The theorems proved in this chapter are separated from those of Chapter 6 on account of their slightly more technical nature and the fact that we do not use them in an essential way in what follows. Both theorems assert the exactness of certain sequences. The first attempts to answer the question: given the homology groups of two subcomplexes L1, L2 of a simplicial complex K (not necessarily disjoint), what are the homology groups of L1 ∪ L2? They obviously depend on the homology groups of L1 ∩ L2 as well, and moreover on the way in which L1 and L2 are stuck together – for example two cylinders can be stuck together to give either a torus or a Klein bottle (see 7.5). The result is not an explicit formula for Hp(L1 ∪ L2) but an exact sequence which, with luck, will give a good deal of information. An example where it does not give quite enough information to determine a homology group of L1 ∪ L2 is mentioned in 7.6(3).

The other theorem proved in this chapter, the exactness of the ‘homology sequence of a triple’, is a generalization of the homology sequence of a pair (see 6.1).

The Mayer–Vietoris sequence

Let K be an oriented simplicial complex of dimension n, and let L1, L2 be subcomplexes of K with L1 ∪ L2 = K. Write L1 ∩ L2 = L and, as usual, regard Cp(L), Cp(L1) and Cp(L2) as subgroups of Cp(K), for each p.