Tag Archives: topology

möbius doble cover


The set E=M\ddot{o}\stackrel{\sim}\times I is an orientable 3-manifold with boundary. In the illustration we see in orange the möbius band at \frac{1}{2} and a small regular neigbourhood of her removed without her, i.e., if Q={\cal{N}}(M\ddot{o}\times \frac{1}{2})\smallsetminus (M\ddot{o}\times \frac{1}{2})\subset E, then which is E\smallsetminus Q?

 

 

 

 

 

 

 

 

 

 

 

 

 

the last step is M\ddot{o}\times\frac{1}{2} in orange, and M\ddot{o}\stackrel{\sim}\times I without M\ddot{o}\stackrel{\sim}\times (\frac{1}{2}-\varepsilon, \frac{1}{2}+\varepsilon), for \varepsilon=|\varepsilon|\to \frac{1}{2}

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elementary classes of 3-manifolds


Strangely, there are very few places that offer simple types or simple constructions of these topological beings. Sometimes innocent guessing make us to think that only \mathbb{R}^3 is the only space we need to appreciate all the available 3d-complexity.

Today there are: “obvious ones”

To learn more,  follow to the elementary classes of 3-manifolds lists.

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Moxi: the cartesian product of the mobius-strip and an interval


MoxI is a solid Klein bottle, the genus one nonorientable handlebody. Here we see it with some typical curves on

MoreandMore

look a rough view

another earlier

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two and three dimensional manifolds


when any someone who is asked to comment about: how many surface’s shapes are there? he or she  could tell you this: if only closed surfaces are considered, there’re two types:

  • orientable
  • non orientable

Orientable are the sphere (the 2-sphere), the torus, double torus, etc… But non orientable are the projective plane: an abstract simple construction given by sewing a mobiusband with a 2-disk, D, by its similar border. The Kleinbottle is the surface got from sewing two mobiusbands along with their boundaries.

Let’s abstract with the symbols O_g to the orientable surface of g-genus and by N_k the non orientable surface of k-genus.

With these we have an algebra of sets that looks like that

  • N_1=M\ddot{o}\cup_{\partial}D
  • N_2=M\ddot{o}\cup_{\partial}M\ddot{o}
  • N_3={O_1}_o\cup_{\partial}M\ddot{o}={N_2}_o\cup_{\partial}M\ddot{o}
  • N_4={N_3}_o\cup_{\partial}M\ddot{o}

where we have abstracted: F_o=F\setminus{\rm{int}}D, for a punctured surface F, a closed surface with a open disk removed. And \cup_{\partial} for sewing in the border

Even to them who know those ideas come into surprise when they are told that there’s no only one three dimensional shape: is \mathbb{R}^3 the only three dimensional manifolds in math?

Even if you think and say that S^2\times S^1 is another together with others little more easy to imagine. 

Would you like to see a sample of the great variety of them? urgently follow this link here please!

This link gonna give you a rough idea of several types of 3-spaces looking a little more close at them,… some are  now celebrated in 3d-topological technology…

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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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