# 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 $\varepsilon=|\varepsilon|\to \frac{1}{2}$

Filed under math

## 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”

Filed under 3-manifold, 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

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

## 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

Filed under fiber bundle, math, topology

## 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…