Alphabeta MathBraids as Fundamental Groups of 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
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 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
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Based motions of an unordered point configuration
Lemma
An interior configuration loop traces a geometric braid
Lemma
A based configuration-loop homotopy traces a braid isotopy
Lemma
A geometric braid slices to an interior configuration loop
Lemma
Tracing and slicing are inverse on relative classes
Lemma
Raw slicing reverses geometric stacking products
Theorem
Geometric braid classes and the unordered configuration fundamental group
Corollary
Pure geometric braids and ordered configuration loops
Proposition
The geometric endpoint permutation matches covering monodromy
Theorem
The geometric and configuration braid models agree at the fixed base configuration
Sources
- Juan Gonzalez-Meneses, Basic results on braid groups, §§1.1–1.3, printed pp. 3–6
- Joan S. Birman and Tara E. Brendle, Braids: A Survey, §1.1, author manuscript pp. 3–5
- Juan Gonzalez-Meneses, Basic results on braid groups, §1.3, printed p. 5
- Juan Gonzalez-Meneses, Basic results on braid groups, §§1.1–1.3, printed pp. 3–5
- Joan S. Birman and Tara E. Brendle, Braids: A Survey, §1.1, author manuscript pp. 3–4
- Juan Gonzalez-Meneses, Basic results on braid groups, §2.1, equation (2.1), printed p. 11