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 $S^2/S^0$?
Best Answer
Take the CW-decomposition to be one 0-cell and one 1-cell and one 2-cell, where the boundary of the 2-cell maps onto the 0-cell and the boundary of the 1-cell also maps onto the 0-cell. Then by van-Kampen on $X=A\cup B$, with $A$ slightly containing more than the 2-cell and $B$ slightly containing more than the 1-cell, we have $A\cap B\simeq\lbrace\text{0-cell}\rbrace$ and $\pi_1A\cong\pi_1S^2=0$ and $\pi_1B\cong\pi_1S^1=\mathbb{Z}$ and hence $\pi_1X\cong\mathbb{Z}$.
The problem with your given cell structure is that you haven't actually told us the attaching maps... for instance, where is your 1-cell attaching to? By definition, it must attach to cells of dimension $<1$, i.e. to your given 0-cell.