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I'm trying to prove that if two (continuous) maps $f, g : \mathbb{S}^{n} \to \mathbb{S}^{n}$ are such that $f(x) \neq -g(x)$ for any $x \in \mathbb{S}^{n}$, then $f$ and $g$ are homotopic. But I can't seem to have achieved any satisfactory results.

Could anyone please give me a short clue as to how to attack this problem?

Update: I, like anyone who loves mathematics, am not asking the complete solution. Rather, I am curious which ideas are used in solving the problem.

Thank you

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Here is a roundabout solution. The shortest path between two points on a sphere is called a geodesic. Is given by the intersection of the sphere and a plane passing through p, q, and the origin.

Now, if p, q are antipodal, then there are infinitely many such geodesics. However, if they are not, then there is a unique shortest such geodesic.

What can you do with that knowledge?

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  • $\begingroup$ Well, we can apply the notion of the unique shortest path between two points to the sphere. $\endgroup$
    – Behrooz
    Commented Nov 16, 2014 at 20:31
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    $\begingroup$ Please do not answer off-topic questions. $\endgroup$
    – Steven Landsburg
    Commented Nov 16, 2014 at 21:27

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