[[Sphere space]]
# Fundamental group of a sphere

The fundamental group $\pi_{1}(\mathbb{S}^n, s_{0})$ of a sphere is $\mathbb{Z}$ iff $n = 1$ and trivial if $n \neq 1$.  #m/thm/homotopy 

> [!check]- Proof
> For $n = 0$ the sphere fails to be [[Path connectedness|path-connected]] as it consists of two disjoint points,
> hence for any $p \in \mathbb{S}^0$ there exists only one loop with that basepoint,
> thus $\pi_{1}(\mathbb{S}^0) \cong \{ e \}$.
> 
> For $n=1$ we regard a [[continuous loop]] $\alpha$ as an endomorphism $\alpha \in \Top(\mathbb{S}^1, \mathbb{S}^1)$.
> We claim that [[Degree of a circle endomorphism]] constitutes an isomorphism
> $$
> \begin{align*}
> \Phi : \pi_{1}(\mathbb{S}^1, 1) &\to \mathbb{Z} \\
> [\alpha] &\mapsto \deg \alpha
> \end{align*}
> $$
> This is well-defined and injective since [[Circle endomorphisms are homotopic iff they are of equal degree]], and it is surjective because $\alpha : z \mapsto z^m$ has degree $m$.
> Let $\alpha_{1}, \alpha_{2}$ be paths with base $1$ and
> let $\varphi_{1}, \varphi_{2} : [0,1] \to \mathbb{R}$ be the required continuous functions so that the following diagram commutes in $\Top$ for $i=1,2$:
> 
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> 
> then the corresponding lift for the concatenated path $\alpha_{1} * a_{2}$ is given by
> $$
> \begin{align*}
> \chi(t) = \begin{cases}
> \varphi_{1}(2t) & 0 \leq t \leq \frac{1}{2} \\ \\
> \varphi_{1} (1) + \varphi_{2}\left( 2t-1 \right) & \frac{1}{2} \leq t \leq 1
> \end{cases}
> \end{align*}
> $$
> and hence $\Phi[\alpha_{1}][\beta_{1}] = \chi(1) = \varphi_{1}(1) + \varphi_{2}(1) = \Phi[\alpha_{1}] + \Phi[\alpha_{2}]$.
> Hence $\Phi$ is an isomorphism, so $\pi_{1}(\mathbb{S}^1, 1) = \mathbb{Z}$.
> 
> For $n \geq 2$ see [[Seifert-Van Kampen-Brown theorem]].
> <span class="QED"/>

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