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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 $S^2$. |
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First, note that $\mathbb R^4 \setminus \mathbb R \simeq S^2\times \mathbb R^2$. So you are asking for an immersion $S^3\to S^2\times \mathbb R^2$ representing the generator $\eta$ of $\pi_3(S^2\times \mathbb R^2)=\pi_3(S^2)=\mathbb Z$. I'm guessing that your immersion doens't exist, and that you need to consider maps $S^3\to S^2\times \mathbb R^n$ with larger $n$, in order to represent $\eta$ by an immersion. But if you are willing to go a little bit up in dimension, and consider maps $S^3\to S^2\times \mathbb R^4$, then you can even find an embedding representing $\eta$. It is given by $(H,I):S^3\to S^2\times \mathbb R^4$, where $H:S^3\to S^2$ is the Hopf map, and $I: S^3 \rightarrow {\mathbb R}^4$ is the standard inclusion. |
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