Two explicit fibrations of $L(2,1)=\mathbb{RP}^3$ are supplied by <Seifert fibrations of real projective 3-space>:

* The ordinary <Hopf fibration> descends from $S^3$ to a circle bundle over $S^2$ with \b[zero exceptional fibers].
* The weighted circle action induced by $(z_1,z_2)\mapsto(e^{it}z_1,e^{3it}z_2)$ has quotient orbifold $S^2$ and \b[one exceptional fiber of multiplicity three].

Thus these are Seifert fibered descriptions of the same filled manifold with different numbers of exceptional fibers.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 2 / c / Solution

{c}
{title2=$\mathbb{RP}^3$}

The Hopf circle action on $S^3$ descends through the antipodal quotient to a free circle action on $\mathbb{RP}^3$, giving a Seifert fibration over $S^2$ with no exceptional fibers. The weighted action
$$
e^{it}(z_1,z_2)=(e^{it}z_1,e^{3it}z_2)
$$
also descends and, after dividing by its generic order-two kernel, gives a fibration over $S^2$ with one exceptional fiber of multiplicity three.


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 Incoming links: Seifert fibrations of real projective 3-space