# 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

$(-z)^a\bar{w}^b=-z^aw^b$

$(-z)^c\bar{w}^d=\bar{z}^c\bar{w}^d$

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 ${\cal{MCG}}(K)=\mathbb{Z}_2\oplus\mathbb{Z}_2$.

Filed under algebra, topology

## 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…