Everything about Chain Complex totally explained
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 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:
» :
.
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's described, for example, in the book of Bott and Tu. See
Homotopy category of chain complexes for further information.
Further Information
Get more info on 'Chain Complex'.
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