Guide to CW Complexes and n-Cells

[alyscia]

I hope I am posting in the right section. I am having a lot of trouble understanding what exactly CW complexes look like. In particular, I am not sure of the definition of an n-cell (in terms of topology, not geometry), and the inductive step of building a CW complex (from X_{n-1} to X_n).

Hope my question makes sense. If anyone has any diagrams to share, that would be invaluable to me. Thanks.

A

 

[eastside00_99]

Your question makes perfect sense. I find it difficult to understand the definition too in the sense of visualizing it. But, it is just an abstract way to talk about dimension of a topological space. The canonical example is the 2-sphere. You start with two points -- the disjoint union of which is the 0-sphere. Then the next cells, the 1-cells, are given by two semicircles. The disjoint union of which gives you the 1-sphere. Then then you have 2 2-cells given by the two hemispheres of the sphere whose disjoint union is the 2-sphere. So, intuitively, the n-cells are at least a collection of disjoint sets [tex]\{D_{\alpha}^{n}\})_{\alpha} [/itex] each of which properly contain a disjoint collection of (n-1)-cells which comes with a natural way of identifying different (n-1)-cells. The wikipedia article on this is pretty good. If you run through it with the example of the sphere in mind, then you should be ok. For example, the a one dimensional ball can be thought of as a semicircle and we have the continuous function f: d(D^1) = S^0 --> X^0. Here X^0 are the two starting points. f gives us the gluing. This canonical example serves also as an example of a regular cw complex because the functions will be homeomorphism.