Non-fundamental homotopy groups of a circle are trivial

For $𝑛 \geq 2$ the homotopy group $𝜋 𝑛 (𝕊 1, 1)$ is the trivial group. For $𝑛 = 1$ we have $ℤ$ ; see fundamental group of a sphere.

Proof

We use the spherical characterisation of the homotopy groups. Let $𝑛 > 2$ and $𝛼 : (𝕊 𝑛, 1) \to (𝕊 1, 1)$ be a continuous $𝑛$ -dimensional loop. Since $𝕊 𝑛$ is locally path connected and has a trivial fundamental group

𝜋 1 𝛼 (𝜋 1 (𝕊 𝑛, 1)) \subseteq 𝜋 1 e x (𝜋 1 (ℝ, 0))

Hence there exists unique lift $˜ 𝑎 : (𝕊 𝑛, 1) \to (ℝ, 0)$ such that $e x ˜ 𝛼 = 𝛼$ . But any such $˜ 𝛼$ is null-homotopic by way of $𝐻 (𝑥, 𝑡) = 𝑡 ˜ 𝛼 (𝑥)$ , and hence $𝛼 = e x ˜ 𝛼$ is null-homotopic by way of $e x 𝐻$ . Therefore any $𝛼$ is null-homotopic and the homotopy group is trivial.

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