Chain Complex -- from Wolfram MathWorld

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Chain Complex

A chain complex is a sequence of maps

(1)

where the spaces may be Abelian groups or modules. The maps must satisfy . Making the domain implicitly understood, the maps are denoted by , called the boundary operator or the differential. Chain complexes are an algebraic tool for computing or defining homology and have a variety of applications. A cochain complex is used in the case of cohomology.

Elements of are called chains. For each , the kernel of is called the group of cycles,

$Z_p={c in C_p:partial(c)=0}.$

(2)

The letter is short for the German word for cycle, "Zyklus." The image is contained in the group of cycles because . It is called the group of boundaries.

$B_p={c in C_p:( exists b in C_(p+1):partial(b)=c)}.$

(3)

The quotients are the homology groups of the chain.

For example, the sequence

(4)

where every space is and each map is given by multiplication by 4 is a chain complex. The cycles at each stage are $Z_p={0,2,4,6}$ and the boundaries are $B_p={0,4}$ . So the homology at each stage is the group of two elements . A simpler example is given by a linear transformation , which can be extended to a chain complex by the zero vector space and the zero map. Then the nontrivial homology groups are and .

The terminology of chain complexes comes from the calculation for homology of geometric objects in a topological space, like a manifold. For example, in the figure above, let and denote the points, and and denote the oriented segments, which are the chains. The boundary of is , and the boundary of is .

The group is the free Abelian group and the group is the free Abelian group . The boundary operator is

(5)

The other groups are the trivial group, and the other maps are the zero map. Then is generated by and is the trivial subgroup. So is the rank one free Abelian group isomorphic to . The zero-dimensional case is slightly more interesting. Every element of has no boundary and so is in while the boundaries are generated by . Hence, is also isomorphic to . Note that the result is not affected by how the circle is cut into pieces, or by how many cuts are used.

This entry contributed by Todd Rowland

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Rowland, Todd. "Chain Complex." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/ChainComplex.html