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


look a rough view

another earlier


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…


Filed under math

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.


Filed under fiber bundle, geometry, topology


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


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,,


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