at.algebraic topology - Circle bundles over $RP^2$

I think that they have Seifert fiber space presentation as:
$(On,1|(1,b))$.

Or
$(On,1|(1,b),(a_1,b_1),...,(a_r,b_r))$, if you allow an orbifold with cone points in $RP^2$.

You can look at the cases by decomposing $RP^2=Mocup_{partial}D$, so the orientable 3-manifold will be the

1) orientable $Q=Motilde{times}S^1$, the twisted circle bundle over the mobius band, very well known being equivalent to the orientable I-bundle over the Klein bottle, with boundary a torus $T$,
2) and a Dehn-filling in the remaining disk $D$, with a whichever fibered solid torus or tori.

We could say that $(On,1mid (1,b))=Qcup_T W(1,b)$, for a fibered $(1,b)$ solid torus $W$

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