# Tag Archives: fiber bundle

## a pinch of beams

a little taste of math bundles for my friends: hit

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## a non orientable 3d-manifold with boundary

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.

## 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