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All four classical braid models realize the Artin presentation

Statement

Assume AC and $n \geq 1$ . Write $B_{n}^{A r t i n}$ for the Artin-presentation group of The braid group by Artin presentation, $G_{n} = B_{n}^{g e o m}$ for the geometric braid group at the base tuple $Q_{n}$ of The elementary geometric half twist, its support disc, and its opposite, $B_{n}^{c o n f} : = π_{1} (C_{n} (D^{2}), [Q_{n}]), π_{1} (C_{n} (int D^{2}), [Q_{n}])$ for the unordered configuration-space fundamental groups of Unordered configuration spaces $C_{n} (X)$ at the same base configuration, the second identified with the first by the open-to-closed inclusion, and $Mod (D^{2}, Q_{n}; \partial D^{2})$ for the boundary-fixed punctured-disk mapping class group of Boundary-fixed mapping class group of a punctured disk. Then:

the four models $B_{n}^{A r t i n}$ , $G_{n}$ , $B_{n}^{c o n f} ≅ π_{1} (C_{n} (int D^{2}), [Q_{n}])$ and $Mod (D^{2}, Q_{n}; \partial D^{2})$ are pairwise connected by the canonical isomorphisms: the completeness isomorphism $φ_{n}$ of The Artin presentation is complete for geometric braids, the published inverse-loop isomorphism of The geometric and configuration braid models agree at the fixed base configuration, the open-to-closed identification, and the AC-dependent boundary-fixed mapping-class isomorphism of Braid group as boundary-fixed punctured-disk mapping classes (an in-run batch-20 scaffold, not a published supplier);
under these identifications, for every $1 \leq i \leq n - 1$ the Artin generator $σ_{i}$ corresponds to the class $[σ_{i}]$ of the elementary geometric half twist, to the configuration loop class $Φ ([σ_{i}])$ whose endpoint monodromy is the adjacent transposition $(i i + 1)$ , and to the class $[H_{i}]$ of the half twist supported near the $i$ -th and $(i + 1)$ -st punctures, that is, of the explicit boundary-fixed homeomorphism supported in the disc $U_{i}$ and exchanging $q_{i}$ and $q_{i + 1}$ ;
consequently each of the four models carries the Artin presentation with these corresponding generators: for each model the assignment $σ_{i} \mapsto$ its generator extends to a group isomorphism from $B_{n}^{A r t i n}$ onto the model, so the model is presented by the generators $σ_{1}, \dots, σ_{n - 1}$ subject to the two Artin relations and to no further relations.

For $n = 1$ there is no generator, all four groups are trivial, and clauses 2 and 3 are vacuous.

Facts & Assumptions

Given: AC, an integer $n \geq 1$ , the Artin-presentation group $B_{n}^{A r t i n} = ⟨ X ∣ R ⟩$ of The braid group by Artin presentation with generating set $X = {σ_{1}, \dots, σ_{n - 1}}$ and its two families of Artin relators interpreted in the sense of Group presentation by generators and relations, the four models of the statement, and an index $i$ with $1 \leq i \leq n - 1$ .

[F1]

For every $n \geq 1$ the surjection $φ_{n} ⁣ : B_{n}^{A r t i n} \to G_{n}$ of The Artin presentation surjects onto the geometric braid group is an isomorphism, and it carries each generator $σ_{i}$ to the class $[σ_{i}]$ of the elementary geometric half twist of The elementary geometric half twist, its support disc, and its opposite; the endpoint permutation of that class is the transposition of $i$ and $i + 1$ (The Artin presentation is complete for geometric braids, The Artin presentation surjects onto the geometric braid group, The elementary geometric half twist, its support disc, and its opposite).

