4 surface bundles

Four Kleinbottle bundles KMS1.

There are four because the extended mapping class group for the genus two, non orientable surface K the Klein bottleMathworldPlanetmath, is 22.

This group is generated by a Dehn-twist τ about the unique two-sided curve in K and by the y-homeomorphismPlanetmathPlanetmath, both representing two isotopy classes of order two.

These bundles are

  • K×S1, the trivial Cartesian product

  • K×τS1,

  • K×yS1=K×IO(0,1)Mo¨×S1,

  • K×yτS1.

Where K×IO is the orientable twisted I-bundle over K, among the three I-bundles over K.The symbol (0,1) is used to indicate that, the meridian in (Mo¨×S1) is attached to the meridian of (K×IO), both 2-tori. Mo¨ is the Möbius band.

Now, since those monodromies are periodic then they are also homeomorphicMathworldPlanetmath respectively to the Seifert fiber spaces

  • (NnI,2|0)=K×S1,

  • (NnI,2|1)=(K×S1intW)(1,1)W,

  • (NnII,2|0)=K×yS1=K×IO(0,1)Mo¨×S1 and

  • (NnII,2|1)=(K×yS1intW)(1,1)W

Where W is a solid torus in the space and (1,1) is the Dehn surgeryMathworldPlanetmath: meridian of W to the longitude of (K×S1intW).

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