# 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

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

• $K\times_{\tau}S^{1}$,

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

• $K\times_{y\tau}S^{1}$.

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

• $(NnI,2|0)=K\times S^{1}$,

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

• $(NnII,2|0)=K\times_{y}S^{1}=K\lx@stackrel{{\scriptstyle\sim}}{{\times}}I^{O}% \cup_{(0,1)}M\ddot{o}\times S^{1}$ 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.

Title 4 surface bundles 4SurfaceBundles 2013-03-22 16:01:40 2013-03-22 16:01:40 juanman (12619) juanman (12619) 12 juanman (12619) Feature msc 55R10 SurfaceBundleOverTheCircle