Circle bundles over $RP^2$ - MathOverflow

Circle bundles over $RP^2$

2009-11-19T18:01:47Z

Does anybody know if orientable, closed $3$-manifolds that are circle bundles over $RP^2$ have been classified? One can determine the isomorphism classes of bundles using obstruction theory, but I am interested in what total spaces can appear. I am not assuming the bundle is principal.

Thank you.

Answer by Danny Calegari for Circle bundles over $RP^2$

2009-11-19T18:13:53Z

Such manifolds are examples of Seifert fibered spaces, which have, indeed, been classified. A good reference is Montesinos "Classical Tessellations and Three-Manifolds". Basically, such manifolds (over any nonorientable surface base) are classified by their Euler class, which measures the obstruction to the existence of a section.

Answer by HW for Circle bundles over $RP^2$

2009-11-19T18:17:18Z

According to Scott's paper "The geometries of 3-manifolds", the only closed manifolds that admit an $S^2\times R$ geometry are the two $S^2$ bundles over $S^1$, $P^2\times S^1$ and $P^3\sharp P^3$. It seems like the last one is the only candidate (unless I'm missing a way in which $S^2\times S^1$ can be viewed as such a bundle).

(How do I get a connect sum symbol in maths mode!?)

EDIT: Used \sharp as Richard suggested.

FURTHER EDIT: As pointed out below, I spoke too soon. Of course such bundles could have spherical geometry too. Scott's paper is a nice reference for that too!

Answer by ivane for Circle bundles over $RP^2$

2009-11-30T16:34:16Z

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 $RP^2=Mo\cup_{\partial}D$, so the orientable 3-manifold will be the

1) orientable $Q=Mo\tilde{\times}S^1$, 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 $(On,1\mid (1,b))=Q\cup_T W(1,b)$, for a fibered $(1,b)$ solid torus $W$