I am watching a video here:
http://youtu.be/KVjRILkbILA?t=1m42s
one can see that the author explains that:
$$\rho_{\alpha}:S^1 \rightarrow e^0$$
The problem that I am having is how he got there, especially as he says that the attachment map for $X^n$ is $\rho_{\alpha}: S^{n-1} \rightarrow X^{n-1}$. If I am originally looking at $X^2$ then this would lead to $\rho_{\alpha}: S^1 \rightarrow X^1$. However isn't $X^0 = e^0$ and not $X^1$?
Furthermore by definition $$X^n = \frac{X^{n-1} \coprod D_{\alpha}^n}{x \sim \rho_{\alpha} (x)}$$
If I do this then we build $X^2$ from $X^1$ by the attachment map , as he says, then we attach a 2-cell to it, which is just attaching a rubber sheet into $S^1$ to get $D^2$; so $X^2 = \frac{X^{1} \coprod D_{\alpha}^2}{x \sim \rho_{\alpha}(x)}$. Then somehow he gets ${X^2 = X^{1} \coprod e^2}$. The $X^1$ is still there and so it seems as if he is modding only $D_{\alpha}^2$ by $x \sim \rho_{\alpha}(x)$, thus $\frac{D_{\alpha}^2}{x \sim \rho_{\alpha}(x)}$ I don't see how this creates a 2-cell $e^2$ as $e^2$ is just the boundary $\partial D^2 = S^1$ removed.
In the end though he says that $S^2$ can be built by only $e^0$ and $e^2$, I am having quite a hard time seeing this. He also says that we can build $X^1$ from $X^0$ by attaching 1-cells, so where are the $e^1$'s?
Thanks much for any insight,
Brian
$\mathbb{S}^2$ can be built from only $e^0$ and $e^2$ simply by considering the cell decomposition $\mathbb{S}^2=(\mathbb{S}^2\setminus \{\infty\})\cup \{\infty\}$ (the zero cell is $\infty$). Generally, the $n$-sphere $\mathbb{S}^n$ admits a cell decomposition consisting of a single $n$-cell and a single $0$-cell (the point at $\infty$) as one can check. (Of course, the zero cell needn't be the point at $\infty$; you can pick any point you like on $\mathbb{S}^n$ to be the zero cell and the complement of this point to be the $n$-cell. The fact that the complement of this point is indeed an $n$-cell, i.e., that it is indeed isomorphic to an $n$-ball is a consequence of stereographic projection.)
If you could tell me what your symbols $X^0$, $X^1$ etc. are, then I am happy to elaborate further on the other points on which you might be stuck.
I hope this helps!