Tag Archives: non orientable

reflector circle at a punctured torus


Sea T_o un toro 2-dimensional donde hemos removido un disco cerrado,
sea S=T_o\cup_{\partial}\bar{A} donde \bar{A} es aro S^1\times I con S^1\times 0 es la frontera
de una “vecindad” de una curva cerrada simple reflectora, y S^1\times 1 como la curva reflectora. Tal “aro reflector”, \bar{A} tiene como^ orbifold – grupo fundamental a \pi_1(\bar{A})=\Bbb{Z}\times{\Bbb{Z}}_2. Entonces el producto amalgamado es:

reflector circle

reflector circle

Esto es divertido por que es bien sabido que la superficie cerrada de género tres no orientable,  N_3 tiene presentación parecida a esta última.

Observemos que las respectivas abelianizaciones son \mathbb{Z}^2+\mathbb{Z}_2

Entonces es ¿cierto o no qué el concepto de curva reflectora dado por P.Scott no sea el mismo que el de curva reflectora en una superficie no orientable?

Recuerde que, poner una curva reflectora a una superficie orientable es para hacer una superficie no orientable de tipo T\#\cdots\#T\#{\Bbb{R}}P^2 de género impar, donde T es el toro 2D.

^ footnote{Ref[P. Scott en 424p. “The Geometries of 3-Manifolds“, 1983]}

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Filed under algebra, cucei math, group theory, low dimensional topology, topology

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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Filed under cucei math, fiber bundle

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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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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Filed under fiber bundle, geometry, topology

möbius


moebio10In math there is a geometric-topologic construction which is a source of fun and surprises. It is surface. It is a two dimensional object which can be embed in the three dimensional euclidean space, \mathbb{R}^3. It is a model to illustrate the concept of twisted bundle. In fact an I-bundle, an interval bundle over the circle.  The möbius band or möbius strip is the principal element to construct all the non orientable surfaces: N_1 the projective plane; N_2 the Klein bottle; N_3; N_4,… etc.

What happens if we remove the core central simple closed curve? look at the 2nd image.

I had spended many eons to popularize the symbol

M\ddot{o}

for her. Isn’t this is beautiful?moebio20

The möbius band is employed to manufature many exotic non orientable three dimensional spaces. By the record, talking about 3d spaces:  \mathbb{R}^3 isn’t the only one.

For example M\ddot{o}\times I is a solid Klein bottle and an innocent quiz is: can you see what is the bounding surface of this I-bundle?

For a bundle of additional properties check this link

The images were rendered in Mathematica v.5

La siguiente figura te explica como ver una construcción tridimensional que involucra a la banda…

Mö x I

Aquí estamos usando N_2 para la superficie de género dos no orientable,,

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Filed under cucei math, fiber bundle, low dimensional topology, topology