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The pure braid extension splits as a semidirect product

Statement

Assume the Axiom of Choice and let $n \geq 2$ , with the notation $P B_{n}$ , $P B_{n - 1}$ , $F_{n - 1}$ and the forgetful homomorphism $φ$ of The Fadell-Neuwirth short exact sequence for pure braids. Then the section of the planar forgetful map from A choice-free continuous section of planar coordinate forgetting, adjusted at the basepoint by a path in the puncture fibre, induces a group homomorphism $s : P B_{n - 1} ⟶ P B_{n} with φ \circ s = {id}_{P B_{n - 1}},$ and consequently the extension splits: $P B_{n} ≅ F_{n - 1} ⋊ P B_{n - 1},$ the semidirect product formed with the action of $P B_{n - 1}$ on the free kernel $F_{n - 1}$ given by conjugation with the chosen section, $g \cdot x = s (g) x s (g)^{- 1}$ . The action depends on the chosen section and the path that adjusts it; no trivial action and no direct-product decomposition are asserted.

Facts & Assumptions

Given: the Axiom of Choice, an integer $n \geq 2$ , a base configuration $q = (q_{1}, \dots, q_{n}) \in F_{n} (int D^{2})$ with $q^{'} = (q_{1}, \dots, q_{n - 1})$ , and the fibre $M_{n - 1} = int D^{2} ∖ {q_{1}, \dots, q_{n - 1}}$ of the last-coordinate forgetful map.

[A1]

The Axiom of Choice holds (The Axiom of Choice).

[F1]

In ZF, AC implies DC (AC implies DC implies countable choice).

[F2]

Under AC the forgetful map $φ : P B_{n} \to P B_{n - 1}$ of the closed-disc convention sits in the short exact sequence $1 \to F_{n - 1} \overset{κ}{\to} P B_{n} \overset{φ}{\to} P B_{n - 1} \to 1$ , where $κ$ is induced by the inclusion $x \mapsto (q_{1}, \dots, q_{n - 1}, x)$ of the fibre and $φ$ is induced by forgetting the last coordinate; in the open-disc model these maps are $i_{*}$ and $p_{*}$ for the fibration $p : F_{n} (int D^{2}) \to F_{n - 1} (int D^{2})$ , and the inclusion $ι_{*}^{F}$ identifies the two models (The Fadell-Neuwirth short exact sequence for pure braids).

[F3]

For $n \geq 2$ , every path $α$ in $M_{n - 1}$ from $q$ to the value $s^{'} (q^{'})$ of the transported planar section induces by path conjugation a homomorphism $σ : π_{1} (F_{n - 1} (int D^{2}), q^{'}) \to π_{1} (F_{n} (int D^{2}), q)$ with $p_{*} \circ σ = id$ , and the conjugation isomorphism is the one of Conjugating loop classes by a path is an isomorphism of fundamental groups; the construction uses no choice principle (A choice-free continuous section of planar coordinate forgetting).

[F4]

For a short exact sequence $1 \to N \overset{i}{\to} G \overset{π}{\to} H \to 1$ : a homomorphic section $s : H \to G$ of $π$ exists exactly when the extension splits, exactly when $G ≅ (\ker π) ⋊ H$ compatibly with the injection and quotient, and for a given section the action is $h \cdot x = s (h) x s (h)^{- 1}$ (Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent).

[F5]

For a group extension $1 \to N \to E \to Q \to 1$ that admits a homomorphic section, the extension is equivalent to $1 \to N \to N ⋊ Q \to Q \to 1$ for the corresponding action (A group extension splits exactly when it has a complement or a compatible semidirect-product model, and a kernel retraction forces a direct product).

[F6]

Induced maps on fundamental groups are functorial, $(g \circ f)_{*} = g_{*} \circ f_{*}$ , and for the inclusions and forgetful maps of the two models the identity $p^{D} \circ ι^{F} = ι^{F} \circ p$ of maps holds because both sides forget the last coordinate (Induced fundamental-group maps are well defined, functorial and invariant under based homotopy).

