In

_{n}'' is abelian all its subgroups are normal. Then because $\backslash ker(\backslash partial\_n)$ is a subgroup of ''C_{n}'', $\backslash ker(\backslash partial\_n)$ is abelian, and since $\backslash mathrm(\backslash partial\_)\; \backslash subseteq\backslash ker(\backslash partial\_n)$ therefore $\backslash mathrm(\backslash partial\_)$ is a _{n}''(''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.
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.
The reduced homology groups of a chain complex ''C''(''X'') are defined as homologies of the augmented chain complex
:$\backslash dotsb\; \backslash overset\; C\_n\; \backslash overset\; C\_\; \backslash overset\; \backslash dotsb\; \backslash overset\; C\_1\; \backslash overset\; C\_0\; \backslash overset\; \backslash Z\; 0$
where the boundary operator $\backslash epsilon$ is
:$\backslash epsilon\; \backslash left(\backslash sum\_i\; n\_i\; \backslash sigma\_i\backslash right)\; =\; \backslash sum\_i\; n\_i$
for a combination $\backslash sum\; n\_i\; \backslash sigma\_i,$ of points $\backslash sigma\_i,$ which are the fixed generators of ''C''_{0}. The reduced homology groups $\backslash tilde\_i(X)$ coincide with $H\_i(X)$ for $i\; \backslash neq\; 0.$ The extra $\backslash Z$ in the chain complex represents the unique map $;\; href="/html/ALL/l/emptyset.html"\; ;"title="emptyset">emptyset$ from the empty simplex to ''X''.
Computing the cycle $Z\_n(X)$ and boundary $B\_n(X)$ 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_{n}''(''X'') of a '' simplicial complex'' ''X'' are defined using the simplicial chain complex ''C''(''X''), with ''C_{n}''(''X'') the _{n}''(''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

_{n}'' is the _{n}'' to ''C_{n−1}'' is called the and sends the simplex
:$\backslash sigma\; =\; (\backslash sigma;\; href="/html/ALL/l/.html"\; ;"title="">$
to the formal sum
:$\backslash partial\_n(\backslash sigma)\; =\; \backslash sum\_^n\; (-1)^i\; \backslash left\; (\backslash sigma;\; href="/html/ALL/l/.html"\; ;"title="">$
which is considered 0 if $n\; =\; 0.$ This behavior on the generators induces a homomorphism on all of ''C_{n}'' as follows. Given an element $c\; \backslash in\; C\_n$, write it as the sum of generators $c\; =\; \backslash sum\_\; m\_i\; \backslash sigma\_i,$ where $X\_n$ is the set of ''n''-simplexes in ''X'' and the ''m_{i}'' are coefficients from the ring ''C_{n}'' is defined over (usually integers, unless otherwise specified). Then define
:$\backslash partial\_n(c)\; =\; \backslash sum\_\; m\_i\; \backslash partial\_n(\backslash sigma\_i).$
The dimension of the ''n''-th homology of ''X'' turns out to be the number of "holes" in ''X'' at dimension ''n''. It may be computed by putting matrix representations of these boundary mappings in

_{n}'' to be the free abelian group (or free module) whose generators are all continuous maps from ''n''-dimensional _{''n''} arise from the boundary maps of simplexes.

_{n}'' can be viewed as a covariant _{n}'' are covariant ^{n}'') form ''contravariant'' functors from the category that ''X'' belongs to into the category of abelian groups or modules.

_{n}'' are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the ''

^{n}'' to itself, then there is a fixed point $a\; \backslash in\; B^n$ with $f(a)\; =\; a.$
*

Linbox

is a C++ library for performing fast matrix operations, including

Gap

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Maple

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are also written in C++. All three implement pre-processing algorithms based on

Kenzo

is written in Lisp, and in addition to homology it may also be used to generate presentations of

''Homology group'' at Encyclopaedia of Mathematics

N.J. Windberger intro to algebraic topology, last six lectures with an easy intro to homology

Algebraic topology Allen Hatcher - Chapter 2 on homology {{Authority control Homology theory,

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, homology is a general way of associating a sequence of algebraic objects, such as abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

s or modules, with other mathematical objects such as topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

s. Homology groups were originally defined in algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...

