a little taste of math bundles for my friends: hit
Tag Archives: fiber bundle
The picture on the right is representation and a description of its parts of a 3d chunk, it is the trivial -bundle over the möbius-strip, . It is useful to determine which 3d spaces are non orientable.
A 3d-space is non orientable if it has a simple closed curve whose 3d regular neighborhood is homeomorphic to the model of the picture.
If we denote by the core of the möbius strip, you can deduce for the curve (the black pearled one) that its tangent bundle can’t be embedded in the tangent bundle of or in any other tangent bundle of an orientable one.
Remember is the Klein bottle. Let’s the image talks by itself.
Ah, il nome di questo grande pezzo è la bottiglia solida di Klein.
Three dimensional -bundles which must be based on a two-manifolds were studied since 1930’s by H. Seifert. Commonly dubbed Seifer fiber spaces, they only include bundles by using singular fibers above cone points in their Zerlegungfläschen, that is 2D-cone-orbifolds
A most general point of view is considered when we ask for 3D-manifolds foliated by circles, these must be called Seifert-Scott fiber spaces and it is needed to consider reflector lines or circles in the orbit-surface
Recently, it was unveiled a -bundle over by using the monodromy , yes, the minus-identity auto-homeomorphism, of the genus-three-non-orientable-surface .
Its Orlik-Raymond presentation is , remember, corresponds to the class in Seifert symbols
So, in this compact 3-manifold we can see a double structure, as a surface bundle:
or as a circle bundle
where is a 2-orbifold with three cone-points and a reflector circle
Do you want a lot of really new problems in la topology? consider tri-dimensional fiber bundles , of the form
where by taking the exact combinations on the dimension of the fiber and the dimension of the base , to be , you will get many possibilities.
For example, if is a two-manifold (a surface) then you must choose to get non trivial surface bundles. Knowing that the mapping class group of the surface , classify the possible ‘s and since increases (depending which of three types of auto-homeomorphismus: periodic, reducible or pseudo-Anosov) its complexity as the genus of rises, then you will have a “bundle” of questions to tackle, to amuse, even to gain a PhD…