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Applications of chain complexes usually define and apply their Homology Group s ( Cohomology Group s 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 Group s or Modules ... ''A''-2, ''A''-1, ''A''0, ''A''1, ''A''2, ... connected by Homomorphism s (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 are usually written out as:

::\cdots o
A_{n+1} \begin{matrix} d_{n+1} \ o \ \, \end{matrix}
A_n \begin{matrix} d_n \ o \ \, \end{matrix}
A_{n-1} \begin{matrix} d_{n-1} \ o \ \, \end{matrix}
A_{n-2} o \cdots o
A_2 \begin{matrix} d_2 \ o \ \, \end{matrix}
A_1 \begin{matrix} d_1 \ o \ \, \end{matrix}
A_0 \begin{matrix} d_0 \ o \ \, \end{matrix}
A_{-1} \begin{matrix} d_{-1} \ o \ \, \end{matrix}
A_{-2} \begin{matrix} d_{-2} \ o \ \, \end{matrix}
\cdots


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 Group s or Modules ''A''-2, ''A''-1, ''A''0, ''A''1, ''A''2, ... connected by Homomorphism s ''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'':

::
\cdots o
A_{-2} \begin{matrix} d_{-2} \ o \ \, \end{matrix}
A_{-1} \begin{matrix} d_1 \ o \ \, \end{matrix}
A_0 \begin{matrix} d_0 \ o \ \, \end{matrix}
A_1 \begin{matrix} d_1 \ o \ \, \end{matrix}
A_2 o \cdots o
A_{n-1} \begin{matrix} d_{n-1} \ o \ \, \end{matrix}
A_n \begin{matrix} d_n \ o \ \, \end{matrix}
A_{n+1} o \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

d


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 Group s.


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

::\partial_n: C_n(X) o C_{n-1}(X): \, (\sigma: {Link without Title} o X) \mapsto



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


CHAIN MAPS


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 Functor s 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. See Homotopy Category Of Chain Complexes for further information.


SEE ALSO



REFERENCES