Fundamental group of a sphere

The fundamental group $𝜋 1 (𝕊 𝑛, 𝑠 0)$ of a sphere is $ℤ$ iff $𝑛 = 1$ and trivial if $𝑛 \neq 1$ . homotopy

Proof

For $𝑛 = 0$ the sphere fails to be path-connected as it consists of two disjoint points, hence for any $𝑝 \in 𝕊 0$ there exists only one loop with that basepoint, thus $𝜋 1 (𝕊 0) ≅ {𝑒}$ .

For $𝑛 = 1$ we regard a continuous loop $𝛼$ as an endomorphism $𝛼 \in 𝖳 𝗈 𝗉 (𝕊 1, 𝕊 1)$ . We claim that Degree of a circle endomorphism constitutes an isomorphism
$Φ : 𝜋 1 (𝕊 1, 1) \to ℤ [𝛼] \mapsto d e g 𝛼$
This is well-defined and injective since Circle endomorphisms are homotopic iff they are of equal degree, and it is surjective because $𝛼 : 𝑧 \mapsto 𝑧 𝑚$ has degree $𝑚$ . Let $𝛼 1, 𝛼 2$ be paths with base $1$ and let $𝜑 1, 𝜑 2 : [0, 1] \to ℝ$ be the required continuous functions so that the following diagram commutes in $𝖳 𝗈 𝗉$ for $𝑖 = 1, 2$ :

then the corresponding lift for the concatenated path $𝛼 1 * 𝑎 2$ is given by
$𝜒 (𝑡) = ⎧ { {⎨ { {⎩ 𝜑 1 (2 𝑡) 0 \leq 𝑡 \leq 12 𝜑 1 (1) + 𝜑 2 (2 𝑡 - 1) 12 \leq 𝑡 \leq 1$
and hence $Φ [𝛼 1] [𝛽 1] = 𝜒 (1) = 𝜑 1 (1) + 𝜑 2 (1) = Φ [𝛼 1] + Φ [𝛼 2]$ . Hence $Φ$ is an isomorphism, so $𝜋 1 (𝕊 1, 1) = ℤ$ .

For $𝑛 \geq 2$ see Seifert-Van Kampen-Brown theorem.

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Fundamental group of a sphere

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