Let $\alpha:S^n\rightarrow S^n$ be the antipodal map. Why is $\alpha \not\simeq id$ if $n$ is even?

Question

I defined the homotopy $H:S^n\times [0,1]\rightarrow S^n$ as $H(x,t)=-x+t2x$. Why does this function not define a homotopy?

Answer 1

Because, for instance, $H\left(x,\frac12\right)=0$ and $0\notin S^n$.

Answer 2

To give a hint for the question in the title:

Cosinder the degree of the identity map and the antipodal map. If you can show that these two do $\textbf{not}$ agree, then you have shown that $\alpha \not\simeq \text{id}$ (why?).