Let $\gamma_0$ and $\gamma_1$ be two paths in $\mathbb{S}^2$ (the 2-d unit sphere in $\mathbb{R}^3$). Let both $\gamma_0$ and $\gamma_1$ start at $p \in \mathbb{S}^2$ and end in $q \in \mathbb{S}^2$. What is an explicit formula that is a homotopy from $\gamma_0$ to $\gamma_1$, using intermediate curves that must all lie on $\mathbb{S}^2$ and connect $p$ to $q$.
Constructing homotopies on S2
3
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general-topology
homotopy-theory
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1stereographic projection to the plane, $t\gamma_1+(1-t)\gamma_0$ go from the plane back to the sphere. – 2012-08-09
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0@butt But what if one of the curves passes through the north pole? I don't think there is a nice explicit formula unless you assume the curves aren't surjective. – 2012-08-10
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0This seems a good case where the Seifert-van Kampen Theorem tells you that a homotopy exists, but does not give an explicit formula, as the proof of the theorem involves subdivisions. Why should an explicit formula be expected? – 2012-08-10