Editing Homology (mathematics) (section)

== Construction of homology groups ==
The following text describes a general algorithm for constructing the homology groups. It may be easier for the reader to look at some simple examples first: [[graph homology]] and [[simplicial homology]].

The general construction begins with an object such as a topological space ''X'', on which one first defines a {{em|[[chain complex]]}} ''C''(''X'') encoding information about ''X''. A chain complex is a sequence of abelian groups or modules <math>C_0, C_1, C_2, \ldots</math>. connected by [[group homomorphism|homomorphisms]] <math>\partial_n : C_n \to C_{n-1},</math> which are called '''boundary operators'''.<ref name="Hatcher 2002 106" /> That is,
: <math>
\dotsb
\overset{\partial_{n+1}}{\longrightarrow\,} C_n
\overset{\partial_n}{\longrightarrow\,} C_{n-1}
\overset{\partial_{n-1}}{\longrightarrow\,} \dotsb
\overset{\partial_2}{\longrightarrow\,} C_1
\overset{\partial_1}{\longrightarrow\,} C_0
\overset{\partial_0}{\longrightarrow\,} 0
</math>
where 0 denotes the trivial group and <math>C_i\equiv0</math> for ''i'' < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all ''n'',
: <math>\partial_n \circ \partial_{n+1} = 0_{n+1, n-1},</math>
i.e., the constant map sending every element of <math>C_{n+1}</math> to the group identity in <math>C_{n-1}.</math>

The statement that the boundary of a boundary is trivial is equivalent to the statement that <math>\mathrm{im}(\partial_{n+1})\subseteq\ker(\partial_n)</math>, where <math>\mathrm{im}(\partial_{n+1})</math> denotes the [[image (mathematics)|image]] of the boundary operator and <math>\ker(\partial_n)</math> its [[kernel (algebra)|kernel]]. Elements of <math>B_n(X) = \mathrm{im}(\partial_{n+1})</math> are called '''boundaries''' and elements of <math>Z_n(X) = \ker(\partial_n)</math> are called '''cycles'''.

Since each chain group ''C<sub>n</sub>'' is abelian all its subgroups are normal. Then because <math>\ker(\partial_n)</math> is a subgroup of ''C<sub>n</sub>'', <math>\ker(\partial_n)</math> is abelian, and since <math>\mathrm{im}(\partial_{n+1}) \subseteq\ker(\partial_n)</math> therefore <math>\mathrm{im}(\partial_{n+1})</math> is a [[normal subgroup]] of <math>\ker(\partial_n)</math>. Then one can create the [[quotient group]]
: <math>H_n(X) := \ker(\partial_n) / \mathrm{im}(\partial_{n+1}) = Z_n(X)/B_n(X),</math>
called the '''''n''th homology group of ''X'''''. The elements of ''H<sub>n</sub>''(''X'') are called '''homology classes'''. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be '''homologous'''.<ref>{{Harvnb|Hatcher|2002|pp=105–106}}</ref>

A chain complex is said to be [[exact sequence|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.<ref>{{Harvnb|Hatcher|2002|p=113}}</ref>

The [[Reduced homology|reduced homology groups]] of a chain complex ''C''(''X'') are defined as homologies of the augmented chain complex<ref>{{Harvnb|Hatcher|2002|p=110}}</ref>
: <math>
\dotsb
\overset{\partial_{n+1}}{\longrightarrow\,} C_n
\overset{\partial_n}{\longrightarrow\,} C_{n-1}
\overset{\partial_{n-1}}{\longrightarrow\,} \dotsb
\overset{\partial_2}{\longrightarrow\,} C_1
\overset{\partial_1}{\longrightarrow\,} C_0
\overset{\epsilon}{\longrightarrow\,} \Z
{\longrightarrow\,} 0
</math>
where the boundary operator <math>\epsilon</math> is
: <math>\epsilon \left(\sum_i n_i \sigma_i\right) = \sum_i n_i</math>
for a combination <math>\sum n_i \sigma_i,</math> of points <math>\sigma_i,</math> which are the fixed generators of ''C''<sub>0</sub>. The reduced homology groups <math>\tilde{H}_i(X)</math> coincide with <math>H_i(X)</math> for <math>i \neq 0.</math> The extra <math>\Z</math> in the chain complex represents the unique map <math>[\emptyset] \longrightarrow X</math> from the empty simplex to ''X''.

Computing the cycle <math>Z_n(X)</math> and boundary <math>B_n(X)</math> groups is usually rather difficult since they have a very large number of generators. On the other hand, there are tools which make the task easier.

The ''[[simplicial homology]]'' groups ''H<sub>n</sub>''(''X'') of a ''[[simplicial complex]]'' ''X'' are defined using the simplicial chain complex ''C''(''X''), with ''C<sub>n</sub>''(''X'') the [[free abelian group]] generated by the ''n''-simplices of ''X''. See [[simplicial homology]] for details.

The ''[[singular homology]]'' groups ''H<sub>n</sub>''(''X'') are defined for any topological space ''X'', and agree with the simplicial homology groups for a simplicial complex.

Cohomology groups are formally similar to homology groups: one starts with a [[cochain complex]], which is the same as a chain complex but whose arrows, now denoted <math>d_n,</math> point in the direction of increasing ''n'' rather than decreasing ''n''; then the groups <math>\ker\left(d^n\right) = Z^n(X)</math> of ''cocycles'' and <math>\mathrm{im}\left(d^{n-1}\right) = B^n(X)</math> of {{em|coboundaries}} follow from the same description. The ''n''th cohomology group of ''X'' is then the quotient group
: <math>H^n(X) = Z^n(X)/B^n(X),</math>
in analogy with the ''n''th homology group.