[F2]

The inverse-loop slicing map $Φ ⁣ : G_{n} \to B_{n}^{c o n f} = π_{1} (C_{n} (D^{2}), [Q_{n}])$ , $Φ ([β]) = (ι_{*}^{C} [S (β)])^{- 1}$ , is a group isomorphism intertwining the endpoint maps, $π_{c o n f} (Φ ([β])) = π_{g e o} ([β])$ for every $[β] \in G_{n}$ ; and the open-to-closed inclusion induces an isomorphism $ι_{*}^{C} ⁣ : π_{1} (C_{n} (int D^{2}), [Q_{n}]) \to π_{1} (C_{n} (D^{2}), [Q_{n}])$ at the same basepoint (The geometric and configuration braid models agree at the fixed base configuration, The geometric endpoint permutation matches covering monodromy, The interior-disc and closed-disc configuration spaces are homotopy equivalent).

[F3]

The composite $Ψ : = δ \circ (ι_{*}^{C})^{- 1} \circ Φ ⁣ : G_{n} \to Mod (D^{2}, Q_{n}; \partial D^{2})$ is a group isomorphism, and for every $1 \leq i \leq n - 1$ it satisfies $Ψ ([σ_{i}]) = [H_{i}]$ , where $H_{i}$ is the explicit boundary-fixed homeomorphism supported in the support disc $U_{i}$ of The elementary geometric half twist, its support disc, and its opposite and exchanging $q_{i}$ and $q_{i + 1}$ . The theorem supplying $Ψ$ is an in-run batch-20 scaffold of this run, not a published supplier (Braid group as boundary-fixed punctured-disk mapping classes, Boundary-fixed mapping class group of a punctured disk).

[F4]

AC holds, and AC implies dependent choice and countable choice (The Axiom of Choice, AC implies DC implies countable choice); this is the hypothesis under which the mapping-class isomorphism of [F3] and the free-kernel suppliers of the completeness theorem of [F1] are available.

[F5]

Presentation transport along an isomorphism. If a group has a presentation $B = ⟨ X ∣ R ⟩ = F (X) / ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ and $θ ⁣ : B \to M$ is a group isomorphism, then the composite $q_{θ} ⁣ : F (X) \to B \overset{θ}{\to} M$ of the quotient map with $θ$ is a surjective homomorphism with kernel $⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ : surjectivity is clear, and $q_{θ} (w) = e_{M}$ holds exactly when $q (w) \in \ker θ = {e_{B}}$ , that is, exactly when $w \in ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ . The first isomorphism theorem therefore gives $M ≅ F (X) / ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)} = ⟨ X ∣ R ⟩$ , the isomorphism carrying the class of each $x \in X$ to $θ ([x])$ (Group presentation by generators and relations, First isomorphism theorem for groups: $G / \ker f ≅ im f$ , The braid group by Artin presentation).

Proof

technique · direct

1.1F1

The abstract and geometric models. By [F1] the map $φ_{n} ⁣ : B_{n}^{A r t i n} \to G_{n}$ is a group isomorphism and $φ_{n} (σ_{i}) = [σ_{i}]$ for every $i$ , so the abstract model and the geometric model are identified generator by generator.

2.1F1F2step 1.1

The two configuration models. By [F2] the inverse-loop slicing map $Φ ⁣ : G_{n} \to B_{n}^{c o n f}$ is a group isomorphism with $π_{c o n f} \circ Φ = π_{g e o}$ , and $ι_{*}^{C}$ is an isomorphism $π_{1} (C_{n} (int D^{2}), [Q_{n}]) \to B_{n}^{c o n f}$ ; hence the composites $Φ \circ φ_{n}$ and $(ι_{*}^{C})^{- 1} \circ Φ \circ φ_{n}$ , being composites of group isomorphisms, are group isomorphisms from $B_{n}^{A r t i n}$ onto $B_{n}^{c o n f}$ and onto $π_{1} (C_{n} (int D^{2}), [Q_{n}])$ respectively. The generator $σ_{i}$ is carried to $Φ ([σ_{i}])$ , whose endpoint monodromy is $π_{c o n f} (Φ ([σ_{i}])) = π_{g e o} ([σ_{i}])$ , the transposition of $i$ and $i + 1$ by [F1] and [F2]; in the open-disc model it is carried to $(ι_{*}^{C})^{- 1} Φ ([σ_{i}])$ , the same configuration loop class read through the inclusion.

