# 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

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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## 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

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

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.

## möbius

In 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?

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?

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