PlanetMath: 4 surface bundles

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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 is ${\mathbb{Z}}_2\oplus{\mathbb{Z}}_2$ .

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

These bundles are

$K\times S^1$ , the trivial Cartesian product
$K\times_{\tau}S^1=$ ,
$K\times_yS^1=K \stackrel{\sim}\times I^O \cup_{(0,1)} M\ddot{o}\times S^1$ ,
$K\times_{y\tau}S^1$ .

Where $K\stackrel{\sim}\times I^O$ is the orientable twisted -bundle over , among the three -bundles over .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 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:

55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

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