. Similar constructions are available in a wide variety of other contexts, such as abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, groups, Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

s, Galois theory, and algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

.
The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

. Loosely speaking, a ''cycle'' is a closed submanifold, a ''boundary'' is a cycle which is also the boundary of a submanifold, and a ''homology class'' (which represents a hole) is an equivalence class of cycles modulo boundaries. A homology class is thus represented by a cycle which is not the boundary of any submanifold: the cycle represents a hole, namely a hypothetical manifold whose boundary would be that cycle, but which is "not there".
There are many different homology theories. A particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has a geometric interpretation as topological spaces do, the ''n''th homology group represents behavior in dimension ''n''. Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proc ...

.
Background

Origins

Homology theory can be said to start with the Euler polyhedron formula, orEuler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...

. This was followed by Riemann's definition of genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial no ...

and ''n''-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.
Homology itself was developed as a way to analyse and classify manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

s according to their ''cycles'' – closed loops (or more generally submanifolds) that can be drawn on a given ''n'' dimensional manifold but not continuously deformed into each other. These cycles are also sometimes thought of as cuts which can be glued back together, or as zippers which can be fastened and unfastened. Cycles are classified by dimension. For example, a line drawn on a surface represents a 1-cycle, a closed loop or $S^1$ (1-manifold), while a surface cut through a three-dimensional manifold is a 2-cycle.
Surfaces

On the ordinarysphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

$S^2$, the cycle ''b'' in the diagram can be shrunk to the pole, and even the equatorial great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry ...

''a'' can be shrunk in the same way. The Jordan curve theorem shows that any arbitrary cycle such as ''c'' can be similarly shrunk to a point. All cycles on the sphere can therefore be continuously transformed into each other and belong to the same homology class. They are said to be homologous to zero. Cutting a manifold along a cycle homologous to zero separates the manifold into two or more components. For example, cutting the sphere along ''a'' produces two hemispheres.
This is not generally true of cycles on other surfaces. The torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not t ...

$T^2$ has cycles which cannot be continuously deformed into each other, for example in the diagram none of the cycles ''a'', ''b'' or ''c'' can be deformed into one another. In particular, cycles ''a'' and ''b'' cannot be shrunk to a point whereas cycle ''c'' can, thus making it homologous to zero.
If the torus surface is cut along both ''a'' and ''b'', it can be opened out and flattened into a rectangle or, more conveniently, a square. One opposite pair of sides represents the cut along ''a'', and the other opposite pair represents the cut along ''b''.
The edges of the square may then be glued back together in different ways. The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram. Up to symmetry, there are four distinct ways of gluing the sides, each creating a different surface:
$K^2$ is the Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...

, which is a torus with a twist in it (The twist can be seen in the square diagram as the reversal of the bottom arrow). It is a theorem that the re-glued surface must self-intersect (when immersed in Euclidean 3-space). Like the torus, cycles ''a'' and ''b'' cannot be shrunk while ''c'' can be. But unlike the torus, following ''b'' forwards right round and back reverses left and right, because ''b'' happens to cross over the twist given to one join. If an equidistant cut on one side of ''b'' is made, it returns on the other side and goes round the surface a second time before returning to its starting point, cutting out a twisted Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Au ...

. Because local left and right can be arbitrarily re-oriented in this way, the surface as a whole is said to be non-orientable.
The projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

$P^2$ has both joins twisted. The uncut form, generally represented as the Boy surface
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-spa ...

, is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim such as ''A'' and ''A′'' are identified as the same point. Again, ''a'' and ''b'' are non-shrinkable while ''c'' is. But this time, both ''a'' and ''b'' reverse left and right.
Cycles can be joined or added together, as ''a'' and ''b'' on the torus were when it was cut open and flattened down. In the Klein bottle diagram, ''a'' goes round one way and −''a'' goes round the opposite way. If ''a'' is thought of as a cut, then −''a'' can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so ''a'' + (−''a'') = 0.
But now consider two ''a''-cycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the ''b''-cycle), and it will come back as −''a''. This is because the Klein bottle is made from a cylinder, whose ''a''-cycle ends are glued together with opposite orientations. Hence 2''a'' = ''a'' + ''a'' = ''a'' + (−''a'') = 0. This phenomenon is called torsion. Similarly, in the projective plane, following the unshrinkable cycle ''b'' round twice remarkably creates a trivial cycle which ''can'' be shrunk to a point; that is, ''b'' + ''b'' = 0. Because ''b'' must be followed around twice to achieve a zero cycle, the surface is said to have a torsion coefficient of 2. However, following a ''b''-cycle around twice in the Klein bottle gives simply ''b'' + ''b'' = 2''b'', since this cycle lives in a torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist, whereas in the projective plane both sides are twisted.
A square is a contractible topological space, which implies that it has trivial homology. Consequently, additional cuts disconnect it. The square is not the only shape in the plane that can be glued into a surface. Gluing opposite sides of an octagon, for example, produces a surface with two holes. In fact, all closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2''n''-gons) can be glued to make different manifolds. Conversely, a closed surface with ''n'' non-zero classes can be cut into a 2''n''-gon. Variations are also possible, for example a hexagon may also be glued to form a torus.
The first recognisable theory of homology was published by Henri Poincaré in his seminal paper " Analysis situs", ''J. Ecole polytech.'' (2) 1. 1–121 (1895). The paper introduced homology classes and relations. The possible configurations of orientable cycles are classified by the Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...

