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The punctured-disk fundamental group is free on the standard meridians

Statement

Let $F_{n} = ⟨ x_{1}, \dots, x_{n} ⟩$ be the free group of Free group on a set of generators on $n$ letters. The assignment $x_{i} \mapsto [x_{i}]$ extends to a group isomorphism $F_{n} \to π_{1} (D^{2} ∖ Q_{n}, d)$ ; equivalently, the classes $[x_{1}], \dots, [x_{n}]$ of the standard meridians of Standard meridians of a punctured disk form a free basis of $π_{1} (D^{2} ∖ Q_{n}, d)$ .

Facts & Assumptions

Given: $n \in N$ , the punctured disk $X = D^{2} ∖ Q_{n}$ , the basepoint $d$ , the standard meridians $x_{i}$ and the flower $W$ of Standard meridians of a punctured disk.

[F1]

$W$ is a deformation retract of $X$ with retraction fixing $d$ , and $π_{1} (W, d)$ is free with basis $[x_{1}], \dots, [x_{n}]$ (The standard flower is a deformation retract with free meridian basis).

[F2]

A deformation retraction onto a subspace $A$ containing the basepoint induces, through inclusion and retraction, mutually inverse isomorphisms between $π_{1} (A, a)$ and $π_{1} (X, a)$ (A retract induces an injection on fundamental groups, and a deformation retract induces an isomorphism).

[F3]

The free group $F_{n}$ on the set ${x_{1}, \dots, x_{n}}$ has the universal property: for every group $G$ and every function $u : {x_{1}, \dots, x_{n}} \to G$ there is a unique homomorphism $\hat{u} : F_{n} \to G$ with $\hat{u} (x_{i}) = u (x_{i})$ ; reduced words form such a free group (Free group on a set of generators, Reduced words form the free group on an alphabet). Free groups on the same set are uniquely isomorphic over the set (Free groups on the same set are uniquely isomorphic compatibly with their generators).

Proof

technique · direct

1.1F1F2F3

The universal-property map. Let $u : {x_{1}, \dots, x_{n}} \to π_{1} (X, d)$ , $u (x_{i}) : = [x_{i}]$ . By [F3] there is a unique homomorphism $\hat{u} : F_{n} \to π_{1} (X, d)$ with $\hat{u} (x_{i}) = [x_{i}]$ . The inclusion $i : W ↪ X$ and the retraction $ρ : X \to W$ of [F1] are based at $d$ , and by [F2] the induced maps $i_{*} : π_{1} (W, d) \to π_{1} (X, d)$ and $ρ_{*}$ are mutually inverse isomorphisms.

1.2F1F3

The basis map. By [F1], $π_{1} (W, d)$ is free on the classes $[x_{1}], \dots, [x_{n}]$ , so the assignment $x_{i} \mapsto [x_{i}] \in π_{1} (W, d)$ extends by [F3] to an isomorphism $ϕ : F_{n} \to π_{1} (W, d)$ : it is the unique homomorphism with $ϕ (x_{i}) = [x_{i}]$ , and the universal property applied to the inverses shows it is bijective (equivalently, $F_{n}$ and $π_{1} (W, d)$ are free on the same set, so [F3]'s uniqueness clause gives the isomorphism).

1.3F2F3

The case $n = 0$ . For $n = 0$ the configuration is empty, $D^{2} ∖ Q_{0} = D^{2}$ is contractible (the straight-line homotopy to the origin), the empty basis is a basis of the trivial group, and the unique map from the trivial free group is an isomorphism; the argument above also covers this case with empty index sets.

2.1step 1.1step 1.2F2F3

Comparison. The composite $i_{*} \circ ϕ : F_{n} \to π_{1} (X, d)$ is a homomorphism with $x_{i} \mapsto i_{*} [x_{i}] = [x_{i}]$ , since $i$ is the inclusion of the subspace containing the loops $x_{i}$ . By the uniqueness clause of [F3] applied to $u$ , $i_{*} \circ ϕ = \hat{u}$ . Since $i_{*}$ and $ϕ$ are bijections, $\hat{u}$ is a group isomorphism.

3.1step 2.1step 1.3∎

Conclusion. Steps 1.1, 1.2 and 2.1 exhibit the isomorphism $\hat{u} : F_{n} \to π_{1} (D^{2} ∖ Q_{n}, d)$ with $x_{i} \mapsto [x_{i}]$ , and step 1.3 covers the empty case; hence $[x_{1}], \dots, [x_{n}]$ is a free basis of $π_{1} (D^{2} ∖ Q_{n}, d)$ .

Remarks

The identification is the one fixed on the whole page: the letters $x_{1}, \dots, x_{n}$ of $F_{n}$ are from now on identified with the classes of the standard meridian loops, and every braid automorphism is computed on this basis.
Asphericity is a separate clause of the flower lemma. The free-basis theorem uses the based deformation retraction and the finite tether-tree collapse; the boundary-product and action calculations use compact cut-disk geometry.

Depends on

Used by

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juan Gonzalez-Meneses, Basic results on braid groups, section 1.6, printed pp. 8-10 (pi_1(D_n) = F_n with free generators x_1,...,x_n) (standard reference, not scraped)
Emil Artin, Theory of Braids, Annals of Mathematics 48 (1947), pp. 101-126, printed p. 111 (the free group of the punctured disk) (standard reference, not scraped)