surjective map $S^n \rightarrow S^n$ of degree zero

Question

Construct a surjective map $S^n \rightarrow S^n$ of degree zero for ah $n\ge 1$.

I’ve been struggling with this exercise from hatcher. I know that if the map is not surjective then the degree is zero, but I have no idea how to approach this one.

Answer 1

Here is an explicit construction that uses (extremely minimal) knowledge of higher homotopy groups (Hatcher also proves it purely axiomatically for homology): The map $1 + -1: S^n \rightarrow S^n$ is surjective and nullhomotopic.

Here is a non explicit construction using point set topology:

The Hahn–Mazurkiewicz theorem tells us that every sphere is the image of the unit interval. Let p denote the projection of $S^n$ onto its first coordinate. Then we can compose p with a surjective map guaranteed by the Hahn-Mazurkiewicz theorem to get a surjective map $S^n \rightarrow S^n$ that factors through a nullhomotopic map. This means it is nullhomotopic.

Here is a hint at an explicit construction not using complicated point set topology or anything about adding maps of spheres:

Project $S^n$ onto $D^n$ then just manipulate the stereographic projection to get a surjective map. It should just be defined piecewise in two parts.