I have to find CW complex for Mobius Strip. I can find a CW complex for a Möbius strip with more cells (one 0-cells, two 1-cells and a single 2-cell). My cells are points $A=(0,0) \approx (1,1)$, part of circle $ABBA$ without point A second 1-cell is $\{(0,t)\approx(1,1-t) \mid 0<t<1\}$ and 2-cell is $\{(x,y) \mid x,y\notin \{0,1\}\}$. (I am not sure this is good explanation, but I hope you understand). I find homeomofisam for this cells. $$f_2(z) = \begin{cases} \frac{\lVert z \rVert_2}{\lVert z \rVert_\infty}z &\quad\text{if } z \ne 0 \\ \text{} 0 &\quad\text{if } z = 0 \\ \end{cases}$$ is attaching mapp for 2-cell $$f_1(x)=\begin{cases} (-x,1) &-1 <x \le 0\\ (x,0)& 0<x<1 \end{cases} $$ is attaching a map for ABBA. It is easy to see that these maps are continuous. And $f(x)=\frac{x+1}{2}$ for second 1-cell. Is this okey?

According to this question, there is CW complex with one 0-cell,one 1-cell and one 2-cell. I think that o-cell is A, 1-cell is circle ABBA, and 2-cell is $Int(M)$, but I can't find homeomorfism for 2-cell. Can someone help?

edit: picture on the second picture with red color is 1-cell on the third is another 1-cell with purple color and on forth picture with blue color is 2-cell


1 Answer 1


According to this question, there is CW complex with one 0-cell,one 1-cell and one 2-cell.

No such CW structure exists on the the Möbius strip. Moreover the linked question doesn't claim that, and the answer that claimed that was deleted.

It is well known that the Euler characteristic of the Möbius strip is zero. Because given a one $0$-cell, two $1$-cells, and one $2$-cell structure (which we already know is a correct CW structure) we have $1-2+1=0$.

While the Euler charateristic of any CW complex with one $0$-cell, one $1$-cell and one $2$-cell is $1-1+1=1$.

The Euler characteristic is a topological (even homotopy) invariant. All CW structures on a given space have to give the same value. And so the number of cells cannot be arbitrary, it has to satisfy certain rules.

Is this okey?

No, it's not. For same reasons as earlier there is no "two 0-cells, two 1-cells and a single 2-cell" structure on the Möbius strip as well, because its Euler characteristic is $2-2+1=1$.

Your description is not really clear to me unfortunately. I'm not exactly sure how you define those two $1$-cells. Still any non-zero Euler characteristic attempt has to fail.


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