# Suspension of a map

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.

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Let $S^+$ be be northen hemisphere of your $n$-sphere $S$. Then $f$ restricts to a map $f:(S^+,\partial S^+)\to(S^+,\partial S^+)$, which gives us a class in $\pi_n(S^+,\partial\ S^+)$. The long exact sequence for homotopy groups corresponding to the pair $(S^+,\partial S^+)$ shows that the boundary map $$\pi_n(S^+,\partial S^+)\to\pi_{n-1}(\partial S^+)$$ is an isomorphism.
(Maybe one can build an exact sequence in homotopy for triples $(X;A,B)$ with $A$ and $B$ subspaces of $X$, and deduce what you want from some general statement) –  Mariano Suárez-Alvarez Oct 10 '11 at 0:04