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_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 $I$ -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

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

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.