My question is related to this one here but is different in that I am wondering about the CW structure on such a space. I am trying to put a CW structure on $S^2/S^0$ and I think that we have $1$ 0 -cell, $1$ 1 - cell and $1$ 2-cell. My $1$ - cell is just some path connecting the north pole to the south pole. However then I run into trouble because by Van - Kampen the fundamental group of $S^2/S^0$ is zero and not $\Bbb{Z}$ as stipulated in the link above.
Question: What is wrong with this cell structure on $S^2/S^0$? It seems to get $\Bbb{Z}$ I would need it to have $2$ 1 -cells but how is this obvious from the definition of $S^2/S^0$?