependency level 4
Independent spherical Artin Garside input
LemmaDependency level 5
Componentwise crossings are inverse elements
LemmaDependency level 5
The Salvetti order complex and its finite CW case
LemmaDependency level 5
The dihedral product complement admits no finite CW model
PropositionDependency level 5
Classical free action and covering exact sequence
PropositionDependency level 5
Irreducible rank-three reflection-arrangement counts
TheoremDependency level 5