s of the manifold (Betti numbers are a refinement of the Euler characteristic). Classifying the non-orientable cycles requires additional information about torsion coefficients.
The complete classification of 1- and 2-manifolds is given in the table.
: Notes
:# For a non-orientable surface, a hole is equivalent to two cross-caps.
:# Any 2-manifold is the connected sum of ''g'' tori and ''c'' projective planes. For the sphere $S^2$, ''g'' = ''c'' = 0.
Generalization

A manifold with boundary or open manifold is topologically distinct from a closed manifold and can be created by making a cut in any suitable closed manifold. For example the disk or 2-ball $B^2$ is bounded by a circle $S^1$. It may be created by cutting a trivial cycle in any 2-manifold and keeping the piece removed, by piercing the sphere and stretching the puncture wide, or by cutting the projective plane. It can also be seen as filling-in the circle in the plane. When two cycles can be continuously deformed into each other, then cutting along one produces the same shape as cutting along the other, up to some bending and stretching. In this case the two cycles are said to be or to lie in the same . Additionally, if one cycle can be continuously deformed into a combination of other cycles, then cutting along the initial cycle is the same as cutting along the combination of other cycles. For example, cutting along a figure 8 is equivalent to cutting along its two lobes. In this case, the figure 8 is said to be homologous to the sum of its lobes. Two open manifolds with similar boundaries (up to some bending and stretching) may be glued together to form a new manifold which is their connected sum. This geometric analysis of manifolds is not rigorous. In a search for increased rigour, Poincaré went on to develop the simplicial homology of a triangulated manifold and to create what is now called a chain complex. These chain complexes (since greatly generalized) form the basis for most modern treatments of homology. In such treatments a cycle need not be continuous: a 0-cycle is a set of points, and cutting along this cycle corresponds to puncturing the manifold. A 1-cycle corresponds to a set of closed loops (an image of the 1-manifold $S^1$). On a surface, cutting along a 1-cycle yields either disconnected pieces or a simpler shape. A 2-cycle corresponds to a collection of embedded surfaces such as a sphere or a torus, and so on.Emmy Noether
Amalie Emmy Noether Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...

and, independently, Leopold Vietoris and Walther Mayer
Walther Mayer (11 March 1887 – 10 September 1948) was an Austrian mathematician, born in Graz, Austria-Hungary. With Leopold Vietoris he is the namesake of the Mayer–Vietoris sequence in topology.. He served as an assistant to Albert Einste ...

further developed the theory of algebraic homology groups in the period 1925–28. The new combinatorial topology formally treated topological classes as abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

s. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part.
The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...

". Algebraic homology remains the primary method of classifying manifolds.
Informal examples

The homology of atopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

''X'' is a set of topological invariants of ''X'' represented by its ''homology groups''
$$H\_0(X),\; H\_1(X),\; H\_2(X),\; \backslash ldots$$
where the $k^$ homology group $H\_k(X)$ describes, informally, the number of holes in ''X'' with a ''k''-dimensional boundary. A 0-dimensional-boundary hole is simply a gap between two components. Consequently, $H\_0(X)$ describes the path-connected components of ''X''.
A one-dimensional sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

$S^1$ is a circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as
$$H\_k\backslash left(S^1\backslash right)\; =\; \backslash begin\; \backslash Z\; \&\; k\; =\; 0,\; 1\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$$
where $\backslash Z$ is the group of integers and $\backslash $ is the trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...

. The group $H\_1\backslash left(S^1\backslash right)\; =\; \backslash Z$ represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.
A two-dimensional sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

$S^2$ has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are
$$H\_k\backslash left(S^2\backslash right)\; =\; \backslash begin\; \backslash Z\; \&\; k\; =\; 0,\; 2\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$$
In general for an ''n''-dimensional sphere $S^n,$the homology groups are
$$H\_k\backslash left(S^n\backslash right)\; =\; \backslash begin\; \backslash Z\; \&\; k\; =\; 0,\; n\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$$
A two-dimensional ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used fo ...

