Hatcher Exercise 2.2.4
We wish to construct a surjective map of degree zero. Since degree is multiplicative with respect to composition, we only need the map to factor through a contractible space.
Consider . Let be the map . This is a projection onto one of the hemispheres of . Then, just let be the quotient map . Then, is a surjective map, and it has degree zero because it factors through the contractible space .