This was inspired by this question.

Let $Y = {\mathbb R}^4 \setminus$a coordinate line, which retracts to ${\mathbb R}^3 \setminus$a point, which retracts to $S^2$.

What is an explicit *immersion* $S^3 \to Y$, whose composition with the above retraction gives the Hopf fibration?

My idea being, perhaps this would make clearer in what sense the $S^3$ is surrounding "a hole" in $S^2$.