<s>[INST] Fundamental group of $S^2 - \{N,S\}$ I would like to calculate the fundamental group $\Pi_1$ of $S^2 - \{N,S\}$, where $N,S$ are the north respectively south pole of $S^2$, but do not see yet how to do this. 
 [/INST] The map \begin{align}\phi:S^2\setminus \{N,S\}&\to S^1\times (-1,1)\\ \begin{pmatrix}x\\ y\\ z\end{pmatrix}&\mapsto \frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix}x\\ y\\ z\sqrt{x^2+y^2}\end{pmatrix}\end{align} is a diffeomorphism.
Its inverse is $$\begin{pmatrix}a\\ b\\ c\end{pmatrix}\mapsto \begin{pmatrix}a\sqrt{1-c^2}\\ b\sqrt{1-c^2}\\ c\end{pmatrix}$$
$S^1$ is a deformation retract of $S^1\times(-1,1)$. 
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