Homological Algebra - Chain Complexes and Homology

Chain Complexes and Homology

The chain complex is the central notion of homological algebra. It is a sequence of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero:

$C_bullet: cdots longrightarrow C_{n+1} stackrel{d_{n+1}}{longrightarrow} C_n stackrel{d_n}{longrightarrow} C_{n-1} stackrel{d_{n-1}}{longrightarrow} cdots, quad d_n circ d_{n+1}=0.$

The elements of Cn are called n-chains and the homomorphisms dn are called the boundary maps or differentials. The chain groups Cn may be endowed with extra structure; for example, they may be vector spaces or modules over a fixed ring R. The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of R-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the category Ab of abelian groups); a celebrated theorem by Barry Mitchell implies the results will generalize to any abelian category. Every chain complex defines two further sequences of abelian groups, the cycles Zn = Ker dn and the boundaries Bn = Im dn+1, where Ker d and Im d denote the kernel and the image of d. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as

Subgroups of abelian groups are automatically normal; therefore we can define the nth homology group Hn(C) as the factor group of the n-cycles by the n-boundaries,

A chain complex is called acyclic or an exact sequence if all its homology groups are zero.

Chain complexes arise in abundance in algebra and algebraic topology. For example, if X is a topological space then the singular chains Cn(X) are formal linear combinations of continuous maps from the standard n-simplex into X; if K is a simplicial complex then the simplicial chains Cn(K) are formal linear combinations of the n-simplices of X; if A = F/R is a presentation of an abelian group A by generators and relations, where F is a free abelian group spanned by the generators and R is the subgroup of relations, then letting C1(A) = R, C0(A) = F, and Cn(A) = 0 for all other n defines a sequence of abelian groups. In all these cases, there are natural differentials dn making Cn into a chain complex, whose homology reflects the structure of the topological space X, the simplicial complex K, or the abelian group A. In the case of topological spaces, we arrive at the notion of singular homology, which plays a fundamental role in investigating the properties of such spaces, for example, manifolds.

On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes, R-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations.

• Two objects X and Y are connected by a map f between them. Homological algebra studies the relation, induced by the map f, between chain complexes associated with X and Y and their homology. This is generalized to the case of several objects and maps connecting them. Phrased in the language of category theory, homological algebra studies the functorial properties of various constructions of chain complexes and of the homology of these complexes.
• An object X admits multiple descriptions (for example, as a topological space and as a simplicial complex) or the complex is constructed using some 'presentation' of X, which involves non-canonical choices. It is important to know the effect of change in the description of X on chain complexes associated with X. Typically, the complex and its homology are functorial with respect to the presentation; and the homology (although not the complex itself) is actually independent of the presentation chosen, thus it is an invariant of X.