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In [[mathematics]] (especially [[algebraic topology]] and [[abstract algebra]]), '''homology''' (in part from [[Greek language|Greek]] ὁμός ''homos'' "identical") is a certain general procedure to associate a [[sequence]] of [[abelian group]]s or [[module (mathematics)|modules]] with a given mathematical object such as a [[topological space]] or a [[group (mathematics)|group]]. See [[singular homology]] for a concrete version for topological spaces, or [[group cohomology]] for a concrete version for groups. 

For a [[topological]] space, the homology groups are generally much easier to compute than the [[homotopy group]]s, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.

The original motivation for defining homology groups is the observation that shapes are distinguished by their ''holes''. But because a hole is "not there", it is not immediately obvious how to define a hole, or how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for defining and categorizing holes in a shape. As it turns out, subtle kinds of holes exist that homology cannot "see" — in which case homotopy groups may be what is needed.

== Construction of homology groups ==
The construction begins with an object such as a topological space ''X'', on which one first defines a ''[[chain complex]]'' ''C(X)'' encoding information about ''X''. A chain complex is a sequence of abelian groups or modules ''C''<sub>0</sub>, ''C''<sub>1</sub>, ''C''<sub>2</sub>, ... connected by [[group homomorphism|homomorphisms]] <math> \partial_n \colon C_n \to C_{n-1},</math> which are called '''boundary operators'''. 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 ''C''<sub>''n'' + 1</sub> to the group identity in ''C''<sub>''n'' - 1</sub>.  This means <math>\mathrm{im}(\partial_{n+1})\subseteq\ker(\partial_n)</math>.

Now since each ''C<sub>n</sub>'' is abelian all its subgroups are normal and because <math>\mathrm{im}(\partial_{n+1})</math> and <math>\ker(\partial_n)</math> are both subgroups of ''C<sub>n</sub>'', <math>\mathrm{im}(\partial_{n+1})</math> is a [[normal subgroup]] of <math>\ker(\partial_n)</math> and one can consider the [[factor group]]
:<math> H_n(X) := \ker(\partial_n) / \mathrm{im}(\partial_{n+1}), \, </math>
called the '''''n''-th homology group of ''X'''''.

We also use the notation <math>\ker(\partial_n)=Z_n(X) </math> and <math>\mathrm{im}(\partial_{n+1})=B_n(X)</math>, so

:<math>H_n(X)=Z_n(X)/B_n(X). \, </math>

Computing these two groups is usually rather difficult since they are very large groups. On the other hand, we do have 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(X)<sub>n</sub>'' the free abelian group generated by the ''n''-simplices of ''X''. 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.

A chain complex is said to be [[exact sequence|exact]] if the image of the (''n''&nbsp;+&nbsp;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.

Cohomology groups are formally similar: one starts with a [[cochain complex]], which is the same as a chain complex but whose arrows, now denoted ''d<sup>n</sup>'' point in the direction of increasing ''n'' rather than decreasing ''n''; then the groups <math>\ker(d^n) = Z^n(X)</math> and <math>\mathrm{im}(d^{n - 1}) = B^n(X)</math> follow from the same description and

:<math>H^n(X) = Z^n(X)/B^n(X), \, </math>

as before.

Sometimes, [[reduced homology|reduced homology groups]] of a chain complex ''C(X)'' are defined as homologies of the augmented complex
:<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 
:<math>\epsilon(\sum_i n_i \sigma_i)=\sum_i n_i</math> 
for a combination ''Σ n<sub>i</sub>σ<sub>i</sub>'' of points σ<sub>i</sub> (fixed generators of ''C<sub>0</sub>''). The reduced homologies <math>\tilde{H}_i(X)</math> coincide with <math>H_i(X)</math> for ''i≠0''.

== Examples ==
The motivating example comes from [[algebraic topology]]: the '''[[simplicial homology]]''' of a [[simplicial complex]] ''X''. Here ''A<sub>n</sub>'' is the [[free abelian group]] or module whose generators are the ''n''-dimensional oriented simplexes of ''X''. The mappings are called the ''boundary mappings'' and send the simplex with [[vertex (geometry)|vertices]]

:<math> (a[0], a[1], \dots, a[n]) \, </math>

to the sum 

:<math> \sum_{i=0}^n (-1)^i(a[0], \dots, a[i-1], a[i+1], \dots, a[n]) </math>

(which is considered 0 if ''n''&nbsp;=&nbsp;0).

