CW complex for Möbius strip I was asked to find a CW complex for the Möbius strip with one 0-cell, two 1-cells, and one 2-cell.
I can find a CW complex for a Möbius strip with more cells (two 0-cells, three 1-cells and a single 2-cell), but this doesn't help me.
I was hinted that I need to attach the 2-cell's boundary to $ab^2$ where $a$,$b$ are the 1-cells. I don't see why this gives me a Möbius strip. Is there a way of coming up with it "automatically" from the fundamental polygon of the Möbius strip i.e. $I \times I$ under the identification $(x,0)\approx (1-x,1)$? There seems to be two 0-cells (two corners) and I'm asked for a single 0-cell so I'm guessing it's unrelated.. if so then what is the way to come up with this?
 A: Try "cutting and regluing" as shown in this picture.
(My picture shows the steps in reverse, so read it from bottom to top!)

A: Perhaps the CW complex that you found, the one with two 0-cells, three 1-cells, and a single 2-cell, might help you after all.
Look at your cells and ask yourself: Do I really need them all? Can I remove some of them, thereby decreasing the number of cells? Or can I merge several of them into fewer cells, thereby decreasing the number of cells?
Here's an example in dimension $1$. Suppose I take the following CW decomposition for the topological space $S^1$, meaning the unit circle in the plane. The $0$ skeleton $X^{(0)}$ has two 0-cells, $X^{(0)}=\{v_0,v_1\}$ where $v_0$ is the point $(1,0)$ and $v_1$ is the point $(-1,0)$; and the $1$-skeleton $X^{(1)}$ equals $X$ itself, having two open 1-cells, namely the open upper half semicircle and the open lower half semicircle of $S^1$
Now I look at the 0-cells, and I focus on the 0-cell $v_1$. I notice that if I continuously moved $v_1$ around the circle in the counterclockwise, making $e_0$ longer and $e_1$ shorter, eventually $v_1$ merges with $v_0$ making a single 0-cell, $e_0$ expands to become $S^1 - \{v_0\}$, and $e_1$ shrinks down into nothingness. In this manner I have visualized a new CW structure on $S^1$ with one 0-cell $v_0$ and one open 1-cell $S^1 - \{v_0\}$.
I suspect that if you carefully examine your CW decomposition of the Möbius band, something similar might happen.
