Characteristic map of a n-cell in a CW complex

Question

I have a problem in understanding the purpose of the definition of a CW complex. What really would help me is to understand the following: Let $\sigma$ be a n-cell and $\Phi_\sigma:\mathbb D^n \to X$ be the characteristic map. Is there any relation between $\Phi_\sigma(\mathbb S^{n-1})$ and $\partial \sigma$ following directly from the Definition? I know, for example, that $\Phi_\sigma(\mathbb S^{n-1})\subset X^{n-1}$, $X^{n-1}$ being the $(n-1)$-Skeleton, but what does this tell me about the boundary of $\sigma$?

Answer 1

If the cell $e^n_\alpha$ is defined as $\Phi_\alpha(\text{int}D^n)$, then you have the following relation:
Since the characteristic map $\Phi_\alpha$ is continuous, you always have $\Phi_\alpha(D^n)=\Phi_\alpha\left(\overline{\text{int}D^n}\right)\subseteq\overline{e^n_\alpha}$. On the other hand, once you have proven that CW-complexes are Hausdorff (and, more generally, normal), you know that compact subsets are closed. Therefore $\Phi(D^n)$ is closed, and since it contains $e^n_\alpha$, it also contains the closure $\overline{e^n_\alpha}$. So in the end, $\Phi_\alpha(D^n)=\overline{e^n_\alpha}$.

But beware that $\partial e_\alpha$ is in general not the same as $\Phi_\alpha\left(S^{n-1}\right)$. For example, $\partial e_\alpha$ can be the cell $e_\alpha$ itself if it's a $2$-cell with a $3$-cell glued to it.

Answer 2

I think it helps to have some pictures. Here is a picture of a $1$-dimensional cell complex.

For a $2$-dimensional example, try attaching a $2$-cell to the unit interval $[0,1]$, itself thought of as a cell complex with two $0$ cells and one $1$-cell. (The mouth in the face.) Here are two examples of attaching a $2$-cell.

In the first the attaching map takes the boundary of the $2$-disc to the point $1$. In the second, it takes the boundary of the $2$-disc to somewhere in the middle of the unit interval.

These pictures are taken from Topology and Groupoids (colour in the e-version). I think it helps also to develop some of the general theory of attaching a space $X$ to a space $B$ by means of a map $f:A \to B$ on a closed subspace $A$ of $X$.

Answer 3

For example, $\partial e_\alpha$ can be the cell $e_α$ itself if it's a 2-cell with a 3-cell glued to it..

seems to come out wrong. If this is correct, then by definition of boundary and characteristic map $\phi_\alpha$:

$$ \partial\Phi_\alpha({\rm int}D^n) = \overline{\Phi_\alpha({\rm int}D^n)} - \Phi_\alpha({\rm int}D^n) = \Phi_\alpha({\rm int}D^n), $$

So $\Phi_\alpha({\rm int}D^n)$ is a clopen subset of CW complex whose boundary is non-empty, contradiction.