If we take the modules to be over a field, then the dimension of the ''n''-th homology of ''X'' turns out to be the number of "holes" in ''X'' at dimension ''n''.

Using this example as a model, one can define a singular homology for any [[topological space]] ''X''. We define a chain complex for ''X'' by taking ''A<sub>n</sub>'' to be the free abelian group (or free module) whose generators are all [[continuous function (topology)|continuous]] maps from ''n''-dimensional [[simplex|simplices]] into ''X''. The homomorphisms <math>\partial_n</math> arise from the boundary maps of simplices. 

In [[abstract algebra]], one uses homology to define [[derived functor]]s, for example the [[Tor functor]]s. Here one starts with some covariant additive functor ''F'' and some module ''X''. The chain complex for ''X'' is defined as follows: first find a free module ''F''<sub>1</sub> and a [[surjective]] homomorphism ''p''<sub>1</sub>: ''F''<sub>1</sub> → ''X''. Then one finds a free module ''F''<sub>2</sub> and a surjective homomorphism ''p''<sub>2</sub>: ''F''<sub>2</sub> → ker(''p''<sub>1</sub>). Continuing in this fashion, a sequence of free modules ''F<sub>n</sub>'' and homomorphisms ''p<sub>n</sub>'' can be defined. By applying the functor ''F'' to this sequence, one obtains a chain complex; the homology ''H<sub>n</sub>'' of this complex depends only on ''F'' and ''X'' and is, by definition, the ''n''-th derived functor of ''F'', applied to ''X''.

== Homology functors ==
Chain complexes form a [[category theory|category]]: A morphism from the chain complex (''d<sub>n</sub>'': ''A<sub>n</sub>'' → ''A''<sub>''n''-1</sub>) to the chain complex (''e<sub>n</sub>'': ''B<sub>n</sub>'' → ''B''<sub>''n''-1</sub>) is a sequence of homomorphisms ''f<sub>n</sub>'': ''A<sub>n</sub>'' → ''B<sub>n</sub>'' such that <math>f_{n-1} \circ d_n = e_{n} \circ f_n </math> for all ''n''. The ''n''-th homology ''H<sub>n</sub>'' can be viewed as a covariant [[functor]] from the category of chain complexes to the category of abelian groups (or modules). 

If the chain complex depends on the object ''X'' in a covariant manner (meaning that any morphism ''X → Y'' induces a morphism from the chain complex of  ''X''  to the chain complex of ''Y''), then the ''H<sub>n</sub>'' are covariant [[functor]]s from the category that ''X'' belongs to into the category of abelian groups (or modules). 

The only difference between homology and [[cohomology]] is that in cohomology the chain complexes depend in a ''contravariant'' manner on ''X'', and that therefore the homology groups (which are called ''cohomology groups'' in this context and denoted by ''H<sup>n</sup>'') form ''contravariant'' functors from the category that ''X'' belongs to into the category of abelian groups or modules.

== Properties ==
If (''d<sub>n</sub>'': ''A<sub>n</sub>'' → ''A''<sub>''n''-1</sub>) is a chain complex such that all but finitely many ''A<sub>n</sub>'' are zero, and the others are finitely generated abelian groups (or finite dimensional vector spaces), then we can define the ''[[Euler characteristic]]''

:<math> \chi = \sum (-1)^n \, \mathrm{rank}(A_n) </math>

(using the [[rank of an abelian group|rank]] in the case of abelian groups and the [[Hamel dimension]] in the case of vector spaces). It turns out that the Euler characteristic can also be computed on the level of homology:

:<math> \chi = \sum (-1)^n \, \mathrm{rank}(H_n) </math>

and, especially in algebraic topology, this provides two ways to compute the important invariant &chi; for the object ''X'' which gave rise to the chain complex.

Every [[short exact sequence]]

:<math> 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 </math>

of chain complexes gives rise to a [[long exact sequence]] of homology groups

:<math> \cdots \rightarrow H_n(A) \rightarrow H_n(B) \rightarrow H_n(C) \rightarrow H_{n-1}(A) \rightarrow H_{n-1}(B) \rightarrow H_{n-1}(C) \rightarrow H_{n-2}(A) \rightarrow \cdots. \,</math>

All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps ''H<sub>n</sub>(C)'' → ''H''<sub>''n''-1</sub>''(A)'' The latter are called ''connecting homomorphisms'' and are provided by the [[snake lemma]].  The snake lemma can be applied to homology in numerous ways that aid in calculating homology groups, such as the theories of ''relative homology'' and ''Mayer-Vietoris sequences''.

