PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Medium Entry average rating: No information on entry rating
4 surface bundles (Feature)
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_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.




"4 surface bundles" is owned by juanman.
(view preamble | get metadata)

View style:

See Also: surface bundle over the circle

Log in to rate this entry.
(view current ratings)

Cross-references: homeomorphisms, longitude, Dehn surgery, torus, solid, Seifert fiber spaces, homeomorphic, periodic, monodromies, Möbius band, orientable, Cartesian product, order, classes, isotopy, y-homeomorphism, curve, generated by, group, Klein bottle, non orientable surface, genus, mapping class group

This is version 9 of 4 surface bundles, born on 2006-06-22, modified 2009-10-27.
Object id is 8070, canonical name is FourSurfaceBundles.
Accessed 1470 times total.

Classification:
AMS MSC55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)