I have learned that the Hopf fibration is a principal $S^1$-bundle over $S^2$. The $S^1$-action on $S^3\subseteq\mathbb{C}^2$ is given by $$w\cdot(z_1,z_2)=(wz_1,wz_2).$$ Now let us consider the representation $$\rho:S^1\to \text{Diff}(S^1), \quad w\mapsto \text{left multiplication by }w^n.$$ What is the associated bundle $E=S^3\times_{S^1}S^1$?

My thought: For $[z_1,z_2,w]\in E$, we have $$[z_1,z_2,w]=[z_1w^{1/n},z_2w^{1/n},1].$$ Does this mean that $E\cong S^{\color{red}3}$ as fibre bundle? If not, where am I wrong?

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    $\begingroup$ I think it's a Lens space. Probably $L(n,1)$. $\endgroup$ – Charlie Frohman Nov 27 '16 at 4:40
  • $\begingroup$ I think so, but I get confused. How can I prove it? $\endgroup$ – Eclipse Sun Nov 27 '16 at 5:09
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    $\begingroup$ Oh, I get it. $[z_1,z_2,1]$ and $[z_1e^{2\pi i/n},z_2e^{2\pi i/n},1]$ are the same point. $\endgroup$ – Eclipse Sun Nov 27 '16 at 5:12
  • $\begingroup$ Yeah, that's it. $\endgroup$ – Charlie Frohman Nov 27 '16 at 5:12

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