Suppose I have a surjective map, say $f$, between two spheres (of dimension $n+1 \geq 2$) such that it takes the closed upper hemisphere to itself and the closed lower hemisphere to itself. Now, I get a map $\hat{f}$ between $S^n$ by restricting $f$ to the equator.
Can someone help me find an explicit homotopy between the suspension of $\hat{f}$ and $f$
Using this I want to establish that the degree of $\hat{f}$ is the same as $f$, which I know to be true via another argument.