Tag Archives: fiber bundle

a pinch of beams


a little taste of math bundles for my friends: hit

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Filed under differential geometry, geometry, mathematics, multilinear algebra, topology

a non orientable 3d-manifold with boundary


Mö x I

Mö x I

The picture on the right is representation and a description of its parts of  a 3d chunk, it is the trivial I-bundle over the möbius-strip, M\ddot{o}\times I. 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 C\ddot{o} the core of the möbius strip, you can deduce for the curve C\ddot{o}\times\{\frac{1}{2}\} (the black pearled one) that its tangent bundle can’t be embedded in the tangent bundle of \mathbb{R}^3 or in any other tangent bundle of an orientable one.

Remember N_2 is the Klein bottle. Let’s the image talks by itself.

Ah, il nome di questo grande pezzo è la bottiglia solida di Klein.

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Circle-bundles over surfaces are more known than surface-bundles over the circle


Three dimensional S^1-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 N_3-bundle over S ^1 by using the monodromy -\mathbb{I}, yes, the minus-identity auto-homeomorphism, of the genus-three-non-orientable-surface N_3.

Its Orlik-Raymond presentation is \{0;(n_2,2,0,1);(1,0)\}, remember, n_2 corresponds to the class No in Seifert symbols

So, in this compact 3-manifold we can see a double structure, as a surface bundle

N_3\subset \{0;(n_2,2,0,1);(1,0)\}\to S^1 

or as a circle bundle

S^1\subset \{0;(n_2,2,0,1);(1,0)\}\to \omega

where \omega is a 2-orbifold with three cone-points and a reflector circle

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3d bundles


Do you want a lot of really new problems in la topology? consider tri-dimensional fiber bundles E, of the form

 F\subset E\to B

where by taking the exact combinations on the dimension of the fiber F and the dimension of the base B,  to be \dim F+\dim B = 3, you will get many possibilities. 

For example, if F is a two-manifold (a surface) then you must choose S^1 to get non trivial surface bundles. Knowing that the mapping class group of the surface {\cal{MCG}}(F),  classify the possible E‘s and since {\cal{MCG}}(F) increases (depending which of three types of auto-homeomorphismus: periodic, reducible or pseudo-Anosov) its complexity as the genus of F rises, then you will have a “bundle” of questions to tackle, to amuse, even to gain a PhD… 

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Filed under 3-manifolds, algebra, fiber bundle, geometry, math, topology