Wednesday, November 14th, 2012 | Author: Konrad Voelkel
We will investigate the notion of cellular objects in a model category; today: the classical case of CW-complexes in the model category of topological spaces with Serre-fibrations as fibrations.
A CW complex is a certain kind of topological space, together with a CW structure, which is a description how to glue the space from spheres (or from affine spaces, if you prefer), called the cells. The acronym CW stands for "closure finite, weak topology", which I will explain soon. CW complexes are a class of spaces broader than simplicial complexes, and they are still combinatorial in nature.
You can read definitions in the Wikipedia, I will only present the inductive definition here:
A CW complex consists of n-skeleta which are Hausdorff topological spaces, together with attaching maps in each degree n, that are maps indexed by $i \in I_n$, such that . By definition, and .
The map describes how to attach an affine space (the "cell") of dimension , considered as unit ball , along its boundary to the space .
The direct limit over all these skeleta carries the weak topology with respect to all the characteristic maps which are the extensions of to the attached inner of the cell. We denote this direct limit by again.
This is also where you can see "weak topology" and "closure finite", the latter means that a set is closed if its intersection with all skeleta is closed, a property of the direct limit topology.
To avoid some confusion: Such a space is just a topological space of CW type (= admits a CW structure), whereas the collection of all the skeleta and attaching maps are the CW structure that turns into a CW complex. There can be different CW structures on the same space, for example the n-sphere can be built from a 0-cell (basepoint) and an n-cell, but one can also introduce intermediate cells from any intermediate sphere.
Some examples where you should be able to come up with CW structures: Tori (possible with a 0-cell, two 1-cells and a 2-cell), projective space (there the attaching map is more interesting), Grassmannians (the famous Schubert cells can be used).
For homotopy theoretic purposes, one can study spaces which may not admit a CW structure, but are homotopy equivalent to a space of CW type. These spaces are called of CW homotopy type, and a prominent example are all smooth manifolds. It is also good to know that all spaces have the weak homotopy type of a CW complex, i.e. every space admits a map to a CW complex such that the map induces isomorphisms on all homotopy groups.
From a CW structure for a space , one can compute the homology and cohomology combinatorically. This is very similar to the Euler identity for planar graphs, that states Edges-Vertices+Faces = Euler-Characteristics (planar graphs are examples of CW complexes).