I'm asked to prove that the following is a CW structure for the 3-sphere, (as a part of an exercise involving defining the Cw structure of the Lens Spaces) I'm asked to prove that the following is a CW decomposition
But in order to visualise it I started computing an homeomorphism between the interior of any $2$-cell and a $2$-disk and guessing where the $2$-cells are attached. Without much results. Same problem for the $3$-cells.
I tried looking in some other alg. top. books like Hatcher's in order to see if there are some explicit computations and I found a more geometric interpretation. Which should help clarify, but I cannot work with it properly:
With Hatcher's description is not clear to me where the $2$-cells are attached and how, moreover it seems that they are attached to an $S^1$ which is not part of the CW structure and by an interior point to one of the $(0,p_j)$, where $p_j$ is the j-th roots of unity. In other words, it seems that it's not a $2$-cell because the identification is not along the boundary and it doesn't attach to the right skeleton.
I want to work with the cell description provided at the beginning of the question, because then it would be easy to formalise it and to show that the rotation defining a lens space is a cellular map. So, someone can explain (or give an hint) about why the sets $e_r^i$'s are cells and how are they attached to the $i-1$-skeleton (e.g. helping on the computations explained above)?