The set is an orientable 3-manifold with boundary. In the illustration we see in orange the möbius band at and a small regular neigbourhood of her removed without her, i.e., if , then which is ?
the last step is in orange, and without , for
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.
MoxI is a solid Klein bottle, the genus one nonorientable handlebody. Here we see it with some typical curves on
look a rough view
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:
- 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…
Three dimensional -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 -bundle over by using the monodromy , yes, the minus-identity auto-homeomorphism, of the genus-three-non-orientable-surface .
Its Orlik-Raymond presentation is , remember, corresponds to the class in Seifert symbols
So, in this compact 3-manifold we can see a double structure, as a surface bundle:
or as a circle bundle
where is a 2-orbifold with three cone-points and a reflector circle