Tag Archives: Klein bottle

mapping class group of the Klein bottle

the Klein bottle is gotten from a quotient in the two-torus S^1\times S^1 via

K=\frac{S^1\times S^1}{(z,w)\sim(-z,\bar{w})}

so to get all the isotopy classes [f]:K\to K one need to consider the auto-homeomorphisms \phi:T\to T of the torus.

It is well known that each of those are determined by a two-by-two matrix of GL_2(\mathbb{Z}), i.e (z,w)\to (z^aw^b,z^cw^d) where \det\left(\begin{array}{cc}a&b\\c&d\end{array}\right)=ad-bc=\pm1, so in seeking those matrices which obey the above gluing conditions we are compeled to analyse



for each couple z,w\in S^1.

Then if we set w=1 this implies (-1)^az^a=-z^a and (-1)^a=-1, so a is odd. Also,  if z=1 then \bar{w}^b=w^b and hence b=0 .

But ad=\pm1, then \left(\begin{array}{cc}\pm1&0\\ 0&\pm1\end{array}\right) 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 {\cal{MCG}}(K)=\mathbb{Z}_2\oplus\mathbb{Z}_2.


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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