Proof

technique · direct

1.1A1F1

Choice bookkeeping. By [F1] the Axiom of Choice [A1] yields DC, so the short exact sequence of [F2] is available; the explicit section itself is choice-free and AC enters only through that published sequence.

1.2F3

The based section in the open model. Fix a path $α$ in $M_{n - 1}$ from $q$ to $s^{'} (q^{'})$ , which exists by [F3] and is a single selection, not an instance of AC; by [F3] the resulting homomorphism $σ : π_{1} (F_{n - 1} (int D^{2}), q^{'}) \to π_{1} (F_{n} (int D^{2}), q)$ satisfies $p_{*} \circ σ = id$ .

1.3F2

The short exact sequence. By [F2] the sequence $1 \to F_{n - 1} \overset{κ}{\to} P B_{n} \overset{φ}{\to} P B_{n - 1} \to 1$ is exact, and the isomorphisms $ι_{*}^{F}$ transport the open-disc maps $i_{*}$ , $p_{*}$ to $κ$ , $φ$ .

1.4F4F5

The splitting criterion. By [F4] a homomorphic section of $φ$ exists exactly when the extension splits, exactly when $P B_{n} ≅ F_{n - 1} ⋊ P B_{n - 1}$ compatibly with $κ$ and $φ$ , with action $g \cdot x = s (g) x s (g)^{- 1}$ for a given section; by [F5] the same conclusion is the semidirect-product model of the split extension.

1.5F6

Naturality of the transport. Both $p^{D}$ and $p$ forget the last coordinate, so $p^{D} \circ ι^{F} = ι^{F} \circ p$ as maps; by the functoriality [F6] the induced maps satisfy $p_{*}^{D} \circ ι_{*}^{F} = ι_{*}^{F} \circ p_{*}$ on fundamental groups at configurations of interior points.

2.1step 1.2step 1.3step 1.5

A section for $φ$ . Define $s : P B_{n - 1} \to P B_{n}$ by $s : = ι_{*}^{F} \circ σ \circ (ι_{*}^{F})^{- 1}$ , where $ι_{*}^{F}$ is the isomorphism of [F2] at the relevant configurations. Then $φ \circ s = p_{*}^{D} \circ ι_{*}^{F} \circ σ \circ (ι_{*}^{F})^{- 1} = ι_{*}^{F} \circ p_{*} \circ σ \circ (ι_{*}^{F})^{- 1} = ι_{*}^{F} \circ (ι_{*}^{F})^{- 1} = {id}_{P B_{n - 1}}$ , using the naturality of step 1.5 and $p_{*} \circ σ = id$ from step 1.2; being a composite of group homomorphisms, $s$ is a homomorphism.

3.1step 1.4step 2.1

The semidirect product. Step 2.1 exhibits a homomorphic section of $φ$ , so the criterion of [F4] applies and the extension of [F2] splits with $P B_{n} ≅ F_{n - 1} ⋊ P B_{n - 1}$ and action $g \cdot x = s (g) x s (g)^{- 1}$ . The decomposition is built from the particular section $s$ and the particular path $α$ , both non-canonical: choosing another path or another section changes the action by an inner automorphism of $F_{n - 1}$ in general, and no trivial action, direct product, or independence-of-choice statement is asserted.

The section is the based version of the explicit planar cross-section, the extension is the published choice-dependent Fadell–Neuwirth sequence, and no claim is made that the splitting is canonical. ∎

Depends on

Used by

PB₃ as F₂ by Z, with its section action Example

Dependency tree · two levels

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Sources

Juan Gonzalez-Meneses, Basic results on braid groups, section 2.1, printed pp. 11-14 (splitting of the pure braid sequence by the explicit cross-section) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 1.3, author manuscript pp. 5-7 (standard reference, not scraped)