- 1. Scope and structure
- 2. Artin presentation
- 3. Configuration spaces
- 4. Mapping classes
- 2 + 3 → 5. Presentation ≅ loops
- 3 + 4 → 6. Loops ≅ mapping classes
A braid can be described by a word, by a collision-free motion of points, or by a deformation class of a punctured disk. This series defines each model and proves that they give isomorphic groups. Throughout, \(n>2\), so \(n\geq3\). We leave the smaller cases outside the scope.
1.1. Three models of the braid group
Write \(D=\{z\in\mathbb C:|z|\leq1\}\), let \(D^\circ\) be its interior, and fix the unordered set \(Q=\{q_1,\ldots,q_n\}\), where \(q_j=-1+2j/(n+1)\).
Definition 1.1. Artin presentation
Worked example for Definition 1.1
Definition 1.2. Configuration-space braid group
Worked example for Definition 1.2
Definition 1.3. Boundary-fixed mapping class group
We use \(s_i\) for abstract generators and \(\sigma_i\) for their elementary configuration-loop counterparts, defined in chapter 3.
Theorem 1.4. Comparison of the three models
For every integer \(n\geq3\), the displayed groups are isomorphic.
Worked example for Theorem 1.4
Proof.
The homomorphism \(s_i\mapsto[\sigma_i]\) and its crossing-word inverse are constructed in Chapter 5. The inverse-endpoint homomorphism to mapping classes and its bijectivity are proved in Chapter 6.
□After these isomorphisms are established, we use \(B_n\) for the braid group in any of the three models. Ordered configurations and homeomorphisms fixing each marked point describe the corresponding pure subgroups.
1.2. Structure of the series
- This chapter: scope, conventions, and dependencies.
- The Artin presentation: words, free groups, normal closures, quotient groups, and presentations.
- Configuration spaces: paths, homotopies, fundamental groups, unordered configurations, and lifted strands.
- Mapping classes: homeomorphisms, boundary-fixed isotopies, and half twists.
- The first equivalence: show that elementary exchanges satisfy the Artin relations, generate every loop, and have no additional relations.
- The second equivalence: lift point motions to disk homeomorphisms and use the Alexander contraction to prove the resulting map is an isomorphism.
1.3. Prerequisites and conventions
We assume basic topology, metric spaces, Euclidean spaces, and elementary group operations. We develop free groups and presentations in chapter 2, the required path and homotopy theory in chapter 3, and isotopies in chapter 4. Further algebraic-topology and general-position tools appear in collapsible sections where they are used.
For loops, \(\alpha*\beta\) means “traverse \(\alpha\), then \(\beta\).” Mapping classes use ordinary composition, \([f][g]=[f\circ g]\). Consequently chapter 6 sends a lifted motion \(F_t\), starting at the identity, to \([F_1^{-1}]\). An elementary positive configuration loop exchanges two adjacent points counterclockwise; its mapping class under this convention is a clockwise half twist.
The open disk and plane configuration models are interchangeable by an explicit homeomorphism, explained in chapter 3. The boundary condition in the mapping-class model remains pointwise fixation.
Sources
Direct definitions are linked above. For the equivalences, see Alphabeta Math's comparison of braid models and Juan González-Meneses, Basic results on braid groups, §§1.2–1.4. The later chapters supply the proofs.
Worked examples
These calculations and proofs use this series’ chronological loop products, inverse-endpoint mapping classes, and counterclockwise positive point exchanges. Related statements link to their examples.
Worked example 1. One exchange in all three descriptions
Applies to: Definition 1.1, Definition 1.2, Definition 1.3, Theorem 1.4.
Proof and calculation.
Take \(n=3\), so \(q=(-1/2,0,1/2)\). The word is \(s_1\). Its configuration loop fixes \(1/2\) and moves \(z_1=-1/4-e^{\pi it}/4,z_2=-1/4+e^{\pi it}/4\). Their separation is \(1/2\), their moduli are at most \(1/2<1\), and their distance to the fixed third point is at least \(3/4-1/4=1/2\). At the endpoints the unordered set is \(Q\); the labels 1 and 2 exchange. At \(t=1/2\), the initially left point is below the initially right point, matching the positive convention.
For the disk model, use the center \(c=-1/4\), inner radius \(r_0=5/16\), outer radius \(R=3/8\), and the chapter-4 cutoff \(b\). The motion \(F_t(c+re^{i\theta})=c+re^{i(\theta+\pi t b(r))}\) fixes the outer boundary, has inverse obtained by negating the added angle, and sends \(Q\) along precisely that configuration loop. Every intermediate map is a homeomorphism because it preserves radius and rotates each circle bijectively, continuously also at the center. Its inverse endpoint is the clockwise half twist \(h_1=F_1^{-1}\). Thus the comparison sends this specific word to this specific point motion and mapping class, rather than identifying signs by a picture. The fact that these maps are bijections on all classes is proved in chapters 5–6; this example checks their construction on a generator.
Worked example 2. Infinite cyclic behavior is visible algebraically and geometrically
Applies to: Definition 1.1, Definition 1.2, Theorem 1.4.
Proof and calculation.
Assign 1 to each Artin generator and \(-1\) to its inverse. Free cancellations change the total by zero, distant relations have sum 2 on each side, and adjacent relations have sum 3 on each side. Hence this total descends to a homomorphism \(e:A_n\to\mathbb Z\). In particular \(e(s_1^m)=m\), so these powers are distinct.
A geometric obstruction for the same example is \(\Delta(S)=\prod_{j<k}(z_j-z_k)^2\), independent of an ordering and nonzero on configurations. For the three-point loop above,
\[\Delta(t)=\frac14e^{2\pi it}\left(\frac9{16}-\frac1{16}e^{2\pi it}\right)^2.\]The first factor winds once about zero. The parenthesized factor has winding zero: shrinking its radius \(1/16\) to zero never reaches its center distance \(9/16\), and gives a constant nonzero loop. Therefore \(\Delta\) winds once, and on the \(m\)-fold loop it winds \(m\) times, with negative powers traversed backward. Winding is homotopy invariant: a nonzero complex path has a continuous argument after subdivision into small disks avoiding zero; for a loop its final argument minus its initial argument is an integer multiple of \(2\pi\). In a homotopy these integers vary continuously, hence are constant. This proves these configuration loops are distinct without assuming the Artin comparison theorem.
Counterexamples and distinctions
Each example refutes the stated stronger claim while retaining the hypotheses explicitly checked in its proof.
Counterexample 1. A permutation is not a braid
Related statements: Definition 1.1, Definition 1.2, Theorem 1.4.
Claim refuted. Two braids with the same endpoint permutation are equal.
Proof.
The identity and \(s_1^2\) both induce the identity permutation, since \((12)^2=1\). But their exponent sums are 0 and 2, which are invariant under every defining relation by the calculation above. Their configuration versions have discriminant winding 0 and 2. Thus they are different braids with identical endpoints for every strand. The square is pure; “pure” does not mean “trivial”.
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