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 Failed to parse (Missing texvc executable; please see math/README to configure.): (A_\bullet, d_\bullet)

is a sequence of abelian groups or modules ... A-2, A-1, A0, A1, A2, ... connected by homomorphisms (called boundary operators) dn : An→An−1, such that the composition of any two consecutive maps is zero: dn o dn+1 = 0 for all n. They are usually written out as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \cdots \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 \cdots \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} A_{-1} \begin{matrix} d_{-1} \\ \to \\ \, \end{matrix} A_{-2} \begin{matrix} d_{-2} \\ \to \\ \, \end{matrix} \cdots

A variant on the concept of chain complex is that of cochain complex. A cochain complex Failed to parse (Missing texvc executable; please see math/README to configure.): (A^\bullet, d^\bullet)

is a sequence of abelian groups or modules A-2, A-1, A0, A1, A2, ... connected by homomorphisms dn : An→An+1, such that the composition of any two consecutive maps is zero: dn+1 o dn = 0 for all n: 

Failed to parse (Missing texvc executable; please see math/README to configure.): \cdots \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} A_1 \begin{matrix} d_1 \\ \to \\ \, \end{matrix} A_2 \to \cdots \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 \cdots.

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 elements of the individual groups of a chain complex are called chains (or cochains in the case of a cochain complex.) The image of d is the group of boundaries, or in a cochain complex, coboundaries. The kernel of d (i.e., 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 (co)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 n-simplices in X, and define the boundary map

Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_n: C_n(X) \to C_{n-1}(X): \, (\sigma: [v_0,\ldots,v_n] \to X) \mapsto (\partial_n \sigma = \sum_{i=0}^n (-1)^i \sigma|[v_0,\ldots, \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 Failed to parse (Missing texvc executable; please see math/README to configure.): (C_\bullet, \partial_\bullet)

is a chain complex; the singular homology Failed to parse (Missing texvc executable; please see math/README to configure.): H_\bullet(X)
is the homology of this complex; that is, 

Failed to parse (Missing texvc executable; please see math/README to configure.): 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_k maps Ω^k(M) to Ω^k+1(M), and d ² = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega^0(M)\ \stackrel{d_0}{\to}\ \Omega^1(M) \to \Omega^2(M) \to \Omega^3(M) \to \cdots.

The homology of this complex is the de Rham cohomology

Failed to parse (Missing texvc executable; please see math/README to configure.): H^0_{\mathrm{DR}}(M, F) = \ker d_0 =

{locally constant functions on M with values in F} Failed to parse (Missing texvc executable; please see math/README to configure.): \cong F

^{#{connected pieces of M}}

Failed to parse (Missing texvc executable; please see math/README to configure.): H^k_{\mathrm{DR}}(M) = \ker d_k / \mathrm{im} \, d_{k-1}.

Chain maps

A chain map f between two chain complexes Failed to parse (Missing texvc executable; please see math/README to configure.): (A_\bullet, d_{A,\bullet})

and Failed to parse (Missing texvc executable; please see math/README to configure.): (B_\bullet, d_{B,\bullet})
is a collection of module homomorphisms Failed to parse (Missing texvc executable; please see math/README to configure.): f_n : A_n \rightarrow B_n
for each n that intertwines with the differentials on the two chain complexes: Failed to parse (Missing texvc executable; please see math/README to configure.):  f_n \circ d_{A,n+1}= d_{B,n+1} \circ f_{n+1}

. Such a map sends cycles to cycles and boundaries to boundaries, and thus descends to a map on homology:Failed to parse (Missing texvc executable; please see math/README to configure.): (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. A particular case is that homotopic maps between two spaces X and Y induce the same maps from homology of X to homology of Y. Chain homotopies have a geometric interpretation; it is described, for example, in the book of Bott and Tu. See Homotopy category of chain complexes for further information.

See also

• Homology
• Differential graded algebra

References

• Bott, Raoul & Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3de:Kettenkomplex

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