Attaching Spaces with Maps

Abstract

The purpose of this chapter is to develop the basic theory of CW complexes and their homology groups. An equivalence relation on a topological space is seen to produce a new space whose points are the equivalence classes. This gives a means of attaching one space to another via a mapping from a sub-space of the first to the second. The case of particular interest is that of attaching a cell to a space via a map defined on the boundary. This leads naturally to the definition of CW complexes. To serve as tools in the study of these spaces, relative homology groups are introduced and the excision theorem is proved. It is shown that the relative groups of adjacent skeletons produce a finitely generated chain complex whose homology is the homology of the space, and this is applied to compute the homology of real projective spaces.