Supplement: The Long Exact Homology Sequence and Applications

A chain complex (or simply a complex ) C∗ is a family of R-modules Cn, n ∈ Z, along with R-homomorphisms dn : Cn → Cn−1 called differentials, satisfying dndn+1 = 0 for all n. A chain complex with only finitely many Cn’s is allowed; it can always be extended with the aid of zero modules and zero maps. [In topology, Cn is the abelian group of nchains, that is, all formal linear combinations with integer coefficients of n-simplices in a topological space X. The map dn is the boundary operator, which assigns to an n-simplex an n − 1-chain that represents the oriented boundary of the simplex.] The kernel of dn is written Zn(C∗) or just Zn; elements of Zn are called cycles in dimension n. The image of dn+1 is written Bn(C∗) or just Bn; elements of Bn are called boundaries in dimension n. Since the composition of two successive differentials is 0, it follows that Bn ⊆ Zn. The quotient Zn/Bn is written Hn(C∗) or just Hn; it is called the n homology module (or homology group if the underlying ring R is Z). [The key idea of algebraic topology is the association of an algebraic object, the collection of homology groups Hn(X), to a topological space X. If two spaces X and Y are homeomorphic, in fact if they merely have the same homotopy type, then Hn(X) and Hn(Y ) are isomorphic for all n. Thus the homology groups can be used to distinguish between topological spaces; if the homology groups differ, the spaces cannot be homeomorphic.] Note that any exact sequence is a complex, since the composition of successive maps is 0.