Circle bundles over $RP^2$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:01:25Z http://mathoverflow.net/feeds/question/6142 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6142/circle-bundles-over-rp2 Circle bundles over $RP^2$ F.G. 2009-11-19T18:01:47Z 2009-12-02T19:48:07Z <p>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.</p> <p>Thank you.</p> http://mathoverflow.net/questions/6142/circle-bundles-over-rp2/6145#6145 Answer by Danny Calegari for Circle bundles over $RP^2$ Danny Calegari 2009-11-19T18:13:53Z 2009-11-19T18:13:53Z <p>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. </p> http://mathoverflow.net/questions/6142/circle-bundles-over-rp2/6146#6146 Answer by HW for Circle bundles over $RP^2$ HW 2009-11-19T18:17:18Z 2009-11-19T18:57:50Z <p>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).</p> <p>(How do I get a connect sum symbol in maths mode!?)</p> <p>EDIT: Used \sharp as Richard suggested.</p> <p>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!</p> http://mathoverflow.net/questions/6142/circle-bundles-over-rp2/7288#7288 Answer by ivane for Circle bundles over $RP^2$ ivane 2009-11-30T16:34:16Z 2009-12-02T19:48:07Z <p>I think that they have <strong>Seifert fiber space</strong> presentation as: $(On,1|(1,b))$.</p></p> <p>Or $(On,1|(1,b),(a_1,b_1),...,(a_r,b_r))$, if you allow an orbifold with cone points in $RP^2$.</p></p> <p>You can look at the cases by decomposing $RP^2=Mo\cup_{\partial}D$, so the orientable 3-manifold will be the</p> 1) orientable $Q=Mo\tilde{\times}S^1$, the <strong>twisted circle bundle over the mobius band</strong>, very well known being equivalent to the <strong>orientable I-bundle over the Klein bottle</strong>, with boundary a torus $T$,</p> 2) and a <strong>Dehn-filling</strong> in the remaining disk $D$, with a whichever fibered solid torus or tori.</p> <p>We could say that $(On,1\mid (1,b))=Q\cup_T W(1,b)$, for a fibered $(1,b)$ solid torus $W$</p>