$B^2$ is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for $H\_0\backslash left(B^2\backslash right)\; =\; \backslash Z$. In general, for an ''n''-dimensional ball $B^n,$
$$H\_k\backslash left(B^n\backslash right)\; =\; \backslash begin\; \backslash Z\; \&\; k\; =\; 0\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$$
The torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not t ...

is defined as a product of two circles $T\; =\; S^1\; \backslash times\; S^1$. The torus has a single path-connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are
$$H\_k(T)\; =\; \backslash begin\; \backslash Z\; \&\; k\; =\; 0,\; 2\; \backslash \backslash \; \backslash Z\; \backslash times\; \backslash Z\; \&\; k\; =\; 1\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$$
The two independent 1-dimensional holes form independent generators in a finitely-generated abelian group, expressed as the product group $\backslash Z\; \backslash times\; \backslash Z.$
For the projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

''P'', a simple computation shows (where $\backslash Z\_2$ is the cyclic group of order 2):
$$H\_k(P)\; =\; \backslash begin\; \backslash Z\; \&\; k\; =\; 0\; \backslash \backslash \; \backslash Z\_2\; \&\; k\; =\; 1\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$$
$H\_0(P)\; =\; \backslash Z$ corresponds, as in the previous examples, to the fact that there is a single connected component. $H\_1(P)\; =\; \backslash Z\_2$ is a new phenomenon: intuitively, it corresponds to the fact that there is a single non-contractible "loop", but if we do the loop twice, it becomes contractible to zero. This phenomenon is called torsion.
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 ''C''(''X'') encoding information about ''X''. A chain complex is a sequence of abelian groups or modules $C\_0,\; C\_1,\; C\_2,\; \backslash ldots$. connected by homomorphisms $\backslash partial\_n\; :\; C\_n\; \backslash to\; C\_,$ which are called boundary operators. That is, :$\backslash dotsb\; \backslash overset\; C\_n\; \backslash overset\; C\_\; \backslash overset\; \backslash dotsb\; \backslash overset\; C\_1\; \backslash overset\; C\_0\; \backslash overset\; 0$ where 0 denotes the trivial group and $C\_i\backslash equiv0$ for ''i'' < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all ''n'', :$\backslash partial\_n\; \backslash circ\; \backslash partial\_\; =\; 0\_,$ i.e., the constant map sending every element of $C\_$ to the group identity in $C\_.$ The statement that the boundary of a boundary is trivial is equivalent to the statement that $\backslash mathrm(\backslash partial\_)\backslash subseteq\backslash ker(\backslash partial\_n)$, where $\backslash mathrm(\backslash partial\_)$ denotes theimage
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...

of the boundary operator and $\backslash ker(\backslash partial\_n)$ its kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...

. Elements of $B\_n(X)\; =\; \backslash mathrm(\backslash partial\_)$ are called boundaries and elements of $Z\_n(X)\; =\; \backslash ker(\backslash partial\_n)$ are called cycles.
Since each chain group ''Cnormal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...

of $\backslash ker(\backslash partial\_n)$. Then one can create the quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...

:$H\_n(X)\; :=\; \backslash ker(\backslash partial\_n)\; /\; \backslash mathrm(\backslash partial\_)\; =\; Z\_n(X)/B\_n(X),$
called the ''n''th homology group of ''X''. The elements of ''Hfree abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...

generated by the ''n''-simplices of ''X''. See simplicial homology for details.
The ''singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...

'' groups ''Hcochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...

, which is the same as a chain complex but whose arrows, now denoted $d\_n,$ point in the direction of increasing ''n'' rather than decreasing ''n''; then the groups $\backslash ker\backslash left(d^n\backslash right)\; =\; Z^n(X)$ of ''cocycles'' and $\backslash mathrm\backslash left(d^\backslash right)\; =\; B^n(X)$ of follow from the same description. The ''n''th cohomology group of ''X'' is then the quotient group
:$H^n(X)\; =\; Z^n(X)/B^n(X),$
in analogy with the ''n''th homology group.
Homology vs. homotopy

Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...

s are similar to homology groups in that they can represent "holes" in a topological space. There is a close connection between the first homotopy group $\backslash pi\_1(X)$ and the first homology group $H\_1(X)$: the latter is the abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...

of the former. Hence, it is said that "homology is a commutative alternative to homotopy". The higher homotopy groups are abelian and are related to homology groups by the Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ...

