Chain complex

In mathematics, a chain complex is a construct originally used in the field of algebraic topology. It is an algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or an algebraic construction such as a simplicial complex. More generally, homological algebra includes the study of chain complexes in the abstract, without any reference to an underlying space. In this case, chain complexes are studied axiomatically as algebraic structures.

Applications of chain complexes usually define and apply their homology groups (cohomology groups for cochain complexes); in more abstract settings various equivalence relations are applied to complexes (for example starting with the chain homotopy idea). Chain complexes are easily defined in abelian categories, also.

Formal definition

A chain complex

(A_\bullet, d_\bullet)

is a sequence of abelian groups or modules A₀, A₁, A₂, ... connected by homomorphisms (called boundary operators) d_n : A_n→A_n−1, such that the composition of any two consecutive maps is zero: d_n o d_n+1 = 0 for all n. They tend to be written out like so:

\ldots \to

A_{n+1} \begin{matrix} d_{n+1} \\ \to \\ \, \end{matrix} A_n \begin{matrix} d_n \\ \to \\ \, \end{matrix} A_{n-1} \begin{matrix} d_{n-1} \\ \to \\ \, \end{matrix} A_{n-2} \to \ldots \to A_2 \begin{matrix} d_2 \\ \to \\ \, \end{matrix} A_1 \begin{matrix} d_1 \\ \to \\ \, \end{matrix} A_0 \begin{matrix} d_0 \\ \to \\ \, \end{matrix} 0.

A variant on the concept of chain complex is that of cochain complex. A cochain complex $(A^\bullet, d^\bullet)$ is a sequence of abelian groups or modules A₀, A₁, A₂, ... connected by homomorphisms d_n : A_n→A_n+1, such that the composition of any two consecutive maps is zero: d_n+1 o d_n = 0 for all n:

0 \to

A_0 \begin{matrix} d_0 \\ \to \\ \, \end{matrix} A_1 \begin{matrix} d_1 \\ \to \\ \, \end{matrix} A_2 \to \ldots \to A_{n-1} \begin{matrix} d_{n-1} \\ \to \\ \, \end{matrix} A_n \begin{matrix} d_n \\ \to \\ \, \end{matrix} A_{n+1} \to \ldots.

The idea is basically the same. In either case, the index i in A_i is referred to as the degree.

A bounded chain complex is one in which almost all the A_i are 0; i.e., a finite complex extended to the left and right by 0's. An example is the complex defining the homology theory of a (finite) simplicial complex. A chain complex is bounded above if all degrees above some fixed degree N are 0, and is bounded below if all degrees below some fixed degree are 0. Clearly, a complex is bounded above and below iff the complex is bounded.

Fundamental terminology

Leaving out the indices, the basic relation on d can be thought of as

d² = 0.

The image of d is the group of boundaries, or in a cochain complex, coboundaries. The subgroup sent to 0 by d is the group of cycles, or in the case of a cochain complex, cocycles. From the basic relation, the (co)boundaries lie inside the (co)cycles. This phenomenon is studied in a systematic way using homology groups.

Examples

Singular homology

Suppose we are given a topological space X.

Define C_n(X) for natural n to be the free abelian group formally generated by singular simplices in X, and define the boundary map

\partial_n: C_n(X) \to C_{n-1}(X): \, (\sigma:

^* \to X) \mapsto

(\partial_n \sigma = \sum_{i=0}^n (-1)^i \sigma|\hat v_i, \ldots, v_n),

where the hat denotes the omission of a vertex. That is, the boundary of a singular simplex is alternating sum of restrictions to its faces. It can be shown ∂² = 0, so $(C_\bullet, \partial_\bullet)$ is a chain complex; the singular homology $H_\bullet(X)$ is the homology of this complex; that is,

H_n(X) = \ker \partial_n / \mbox{im } \partial_{n+1}.

de Rham cohomology

The differential k-forms on any smooth manifold M form an abelian group (in fact an R-vector space) called Ω^k(M) under addition. The exterior derivative d = d_k maps Ω^k(M) → Ω^k+1(M), and d² = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex:

\Omega^0(M) \to \Omega^1(M) \to \Omega^2(M) \to \Omega^3(M) \to \ldots.

The homology of this complex is the de Rham cohomology

H^k_{\mathrm{DR}}(M) = \ker d_{k+1} / \mbox{im } d_k.

Chain maps

A chain map f between two chain complexes

(A_\bullet, d_{A,\bullet})

and

(B_\bullet, d_{B,\bullet})

is a collection of maps

f_n : A_n \rightarrow B_n

for each n that intertwines with the differentials on the two chain complexes:

f_n \circ d_{A,n}= d_{B,n} \circ f_{n+1}

. Such a map sends cycles to cycles and boundaries to boundaries, and thus descends to a map on homology:

(f_n)_*:H_\bullet(A_\bullet, d_{A,\bullet}) \rightarrow H_\bullet(B_\bullet, d_{B,\bullet})

A continuous map of topological spaces induces chain maps in both the singular and de Rham chain complexes described above (and in general for the chain complex defining any homology theory of topological spaces) and thus a continuous map induces a map on homology. Because the map induced on a composition of maps is the composition of the induced maps, these homology theories are functors from the category of topological spaces with continuous maps to the category of abelian groups with group homomorphisms.

Chain homotopy

Chain homotopies give an important equivalence relation between chain maps. Chain homotopic chain maps induce the same maps on homology groups. The notion is modelled on the homotopy concept for mappings of topological space, and reflects the fact that homotopic maps of continuous spaces induce the same maps on their homology. Chain homotopies have a geometric interpretation; it is described, for example, in the book of Bott and Tu.

Let (A_n, d_n) and (B_n, d′_n) be chain complexes and f, g be chain maps from the first to the second. A chain homotopy between f and g is given by a homotopy operator, a sequence of homomorphisms D_n from A_n to B_n+1 such that

f − g = Dd + d′D,

or adorned with full indices, as can be easily reconstructed for diagram chasing,

f_n − g_n = D_n−1d_n + d′_n+1D_n.

The chain maps $f - g$ induce the same maps on homology because $(f-g)$ sends cycles to boundaries, which are zero in homology.

A weaker notion of equivalence of maps between chain complexes is that of a quasi-isomorphism, which is a chain map in which the induced map on homology is an isomorphism. Maps of this kind form the morphisms of a derived category, whose objects are chain complexes.

References

Raoul Bott and Loring Tu, Differential Forms in Algebraic Topology. Springer-Verlag, 1982.

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