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The problem with this argument is that there are loops $\alpha:I\to \mathbb{S}^n$ which are surjective, they cover entire sphere. Such loops are known as space-filling curves and they exist for any $n\geq 1$ by the Hahn-Mazurkiewicz theorem. Such loops always pass through (any) $p\in \mathbb{S}^n$ regardless of applied rotation (or any other homeomorphism). ...

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