, but can be vastly more complicated. For instance, the homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure ...

are poorly understood and are not known in general, in contrast to the straightforward description given above for the homology groups.
As an example, let ''X'' be the figure eight. Its first homotopy group $\backslash pi\_1(X)$ is the group of directed loops starting and ending at a predetermined point (e.g. its center). It is equivalent to the free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu'' ...

of rank 2, which is not commutative: looping around the leftmost cycle and then around the rightmost cycle is different than looping around the rightmost cycle and then looping around the leftmost cycle. In contrast, its first homology group $H\_1(X)$ is the group of cuts made in a surface. This group is commutative, since (informally) cutting the leftmost cycle and then the rightmost cycle leads to the same result as cutting the rightmost cycle and then the leftmost cycle.
Types of homology

The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the overall homology functor for the theory.Simplicial homology

The motivating example comes fromalgebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...

: the simplicial homology of a simplicial complex ''X''. Here the chain group ''Cfree abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...

or module whose generators are the ''n''-dimensional oriented simplexes of ''X''. The orientation is captured by ordering the complex's vertices and expressing an oriented simplex $\backslash sigma$ as an ''n''-tuple $(\backslash sigma;\; href="/html/ALL/l/.html"\; ;"title="">$ of its vertices listed in increasing order (i.e. $\backslash sigma;\; href="/html/ALL/l/.html"\; ;"title="">$Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...

.
Singular homology

Using simplicial homology example as a model, one can define a ''singular homology'' for anytopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

''X''. A chain complex for ''X'' is defined by taking ''Csimplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimensio ...

into ''X''. The homomorphisms ∂Group homology

Inabstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, one uses homology to define derived functors, for example the Tor functors. 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 $F\_1$ and a surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

homomorphism $p\_1\; :\; F\_1\; \backslash to\; X.$ Then one finds a free module $F\_2$ and a surjective homomorphism $p\_2\; :\; F\_2\; \backslash to\; \backslash ker\backslash left(p\_1\backslash right).$ Continuing in this fashion, a sequence of free modules $F\_n$ and homomorphisms $p\_n$ can be defined. By applying the functor ''F'' to this sequence, one obtains a chain complex; the homology $H\_n$ of this complex depends only on ''F'' and ''X'' and is, by definition, the ''n''-th derived functor of ''F'', applied to ''X''.
A common use of group (co)homology $H^2(G,\; M)$is to classify the possible extension groups ''E'' which contain a given ''G''-module ''M'' as a normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...

and have a given quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...

''G'', so that $G\; =\; E\; /\; M.$
Other homology theories

*Borel–Moore homology
In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960.
For reasonable compact spaces, Borel−Moore homology coincides with the usu ...

* Cellular homology
* Cyclic homology
* Hochschild homology
* Floer homology
* Intersection homology
* K-homology
* Khovanov homology
* Morse homology
* Persistent homology
* Steenrod homology
Homology functors

Chain complexes form acategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...

: A morphism from the chain complex ($d\_n\; :\; A\_n\; \backslash to\; A\_$) to the chain complex ($e\_n\; :\; B\_n\; \backslash to\; B\_$) is a sequence of homomorphisms $f\_n\; :\; A\_n\; \backslash to\; B\_n$ such that $f\_\; \backslash circ\; d\_n\; =\; e\_n\; \backslash circ\; f\_n$ for all ''n''. The ''n''-th homology ''Hfunctor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...

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 $X\; \backslash to\; Y$ induces a morphism from the chain complex of ''X'' to the chain complex of ''Y''), then the ''Hfunctor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...

s from the category that ''X'' belongs to into the category of abelian groups (or modules).
The only difference between homology and cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

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 ''HProperties

If ($d\_n\; :\; A\_n\; \backslash to\; A\_$) is a chain complex such that all but finitely many ''AEuler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...

''
:$\backslash chi\; =\; \backslash sum\; (-1)^n\; \backslash ,\; \backslash mathrm(A\_n)$
(using the rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...

