4 surface bundles
Four Kleinbottle bundles $K\mathrm{\subset}M\mathrm{\to}{S}^{\mathrm{1}}$.
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 Dehntwist $\tau $ about the unique twosided curve in $K$ and by the yhomeomorphism^{}, 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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$K{\times}_{y}{S}^{1}=K\stackrel{\sim}{\times}{I}^{O}{\cup}_{(0,1)}M\ddot{o}\times {S}^{1}$,

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$K{\times}_{y\tau}{S}^{1}$.
Where $K\stackrel{\sim}{\times}{I}^{O}$ 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 2tori. $M\ddot{o}$ 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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$(NnI,20)=K\times {S}^{1}$,

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$(NnI,21)=(K\times {S}^{1}\setminus \mathrm{int}W){\cup}_{(1,1)}W$,

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$(NnII,20)=K{\times}_{y}{S}^{1}=K\stackrel{\sim}{\times}{I}^{O}{\cup}_{(0,1)}M\ddot{o}\times {S}^{1}$ and

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$(NnII,21)=(K{\times}_{y}{S}^{1}\setminus \mathrm{int}W){\cup}_{(1,1)}W$
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 $\partial (K\times {S}^{1}\setminus \mathrm{int}W)$.
The non trivial homeomorphisms were given by Per Orlik and Frank Raymond, in 1969.
Title  4 surface bundles 

Canonical name  4SurfaceBundles 
Date of creation  20130322 16:01:40 
Last modified on  20130322 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 