In this chapter and the next we shall prove some general results which will find application in Chapter 9. For the present the motivation lies in the evident difficulty of calculating homology groups directly from the definition, and the consequent need for some help from general theorems. Enough is proved here to enable us to calculate the homology groups of all the closed surfaces described in Chapter 2, without difficulty. We shall also calculate the homology groups of some other standard simplicial complexes such as cones (6.8) and spheres (6.11(3)), and show (6.10) that relative homology groups can be defined in terms of ordinary homology groups.

The method presented here for the calculation of homology groups of closed surfaces is based on the idea of collapsing which was introduced in 3.30. This is not the only available method. From the point of view of this book, the advantage of using it is that, having established the invariance of homology groups under barycentric subdivision (5.7), we are in possession of a rapid and rigorous approach to the results. Another method of wide application comes from the Mayer–Vietoris sequence, presented in Chapter 7, where indications are given as to how the homology groups of closed surfaces can be re-calculated by use of the sequence. For effective and unfettered use of the Mayer–Vietoris sequence one really needs to use something like the topological invariance theorem stated in 5.13 – that is, it is better to forget about particular triangulations altogether.