== History ==
Homology classes were first defined rigorously by [[Henri Poincaré]] in his seminal paper "Analysis situs", ''J. Ecole polytech.'' (2) '''1'''. 1–121 (1895).

The homology group was further developed by [[Emmy Noether]]<ref>{{Harvnb|Hilton|1988|p=284}}</ref><ref>For example [http://smf4.emath.fr/Publications/Gazette/2011/127/smf_gazette_127_15-44.pdf ''L'émergence de la notion de groupe d'homologie'', Nicolas Basbois (PDF)], in French, note 41, explicitly names Noether as inventing the [[homology group]].</ref> and, independently, by [[Leopold Vietoris]] and [[Walther Mayer]], in the period 1925–28.<ref>Hirzebruch,  Friedrich, [http://www.mathe2.uni-bayreuth.de/axel/papers/hierzebruch:emmy_noether_and_topology.ps.gz Emmy Noether and Topology] in {{Harvnb|Teicher|1999|pp=61–63}}.</ref>
Prior to this, topological classes in [[combinatorial topology]] were not formally considered as [[abelian group]]s. The spread of homology groups marked the change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".<ref>[http://math.vassar.edu/faculty/McCleary/BourbakiAlgTop.pdf ''Bourbaki and Algebraic Topology'' by John McCleary (PDF)] gives documentation (translated into English from French originals).</ref>

==Applications==
Notable theorems proved using homology include the following:
* The [[Brouwer fixed point theorem]]: If ''f'' is any continuous map from the ball ''B<sup>n</sup>'' to itself, then there is a fixed point ''a'' ∈ ''B<sup>n</sup>'' with ''f''(''a'') = ''a''.
* [[Invariance of domain]]: If ''U'' is an [[open set|open subset]] of '''R'''<sup>''n''</sup> and ''f'' : ''U'' → '''R'''<sup>''n''</sup> is an [[injective]] [[continuous map]], then ''V'' = ''f''(''U'') is open and ''f'' is a [[homeomorphism]] between ''U'' and ''V''.
* The [[Hairy ball theorem]]: any vector field on the 2-sphere (or more generally, the 2''k''-sphere for any ''k'' ≥ 1) vanishes at some point.
* The [[Borsuk–Ulam theorem]]: any [[continuous function]] from an [[n-sphere|''n''-sphere]] into [[Euclidean space|Euclidean ''n''-space]] maps some pair of [[antipodal point]]s to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)

== See also ==
*[[Simplicial homology]]
*[[Singular homology]]
*[[Cellular homology]]
*[[Homological algebra]]
*[[Cohomology]]

== Notes ==
{{reflist}}

==References==
* [[Henri Cartan|Cartan, Henri Paul]] and [[Samuel Eilenberg|Eilenberg, Samuel]] (1956) ''Homological Algebra'' Princeton University Press, Princeton, NJ, [http://worldcat.org/oclc/529171 OCLC 529171]
* Eilenberg, Samuel and Moore, J. C.  (1965) ''Foundations of relative homological algebra'' (Memoirs of the American Mathematical Society number 55)  American Mathematical Society, Providence, R.I., [http://worldcat.org/oclc/1361982 OCLC 1361982]
* Hatcher, A., (2002) ''[http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology]''  Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
*[http://www.encyclopediaofmath.org/index.php/Homology_group ''Homology group'' at Encyclopaedia of Mathematics]
*{{citation|last=Hilton|first=Peter|year=1988|title=A Brief, Subjective History of Homology and Homotopy Theory in This Century |journal = Mathematics Magazine|volume=60|issue=5|pages=282–291| jstor = 2689545|publisher=Mathematical Association of America}}
*{{citation|title=The Heritage of Emmy Noether|first= M. (ed.)|last =Teicher|series=Israel Mathematical Conference Proceedings|publisher= [[Bar-Ilan University]]/[[American Mathematical Society]]/[[Oxford University Press]]|year= 1999|oclc= 223099225 |isbn= 978-0-19-851045-1}}
*{{planetmath reference|id=3720|title=Homology (Topological space)}}

[[Category:Homology theory|*]]