Let $f: S^n \to S^n$ a continous map. Show that if $f_* : H_n \to H_n$ is not 0 then $f$ is onto.

Question

Let $n>0$ and $f: S^n \to S^n$ a continous map. Let $f_* : H_n(S^n, \mathbb {Z}) \to H_n(S^n, \mathbb {Z})$ the induced homomorfism. Show that if $f_* \not = 0$ then $f$ is onto.

What I have done: Of course $H_n(S^n, \mathbb {Z}) \simeq \mathbb {Z}$. $1$ must be send to some $m$ by $f_*$, which covers $S^n$ exactely $m$ times, thus $f$ must be onto. I don't know how to say this properly.

Answer 1

Suppose that $f$ is not onto.

Then I claim that actually $f$ is nullhomotopic: the reason is that the sphere without a point is contractible (as it is homeomorphic to $\mathbb{R}^n$).

But the induced map on homology is a homotopy invariant.