Suppose I have a function $\phi:\mathbb{R}\rightarrow\mathbb{R}^4-$ {$0$}.
Since $\mathbb{R}^4-$ {$0$} $\cong S^3$ can I map the vectors of $\phi$ in $\mathbb{R}^4-$ {$0$} to $\mathbb{R}^3-$ {$0$} by the Hopf fibration since $\mathbb{R}^3-$ {$0$} $\cong S^2$ ?