CW complex adjunction map

Question

In topology we defined a quotient topology for glueing in the following way: Let $(X,O)$ and $(Y,O)$ be topological spaces and $f:A \subseteq X \rightarrow Y$ a continuous map, then we have that $X \cup_f Y = (X \cup Y)/(a \tilde \ f(a))$ is a quotient space. So far so good. Now, we had to deal with CW complexes. I know the definition, but I don't know how this concept is used to build up a CW complex. Let's take the n-sphere. The only thing I know is that this concept of adjunction is used to build CW-structure, but how exactly is that done? Wikipedia says that we have one 0-cell $e_0$ and one n-cell $e_n$. It does not say how exactly these two look like. I guess $e_0$ is just the northpole and $e_n$ the rest. Actually, if I am correct so far, we don't need the concept of adjunction because we could just write down a homeomorphism $\phi : B(0,1) \subset \mathbb{R}^n \rightarrow S^{n}\backslash\{(0,...,0,1\} \subset \mathbb{R}^{n+1}.$ Something like a stereographic projection should do this(although I did not really think about the details), but how would one build up this complex with the use of adjunction maps?

Answer 1

The word adjunction you're seeing here just means the gluing construction you've already described.

So to clarify your example: the cells of a CW complex are all $n$-dimensional disks, which may be realized as $[0,1]^n$ or as the homeomorphic $\{x\in\mathbb{R}^n:|x|\leq 1\}$. So in the case of $S^n$ we have a $0$-cell and an $n$-cell. We must map the boundary $S^{n-1}$ of the $n$-cell to the union of lower-dimensional cells, in this case just the $0$-cell, so that the map is the collapse of $S^{n-1}$ to a point.