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:
:$\backslash chi\; =\; \backslash sum\; (-1)^n\; \backslash ,\; \backslash mathrm(H\_n)$
and, especially in algebraic topology, this provides two ways to compute the important invariant $\backslash chi$ for the object ''X'' which gave rise to the chain complex.
Every short exact sequence
:$0\; \backslash rightarrow\; A\; \backslash rightarrow\; B\; \backslash rightarrow\; C\; \backslash rightarrow\; 0$
of chain complexes gives rise to a long exact sequence of homology groups
:$\backslash cdots\; \backslash to\; H\_n(A)\; \backslash to\; H\_n(B)\; \backslash to\; H\_n(C)\; \backslash to\; H\_(A)\; \backslash to\; H\_(B)\; \backslash to\; H\_(C)\; \backslash to\; H\_(A)\; \backslash to\; \backslash cdots$
All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps $H\_n(C)\; \backslash to\; H\_(A)$ The latter are called and are provided by the zig-zag lemma
In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.
Statement
In an a ...

. This lemma can be applied to homology in numerous ways that aid in calculating homology groups, such as the theories of relative homology and .
Applications

Application in pure mathematics

Notable theorems proved using homology include the following: * TheBrouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simpl ...

: If ''f'' is any continuous map from the ball ''BInvariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n.
It states:
:If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeo ...

: If ''U'' is an open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

of $\backslash R^n$ and $f\; :\; U\; \backslash to\; \backslash R^n$ is an injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...

, then $V\; =\; f(U)$ is open and ''f'' is a homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...

between ''U'' and ''V''.
* The Hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, i ...

: any continuous vector field on the 2-sphere (or more generally, the 2''k''-sphere for any $k\; \backslash geq\; 1$) vanishes at some point.
* The Borsuk–Ulam theorem
In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are i ...

: any continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...

from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal point
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...

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.)
* Invariance of dimension: if non-empty open subsets $U\; \backslash subseteq\; \backslash R^m$ and $V\; \backslash subseteq\; \backslash R^n$ are homeomorphic, then $m\; =\; n.$
Application in science and engineering

In topological data analysis, data sets are regarded as apoint cloud
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Poin ...

sampling of a manifold or algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. M ...

embedded in Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Finding techniques to robustly calculate homology using various triangulation strategies over multiple length scales is the topic of persistent homology.
In sensor network
Wireless sensor networks (WSNs) refer to networks of spatially dispersed and dedicated sensors that monitor and record the physical conditions of the environment and forward the collected data to a central location. WSNs can measure environmental c ...

s, sensors may communicate information via an ad-hoc network that dynamically changes in time. To understand the global context of this set of local measurements and communication paths, it is useful to compute the homology of the network topology
Network topology is the arrangement of the elements ( links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and contr ...

to evaluate, for instance, holes in coverage.
In dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

s theory in physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...

, Poincaré was one of the first to consider the interplay between the invariant manifold of a dynamical system and its topological invariants. Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentia ...

relates the dynamics of a gradient flow on a manifold to, for example, its homology. Floer homology extended this to infinite-dimensional manifolds. The KAM theorem established that periodic orbits can follow complex trajectories; in particular, they may form braids
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...

that can be investigated using Floer homology.
In one class of finite element methods, boundary-value problems for differential equations involving the Hodge-Laplace operator may need to be solved on topologically nontrivial domains, for example, in electromagnetic simulations. In these simulations, solution is aided by fixing the cohomology class of the solution based on the chosen boundary conditions and the homology of the domain. FEM domains can be triangulated, from which the simplicial homology can be calculated.
Software

Various software packages have been developed for the purposes of computing homology groups of finite cell complexesLinbox

is a C++ library for performing fast matrix operations, including

Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...

; it interfaces with botGap

an

Maple

Chomp

CAPD::Redhom

an

Perseus

are also written in C++. All three implement pre-processing algorithms based on

simple-homotopy equivalence In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions ...

and discrete Morse theory to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebraKenzo

is written in Lisp, and in addition to homology it may also be used to generate presentations of

homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

groups of finite simplicial complexes. Gmsh includes a homology solver for finite element meshes, which can generate Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

bases directly usable by finite element software.
See also

*Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...

* Cycle space
In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs.
This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The dim ...

* De Rham cohomology
* Eilenberg–Steenrod axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...

* Extraordinary homology theory
* Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...

* Homological conjectures in commutative algebra
* Homological connectivity
In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups.
Definitions
Background
''X'' is ''homologically-connected'' if its 0-th homology group equals Z, i.e. H_0(X)\cong \mat ...

* Homological dimension
* Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...

* Künneth theorem
* List of cohomology theories - also has a list of homology theories
* Poincaré duality
Notes

References

* * *. *. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc. * *. *. *. * *.External links

''Homology group'' at Encyclopaedia of Mathematics

N.J. Windberger intro to algebraic topology, last six lectures with an easy intro to homology

Algebraic topology Allen Hatcher - Chapter 2 on homology {{Authority control Homology theory,