Alphabeta MathOrdered and Unordered Configuration Spaces
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Semidirect Products, Automorphism Groups and Split Extensions
- Simple Field Extensions and the Construction of the Complex Numbers
Algebraic Topology
Coxeter Groups
Differential Geometry
Functional Analysis
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- Topology of ℝ
Set Theory
Topology
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Ordered configuration spaces Fₙ(X)
Proposition
The symmetric group acts continuously and freely on Fₙ(X) by permuting labels
Definition
Unordered configuration spaces Cₙ(X)
Lemma
Conjugating loop classes by a path is an isomorphism of fundamental groups
Lemma
The interior-disc and closed-disc configuration spaces are homotopy equivalent
Lemma
Disjoint coordinate neighbourhoods evenly cover the unordered configuration space
Theorem
Ordered configuration spaces cover the unordered ones regularly with deck group Sₙ
Lemma
The closed disk D² is a connected Hausdorff topological 2-manifold with boundary
Definition
The pure braid group PBₙ as the fundamental group of an ordered configuration space
Definition
The configuration braid group Bₙᶜᵒⁿᶠ as the fundamental group of an unordered configuration space
Definition
Endpoint monodromy of an unordered configuration loop as a permutation of the labels
Theorem
The configuration braid short exact sequence 1→ PBₙ→ Bₙᶜᵒⁿᶠ→ Sₙ→ 1
Lemma
Forgetting the last n points is locally trivial with fibre Fₙ of the punctured manifold
Theorem
The Fadell-Neuwirth forgetful map: local triviality, constant fibre, and numerability for configurations in the disk
Sources
- Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.1 and 1.3, printed pp. 3-6
- Fadell-Neuwirth, Configuration Spaces, section II Theorem 1, printed pp. 111-114
- Allen Hatcher, Algebraic Topology, section 1.1, printed p. 28, Proposition 1.5
- Allen Hatcher, Algebraic Topology, section 0, printed p. 3
- Fadell-Neuwirth, Configuration Spaces, section II Theorems 1 and 3, printed pp. 111-114
- Ioan Marcut, Manifolds (2017 lecture notes), sections 14.5 and 15.1
- Nigel Hitchin, Differentiable Manifolds, section 2.2
- Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.1-1.3, printed pp. 3-6
- Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.1-1.3 and 2.1, printed pp. 3-6, 11-13
- Juan Gonzalez-Meneses, Basic results on braid groups, section 2.1 equation (2.1), printed p. 11
- Allen Hatcher, Algebraic Topology, section 1.3, covering spaces and lifting, printed pp. 60-64
- Edward Fadell and Lee Neuwirth, Configuration Spaces, section II Theorem 1 and its proof, printed pp. 111-113
- Najib Idrissi, answer to 'Fadell-Neuwirth fibration', MathOverflow question 500383 (point-moving trivialization)
- Edward Fadell and Lee Neuwirth, Configuration Spaces, section II Theorem 3, printed p. 113
- Allen Hatcher, Algebraic Topology, section 4.2, the Huebsch-Hurewicz paracompact-base strengthening, printed pp. 379-380