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?
Best Answer
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.