Denote $Y$ the space I mentioned in the title. The original problem is Hatcher's Exercise 1.2.12 part (b) which ask to prove $\pi_1(Y) = \langle a,b,c|aba^{-1}b^{-1}cb^\epsilon c^{-1}\rangle$ for $\epsilon=\pm 1$. It seems the cell structure on $Y$ is
But I can't see why this is a cell structure of $Y$. Could you give me some drawing or at least some intuition of this?
I am working through Hatcher and I ran into this same problem. First, it helped to read part of the example he cites in that problem (ex. 1B.13). Hatcher says this: "Suppose we take a torus, delete a small open disk, then identify the resulting boundary circle with the a longitudinal circle of the torus." This space is homeomorphic to the space we are working with in this exercise. (To convince myself I had to draw it out a couple times). With this intuition, the picture you provided makes sense. The torus can be represented as a square with pairs of sides labelled $a$ and $b$ identified. The torus minus a disk will appear as that same square except with a disk removed in the center, just as you have drawn. Since the boundary circle is identified with side $b$ of the square, the 1-skeleton of this shape becomes more apparent as does the path where the required 2-cell is attached. Then corollary 1.28 gives the desired result.