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 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

• $K\times S^1$, the trivial Cartesian product

• $K{\times }_{\tau }{S}^1$,

• $K{\times }_y{S}^1=K\stackrel{\sim }{\times }{I}^O{\cup }_{(0,1)}M\ddot{o}\times S^1$,

• $K{\times }_{y\tau }{S}^1$.

Where $K\stackrel{\sim }{\times }{I}^O$ is the orientable twisted $I$-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

• $(NnI,2|0)=K\times S^1$,

• $(NnI,2|1)=(K\times S^1\setminus \mathrm{int}W){\cup }_{(1,1)}W$,

• $(NnII,2|0)=K{\times }_y{S}^1=K\stackrel{\sim }{\times }{I}^O{\cup }_{(0,1)}M\ddot{o}\times S^1$ and

• $(NnII,2|1)=(K{\times }_y{S}^1\setminus \mathrm{int}W){\cup }_{(1,1)}W$

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.