2.2F3step 1.1

The mapping-class model. By [F3] the composite $Ψ$ is a group isomorphism $G_{n} \to Mod (D^{2}, Q_{n}; \partial D^{2})$ with $Ψ ([σ_{i}]) = [H_{i}]$ , so $Ψ \circ φ_{n}$ is a group isomorphism $B_{n}^{A r t i n} \to Mod (D^{2}, Q_{n}; \partial D^{2})$ carrying $σ_{i}$ to $[H_{i}]$ , the class of the boundary-fixed half twist supported in $U_{i}$ that exchanges $q_{i}$ and $q_{i + 1}$ .

3.1F5step 1.1step 2.1step 2.2

Presentation transport to each model. Let $R$ be the set of the two families of Artin relators, so that $B_{n}^{A r t i n} = ⟨ X ∣ R ⟩ = F (X) / ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ by The braid group by Artin presentation and Group presentation by generators and relations. Apply [F5] to the identity isomorphism of $B_{n}^{A r t i n}$ and to the isomorphisms of steps 1.1, 2.1 and 2.2: the identity on $B_{n}^{A r t i n}$ , $φ_{n}$ , $Φ \circ φ_{n}$ , $(ι_{*}^{C})^{- 1} \circ Φ \circ φ_{n}$ and $Ψ \circ φ_{n}$ . Each of the five models is therefore isomorphic to $F (X) / ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩$ through the composite of the quotient map with that isomorphism, with the class of $σ_{i}$ mapping to the corresponding generator displayed in steps 1.1, 2.1 and 2.2; in particular each model is generated by those $n - 1$ elements and satisfies no relation among them beyond the Artin relators.

4.1F1F3F4step 1.1step 2.1step 2.2step 3.1

Conclusion and the one-strand case. Steps 1.1, 2.1 and 2.2 identify the four models pairwise through the stated isomorphisms and track $σ_{i}$ to the half twist, to the loop of monodromy $(i i + 1)$ and to the supported half twist, and step 3.1 transports the presentation to each of them, proving all three clauses for $n \geq 2$ . For $n = 1$ the index range $1 \leq i \leq n - 1$ is empty, so the generator clauses are vacuous, and $B_{1}^{A r t i n}$ is the trivial group given by the empty presentation by The braid group by Artin presentation; the isomorphisms of [F1]–[F3] then identify the other three models with it, so each of the four models is trivial and carries the empty presentation of the trivial group, which is clause 3 at $n = 1$ . AC enters only through [F3] and through the free-kernel suppliers of the completeness theorem recorded in [F4]. ∎

Remarks

The corollary does not reprove the mapping-class or configuration identifications: it composes them with the completeness theorem and tracks the generator through the composite. Its only genuinely new input beyond the suppliers is the bookkeeping that the generator correspondence survives each composite, which is why the configuration and mapping-class models inherit the Artin presentation.
The mapping-class isomorphism is an in-run batch-20 draft (Braid group as boundary-fixed punctured-disk mapping classes, precheck PASS; not a published supplier), flagged in the dispatch report together with the consuming step 2.2 and the cross-batch edge recorded in frontier-37-owner-30-batch-22.cross-batch-dependencies.json; no published theorem supplies it.
The construction is choice-free apart from the AC hypothesis: the presentations are finite, the free group on $X$ is explicit, and no connecting path or lift is chosen in the composites, all of which use the fixed basepoint $Q_{n}$ of the suppliers.

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Sources

Benson Farb and Dan Margalit, A Primer on Mapping Class Groups, version 5.0 author draft, section 9.1.3 'Mapping class group of a punctured disk', printed p. 256 (PDF p. 266) (standard reference, not scraped)