Proving that hopf map from $S^3 \to S^2 $ is not null homotopic

Question

I want to prove that hopf map from $S^3 \to S^2 $ is not null homotopic. Is there some elementary proof of this fact?

Answer 1

If it were nullhomotopic, what do you know about the homotopy type of its mapping cone? On the other hand, the Hopf map is the attaching map of the $4$-cell in $\Bbb CP^2$, so its mapping cone is just $\Bbb CP^2$.

Answer 2

The exercises in the last chapter of Milnor's book, Topology from the Differentiable Viewpoint, may provide an elementary proof by the use of the idea of Linking number. You prove in a sequence of exercises that the linking number is a homotopy invariant. The only part, honestly, that I have not completely written down in detail is proving that for the hopf fibration the linking number is non zero - this would prove that the hopf map is not homotopic to the constant map.