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Is a Lens space $L(p,q)$, for some particular values of $p$ and/or $q$, homeomorphic to an $S^3$ arising as a Hopf fibration?

For instance, from 1903.04942:

[...] the $S^1$ bundle over $S^2$ with Hopf invariant $N$ -- i.e. the Lens space $L(N,1)$.

This seems like a realistic claim to me, because it would mean, for example, that the circle bundle over $\mathbb{E}^3\setminus0$ whose total space is the single-centre $(N=1)$ Euclidean Taub-NUT space would restrict to an $L(1,1)$ topology on a constant radius surface in the $\mathbb{E}^3$, and indeed it is known that this restriction is an $S^3$ realised as a Hopf fibration (see e.g.hep-th/0208108) (and the topology of the $S^1$ fibration over $S^2$ is $S^3$, and $S^3$ is homeomorphic to $L(1,q)$).

What I'm also confused about is that $L(1,p)\simeq S^3$. From the claim above, it would seem to me that, to have $S^3$ as a Hopf fibration with invariant $N$, I have to set $p=1$, and then the first parameter of the Lens somehow encodes the integer classifying the Hopf fibration?? Thanks.

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I think that what creates some confusion here is the different notations that authors use throughout the literature. I believe that the notation used in the first claim you mention, $L(p,q)$ is defined as a $\textit{quotient of} $ $S^3$ (with $S^3: |z_0|^2 + |z_1|^2 =1$) obtained via the equivalence relation $(z_0,z_1) \sim (e^{i2\pi /p} z_0, e^{i 2 \pi q / p} z_1)$. This is a very common definition (for instance, it is the one you would find on Wikipedia, where you can also find some references). With this definition, then the claim that $L(p,1)$ is the Hopf bundle with Hopf invariant $p$ is correct.
Unfortunately, there is another definition which is sometimes used, where the space $L(t,s)$ is defined as the quotient space of $S^{2t+1}$ obtained via $(z_0,...,z_t) \sim (e^{i2\pi /s} z_0,..., e^{i 2 \pi / s} z_t)$. This definition is given for instance in the famous book by Bott and Tu. I am pointing this out just to make you aware of this annoying two-notations issue in the literature.
Regarding the second claim, i.e. $L(1,p) \simeq S^3$, could you be a bit more specific about the context in which you found it?

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