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4 surface bundles (Feature)
Four Kleinbottle bundles $ K\subset M\to S^1$.

There are four because the extended mapping class group for $ K$ is $ {\mathbb{Z}}_2\oplus{\mathbb{Z}}_2$.

This group is generated by a Dehn-twist $ \tau$ about the unique two-sided curve in $ K$ and by the y-homeomorphism, both representing two isotopy classes of order two.

These bundles are

Where $ K\stackrel{\sim}\times I^O$ is the orientable twisted $ I$-bundle over $ K$, among the three $ I$-bundles over $ K$.The symbol $ \cup_{(0,1)}$ is used to indicate that, the meridian in $ \partial(M\ddot{o}\times S^1)$ is attached to the meridian of $ \partial(K\stackrel{\sim}\times I ^O)$, both 2-tori. $ M\ddot{o}$ is the Möbius band.

Now, since those monodromies are periodic then they are also homeomorphic respectively to the Seifert fiber spaces

  • $ (NnI,2\vert)=K\times S^1$,
  • $ (NnI,2\vert 1)=(K\times S^1\setminus{\rm int} W)\cup_{(1,1)}W$,
  • $ (NnII,2\vert)=K\times_yS^1=K \stackrel{\sim}\times I^O \cup_{(0,1)} M\ddot{o}\times S^1$ and
  • $ (NnII,2\vert 1)=(K\times_y S^1\setminus{\rm int} W)\cup_{(1,1)}W$

Where $ W$ is a solid torus in the space and $ \cup_{(1,1)}$ is the Dehn surgery: meridian of $ \partial W$ to the longitude of $ \partial(K\times S^1\setminus{\rm int} W)$.

The non trivial homeomorphisms were given by Per Orlik and Frank Raymond, in 1969.



"4 surface bundles" is owned by juanman.
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See Also: surface bundle over the circle

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Cross-references: homeomorphisms, Dehn surgery, torus, solid, Seifert fiber spaces, homeomorphic, periodic, monodromies, Möbius band, orientable, Cartesian product, order, classes, isotopy, y-homeomorphism, curve, generated by, group, mapping class group

This is version 8 of 4 surface bundles, born on 2006-06-22, modified 2006-09-14.
Object id is 8070, canonical name is FourSurfaceBundles.
Accessed 870 times total.

Classification:
AMS MSC55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

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