splitting into handlebodies

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 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.

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

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

mapping class group of the Klein bottle

the Klein bottle is gotten from a quotient in the two-torus via

so to get all the isotopy classes one need to consider the auto-homeomorphisms of the torus.

It is well known that each of those are determined by a two-by-two matrix of , i.e where , so in seeking those matrices which obey the above gluing conditions we are compeled to analyse

for each couple .

Then if we set this implies and , so is odd. Also,  if then and hence  .

But , then are the only matrices which preserve the gluing conditions.

It is easy to see that these four matrices form the famous 4-Klein group. So .

 

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, , by its similar border. The Kleinbottle is the surface got from sewing two mobiusbands along with their boundaries.

Let’s abstract with the symbols  to the orientable surface of -genus and by the non orientable surface of -genus.

With these we have an algebra of sets that looks like that

• 
• 
• 
• 

where we have abstracted: , for a punctured surface , a closed surface with a open disk removed. And 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 the only three dimensional manifolds in math?

Even if you think and say that 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…

can you see a solid torus?

The 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 , the twisted interval bundle over the Möbius strip.  In constrast is the genus one non orientable handlebody: the solid Klein bottle. Isn’t difficult to prove that they are the only two -bundles over