Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). See homology theory for more background.

Construction of homology groups

The procedure works as follows: Given the object X, one first defines a chain complex that encodes information about X. A chain complex is a sequence of abelian groups or modules A0, A1, A2... connected by homomorphisms dn : An -> An-1, such that the composition of any two consecutive maps is zero: dn o dn+1 = 0 for all n. This means that the image of the n+1-th map is contained in the kernel of the n-th, and we can define the n-th homology group of X to be the factor group (or factor module)

Related Topics:
Chain complex - Homomorphisms - Image - Kernel - Factor group

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:Hn(X) = ker(dn) / im(dn+1).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A chain complex is said to be exact if the image of the n+1-th map is always equal to the kernel of the n-th map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~