Tag Archives: 3-manifolds

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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a theorem about 3-manifolds and circle-foliations

si M es una tres-variedad cerrada y foliada por uno-esferas sobre un orbifold que tiene más de dos curvo-reflectores, y pero que M no admite al tres-toro como cubriente entonces el SW-género de M es mayor que uno

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Filed under fiber bundle, geometry, low dimensional topology, Mapping class group, 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…


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can you see a solid torus?

dcoato5The image on the right is a solid torus with four lateral annulus in colours: orange, green, yellow and blue.

Solid tori are important elementary type of 3-manifolds. Also called orientable genus one handlebodies and can be fibered -in the sense of Seifert- by longitudinal circles, and in many different ways.

Among three dimensional technologeeks, they are customed to see a solid torus that can be fibered as the M\ddot{o}\stackrel{\sim}\times I, the twisted interval bundle over the Möbius strip.  In constrast M\ddot{o}\times I is the genus one non orientable handlebody: the solid Klein bottle. Isn’t difficult to prove that they are the only two I-bundles over M\ddot{o}

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congreso nacional de la Sociedad Matemática Mexicana

Esta próxima semana del 12-16 de octubre tendremos Congreso Nacional (mexicano) de Matemáticas, el 42avo y  en la ciudad de Zacatecas, además  me es grato comunicarles que el CUCEI tiene presencia con al menos 3 ponencias:

  • Sum of reciprocal central binomial coefficients por David Íñiguez Báez
  • Geometría diferencial de superficies con la derivada covariante por Juan M. Márquez
  • Surface bundles over the circle por Juan M. Márquez

La primera y la tercera son reportes de investigación de correspondientes trabajos de tesis, éstas en las sesiones de combinatoria y matemáticas discretas y la otra en la sesiones de topología algebraica, respectivamente. David va a hablar sobre las relaciones que se dan acerca de las sumas de recíprocos de entre dos importantes secuencias crecientes de números en la combinatoria: números de Catalan y los coeficientes binomiales centrales. Juan hablará del estudio topológico de una variedad de tres dimensiones que surge dentro de los objetos conocidos como fibrados de Seifert (Seifert bundles) y que resulta tienen una característica especial: contiene un circle completo de singularidades en la correspondiente superficie de órbitas.

La segunda es una plática de divulgación acerca de la forma en que, acá en el CUCEI, estudiamos una parte de la geometría diferencial. Ésta, dentro de las sesiones de geometría y geometría algebraica.

El programa recientemente (1 de octubre) ha visto la luz y ya estabamos medio desesperados pues la publicación de esta información ya se había tardado mucho, en fin ya esta ahí y ahora habrá que planear a que cosas vamos a asistir en ese baquete intelectual.

¿Quieres saber qué más habrá allá?  échale un ojo aquí


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Filed under Catalan numbers, differential geometry, mathematics, sum of reciprocals, what is math