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

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- Solution c

Solution

Two explicit fibrations of

L (2, 1) = 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 zero exceptional fibers.
The weighted circle action induced by $(z_{1}, z_{2}) \mapsto (e^{i t} z_{1}, e^{3 i t} z_{2})$ has quotient orbifold $S^{2}$ and one exceptional fiber of multiplicity three.

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

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