4 surface bundles

Four Kleinbottle bundles $K\subset M\to S^{1}$ .

There are four because the extended mapping class group for the genus two, non orientable surface $K$ the Klein bottle, 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

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

Where $K\lx@stackrel{{\scriptstyle\sim}}{{\times}}I^{O}$ 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 $\partial(K\lx@stackrel{{\scriptstyle\sim}}{{\times}}I^{O})$ , 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

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$(NnI,2|0)=K\times S^{1}$ ,
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$(NnI,2|1)=(K\times S^{1}\setminus{\rm int}W)\cup_{(1,1)}W$ ,
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$(NnII,2|0)=K\times_{y}S^{1}=K\lx@stackrel{{\scriptstyle\sim}}{{\times}}I^{O}% \cup_{(0,1)}M\ddot{o}\times S^{1}$ and
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$(NnII,2|1)=(K\times_{y}S^{1}\setminus{\rm int}W)\cup_{(1,1)}W$

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.

Title	4 surface bundles
Canonical name	4SurfaceBundles
Date of creation	2013-03-22 16:01:40
Last modified on	2013-03-22 16:01:40
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	12
Author	juanman (12619)
Entry type	Feature
Classification	msc 55R10
Related topic	SurfaceBundleOverTheCircle