CW complexes

Abstract

Abstract In Section 3.8, where we discussed homotopic equivalences, we showed that if we had a handle decomposition of a surface with one 0-handle and k 1-handles, then we could find a deformation retraction of this surface down to a wedge of k circles. This much of the space can be thought of as being built up from a central point corresponding to the 0-handle and then attaching intervals corresponding to the 1-handles where their end points are all identified to the central point. When the 2-handles are attached, we could compose their attaching maps with this homotopic equivalence and get a space which is built up from a point, some 1-disks attached, and then some 2-disks attached. It turns out that the original surface is homotopic equivalent to this space, which is an example of a two dimensional CW complex. In this chapter we will develop the concepts of CW complexes and apply them to fundamental group calculations as well as discuss homotopy-theoretic ideas concerning them. Our discussion will include finite CW complexes of any dimension, but we will emphasize two-dimensional complexes where the geometry is easier to visualize. We will also discuss important special cases of simplicial complexes and -complexes.