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Winding number identifies the fundamental group of C times with the integers

Statement

For a based loop $γ : [0, 1] \to C^{\times}$ at $1$ , let $r (z) = z / ∣ z ∣$ and define

$W ([γ]) : = \deg (r \circ γ) .$

Then $W$ is the standard isomorphism

$π_{1} (C^{\times}, 1) ≅ (Z, +) .$

If $γ$ is rectifiable, then

$W ([γ]) = n (γ, 0) .$

Thus the analytic winding number of any rectifiable representative is the integer classifying its loop class.

Facts & Assumptions

Given: A based loop $γ : [0, 1] \to C^{\times}$ at $1$ .

[L1]

For a rectifiable based loop, the winding number about $0$ equals the degree of its normalized circle loop (For loops in C times, the winding number about 0 equals the circle degree).

[L2]

Radial normalization is a deformation retraction of $R^{2} ∖ {0} = C^{\times}$ onto the unit circle (For $n \geq 1$ , radial normalisation is a deformation retraction of $R^{n} ∖ {0}$ onto $S^{n - 1}$ ).

[L3]

A deformation retract induces mutually inverse fundamental-group isomorphisms between the retract and the ambient space (A retract induces an injection on fundamental groups, and a deformation retract induces an isomorphism).

[L4]

The unit circle has fundamental group $Z$ with the standard trigonometric generator (The trigonometric loops give $π_{1} ({(x, y) : x^{2} + y^{2} = 1}, (1, 0)) ≅ Z$ ).

Proof

technique · direct

1.1givenL2L3

Let $C = {z \in C : ∣ z ∣ = 1} \subseteq C^{\times}$ and let $r : C^{\times} \to C$ be radial normalization, $r (z) = z / ∣ z ∣$ . Specializing [L2] to $n = 2$ and applying [L3], the induced map $r_{*} : π_{1} (C^{\times}, 1) ⟶ π_{1} (C, 1)$ is an isomorphism. For the given loop $γ$ , its image under $r_{*}$ is the class of the normalized circle loop $α (t) = γ (t) / ∣ γ (t) ∣$ .

2.1step 1.1L1L4∎

Fact [L4] identifies the class of $α$ in $π_{1} (C, 1)$ with the integer $\deg (α)$ . Hence $W ([γ]) = \deg (r \circ γ)$ is exactly the composite of the isomorphism $r_{*}$ from step 1.1 with the standard identification $π_{1} (C, 1) ≅ Z$ , and is therefore the standard isomorphism $π_{1} (C^{\times}, 1) ≅ (Z, +) .$ If $γ$ is rectifiable, [L1] gives $W ([γ]) = \deg (α) = n (γ, 0)$ .

Winding number identifies the fundamental group of C times with the integers

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources