# Homology (mathematics)

In mathematics, the term homology[a], originally developed in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups. This operation, in turn, allows one to associate various named homologies or homology theories to various other types of mathematical objects. Lastly, since there are many homology theories for topological spaces that produce the same answer, one also often speaks of the homology of a topological space. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.) There is also a related notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space.

### Homology of Chain Complexes

To take the homology of a chain complex, one starts with a chain complex, which is a sequence ${\displaystyle (C_{\bullet },d_{\bullet })}$ of abelian groups ${\displaystyle C_{n}}$ (whose elements are called chains) and group homomorphisms ${\displaystyle d_{n}}$ (called boundary maps) such that the composition of any two consecutive maps is zero:

${\displaystyle C_{\bullet }:\cdots \longrightarrow C_{n+1}{\stackrel {d_{n+1}}{\longrightarrow }}C_{n}{\stackrel {d_{n}}{\longrightarrow }}C_{n-1}{\stackrel {d_{n-1}}{\longrightarrow }}\cdots ,\quad d_{n}\circ d_{n+1}=0.}$

The ${\displaystyle n}$th homology group ${\displaystyle H_{n}}$ of this chain complex is then the quotient group ${\displaystyle H_{n}=Z_{n}/B_{n}}$ of cycles modulo boundaries, where the ${\displaystyle n}$th group of cycles ${\displaystyle Z_{n}}$ is given by the kernel subgroup${\displaystyle Z_{n}:=\ker d_{n}:=\{c\in C_{n}\,|\;d_{n}(c)=0\}}$, and the ${\displaystyle n}$th group of boundaries ${\displaystyle B_{n}}$ is given by the image subgroup${\displaystyle B_{n}:=\mathrm {im} \,d_{n+1}:=\{d_{n+1}(c)\,|\;c\in C_{n+1}\}}$. One can optionally endow chain complexes with additional structure, for example by additionally taking the groups ${\displaystyle C_{n}}$ to be modules over a coefficient ring ${\displaystyle R}$, and taking the boundary maps ${\displaystyle d_{n}}$ to be ${\displaystyle R}$-module homomorphisms, resulting in homology groups ${\displaystyle H_{n}}$ that are also quotient modules. Tools from homological algebra can be used to relate homology groups of different chain complexes.

### Homology Theories

To associate a homology theory to other types of mathematical objects, one first gives a prescription for associating chain complexes to that object, and then takes the homology of such a chain complex. For the homology theory to be valid, all such chain complexes associated the same mathematical object must have the same homology. The resulting homology theory is often named according to the type of chain complex prescribed. For example, singular homology, Morse homology, Khovanov homology, and Hochschild homology are respectively obtained from singular chain complexes, Morse complexes, Khovanov complexes, and Hochschild complexes. In other cases, such as for group homology, there are multiple common methods to compute the same homology groups.

In the language of category theory, a homology theory is a type of functor from the category of the mathematical object being studied to the category of abelian groups and group homomorphisms, or more generally to the category corresponding to the associated chain complexes. One can also formulate homology theories as derived functors on appropriate abelian categories, measuring the failure of an appropriate functor to be exact. One can describe this latter construction explicitly in terms of resolutions, or more abstractly from the perspective of derived categories or model categories. Regardless of how they are formulated, homology theories help provide information about the structure of the mathematical objects to which they are associated, and can sometimes help distinguish different objects.

### Homology of a Topological Space

Perhaps the most familiar usage of the term homology is for the homology of a topological space. For sufficiently nice topological spaces and compatible choices of coefficient rings, any homology theory satisfying the Eilenberg-Steenrod axioms yields the same homology groups as the singular homology (see below) of that topological space, with the consequence that one often simply refers to the "homology" of that space, instead of specifying which homology theory was used to compute the homology groups in question.

For 1-dimensional topological spaces, probably the simplest homology theory to use is graph homology, which could be regarded as a 1-dimensional special case of simplicial homology, the latter of which involves a decomposition of the topological space into simplices. (Simplices are a generalization of triangles to arbitrary dimension; for example, an edge in a graph is homeomorphic to a one-dimensional simplex, and a triangle-based pyramid is a 3-simplex.) Simplicial homology can in turn be generalized to singular homology, which allows more general maps of simplices into the topological space. Replacing simplices with disks of various dimensions results in a related construction called cellular homology. There are also other ways of computing these homology groups, for example via Morse homology, or by taking the output of the Universal Coefficient Theorem when applied to a cohomology theory such as Čech cohomology or (in the case of real coefficients) De Rham cohomology.

#### Inspirations for homology (informal discussion)

One of the ideas that led to the development of homology was the observation that certain low-dimensional shapes can be topologically distinguished by examining their holes or cavities. For instance, a figure-eight shape has more holes than a circle ${\displaystyle S^{1}}$, and a 2-torus ${\displaystyle T^{2}}$ (a 2-dimensional surface shaped like an inner tube) has different holes from a 2-sphere ${\displaystyle S^{2}}$ (a 2-dimensional surface shaped like a basketball). Studying topological features such as these led to the notion of the cycles that represent homology classes (the elements of homology groups). For example, the circles making up a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus ${\displaystyle T^{2}}$ and 2-sphere ${\displaystyle S^{2}}$ represent 2-cycles. Cycles form a group under the operation of formal addition, which refers to adding cycles symbolically rather than combining them geometrically. Reversing the orientation of a cycle corresponds to multiplying its coefficient by negative 1. Any formal sum of cycles is again called a cycle.

#### Cycles and Boundaries (informal discussion)

The defining feature of a cycle is not that it is the boundary of a missing object, but rather that the cycle itself has no boundary. For example, the closed interval ${\displaystyle [0,1]}$ is not a cycle, since its boundary is the disjoint union ${\displaystyle \{0\}\,\amalg \,\{1\}}$, or in the language of chain complexes, ${\displaystyle d_{1}(1\cdot [0,1])=1\cdot \{1\}-1\cdot \{0\}}$. By contrast, the circle ${\displaystyle S^{1}}$ is a cycle because it has "no boundary," by which we mean that its boundary is the empty set. Similarly, a two-dimensional disk ${\displaystyle D^{2}}$ is not a cycle, since its boundary is a circle ${\displaystyle S^{1}}$, but the 2-torus ${\displaystyle T^{2}}$ and 2-sphere ${\displaystyle S^{2}}$ are 2-cycles, since they are 2-dimensional shapes that have no boundary.

One might notice that for each of the 3 preceding examples we gave of cycles, ${\displaystyle S^{1}}$, ${\displaystyle T^{2}}$, and ${\displaystyle S^{2}}$, it is possible to construct another space for which that cycle is the boundary. For example, a 2-sphere ${\displaystyle S^{2}}$ can be embedded in a 3-dimensional space that contains a 3-dimensional ball ${\displaystyle B^{3}}$ whose boundary is that ${\displaystyle S^{2}}$. One sometimes describes this process as "filling in" an ${\displaystyle S^{2}}$ with a ${\displaystyle B^{3}}$, or says that ${\displaystyle S^{2}}$ has a ${\displaystyle B^{3}}$-shaped "hole." Due to these and similar examples, it is sometimes said that cycles can intuitively be thought of as holes. Any cycle at least admits a cone-shaped hole, since the boundary of the cone on a cycle is again that cycle. (For example, a cone on ${\displaystyle S^{1}}$ is homeomorphic to a disk, and its boundary is ${\displaystyle S^{1}}$.) However, it is sometimes desirable to restrict to nicer spaces such as manifolds, and not every cone is homeomorphic to a manifold. Embedded representatives of 1-cycles, 3-cycles, and oriented 2-cycles all admit manifold-shaped holes, but for example the real projective plane ${\displaystyle \mathbb {RP} ^{2}}$ and complex projective plane ${\displaystyle \mathbb {CP} ^{2}}$ have nontrivial cobordism classes and therefore cannot be "filled in" with manifolds.

The boundaries discussed in homology theory are different from the boundaries of "filled in" holes, however, because the former boundaries must be defined in terms of the original topological space, without recourse to attaching any extra pieces to that space. For example, any embedded circle ${\displaystyle C}$ in ${\displaystyle S^{2}}$ bounds some embedded disk ${\displaystyle D}$ in ${\displaystyle S^{2}}$, so such ${\displaystyle C}$ is a boundary in the homology of ${\displaystyle S^{2}}$. (More generally, any continuous map of ${\displaystyle S^{1}}$ to ${\displaystyle S^{2}}$ that extends to a continuous map of ${\displaystyle D^{2}}$ to ${\displaystyle S^{2}}$ can be regarded as a boundary, as long as it is compatible with the homology theory chosen.) By contrast, no embedding of ${\displaystyle S^{1}}$ in the figure-eight ${\displaystyle X_{8}}$ gives a boundary. On the other hand, there are self-intersecting maps of ${\displaystyle S^{1}}$ to ${\displaystyle X_{8}}$ that give boundaries. For example, the map that wraps the circle counter-clockwise around the first lobe of ${\displaystyle X_{8}}$, counter-clockwise around the second lobe of ${\displaystyle X_{8}}$, then clockwise around the first lobe and finally clockwise around the second lobe of ${\displaystyle X_{8}}$ before completing the circle, has nontrivial homotopy class, but as a chain (group element in a chain complex), this can be expressed as ${\displaystyle X_{1}+X_{2}-X_{1}-X_{2}=0}$, where ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are counter-clockwise embedded circles as the respective first and second lobes of ${\displaystyle X_{8}}$. (Any chain equal to zero is also a boundary, since it can be obtained by applying the boundary map to a zero chain, which topologically is like taking the boundary of the empty set.)

We have so far only given examples of boundaries instead of defining them rigorously. That is because the definition of boundary is part of the data of the type of chain complex chosen, which in turn depends on the choice of homology theory, although most of these are compatible with taking a topological boundary by deleting the interior of a space from the closure of a space. Once our chain complex, and hence our definition of boundary, are specified, the cycles are defined to be the elements with zero boundary. Since the boundary of a boundary is always zero in a chain complex, a boundary is always a cycle, regardless of choice of homology theory.

#### Homology groups

Given a topological space ${\displaystyle X}$, the ${\displaystyle n}$th homology group ${\displaystyle H_{n}(X)}$ is then given by the quotient group of ${\displaystyle n}$-cycles (${\displaystyle n}$-dimensional cycles) modulo ${\displaystyle n}$-dimensional boundaries. In other words, the elements of ${\displaystyle H_{n}(X)}$, called homology classes, are equivalence classes whose representatives are ${\displaystyle n}$-cycles, and any two cycles are regarded as equal in ${\displaystyle H_{n}(X)}$ if and only if they differ by the addition of a boundary. This also implies that the "zero" element of ${\displaystyle H_{n}(X)}$ is given by the group of ${\displaystyle n}$-dimensional boundaries, which also includes formal sums of such boundaries.

Returning to the examples discussed above, the two-sphere ${\displaystyle S^{2}}$ has ${\displaystyle H_{0}(S^{2})=\mathbb {Z} }$, ${\displaystyle H_{1}(S^{2})=0}$, and ${\displaystyle H_{2}(S^{2})=\mathbb {Z} }$, with ${\displaystyle H_{n}(S^{2})=0}$ for ${\displaystyle n>2}$. The figure-eight ${\displaystyle X_{8}}$ has ${\displaystyle H_{0}(X_{8})=\mathbb {Z} }$, ${\displaystyle H_{1}(X_{8})=\mathbb {Z} \times \mathbb {Z} }$, and ${\displaystyle H_{n}(X_{8})=0}$ for ${\displaystyle n>1}$. One can take the equivalence classes ${\displaystyle [X_{1}],[X_{2}]}$ as a basis for ${\displaystyle H_{1}(X_{8})}$, and the equivalence class ${\displaystyle [\mathrm {pt} ]}$ of the map of any point into ${\displaystyle X_{8}}$ as a basis for ${\displaystyle H_{0}(X_{8}).}$

## Background

### Origins

Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic.[1] This was followed by Riemann's definition of genus 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.[2]

Homology itself was developed as a way to analyse and classify manifolds 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.[3] 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 ${\displaystyle S^{1}}$ (1-manifold), while a surface cut through a three-dimensional manifold is a 2-cycle.

### Surfaces

Cycles on a 2-sphere ${\displaystyle S^{2}}$
Cycles on a torus ${\displaystyle T^{2}}$
Cycles on a Klein
bottle ${\displaystyle K^{2}}$
Cycles on a hemispherical projective plane ${\displaystyle P^{2}}$

On the ordinary sphere ${\displaystyle S^{2}}$, the cycle b in the diagram can be shrunk to the pole, and even the equatorial great circle 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 ${\displaystyle T^{2}}$ 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. The various ways of gluing the sides yield just four topologically distinct surfaces:

${\displaystyle K^{2}}$ is the Klein bottle, which is a torus with a twist in it (In the square diagram, the twist can be seen 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. 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 ${\displaystyle P^{2}}$ has both joins twisted. The uncut form, generally represented as the Boy surface, 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 is non-shrinkable while c is. If b were only wound once, it would also be non-shrinkable and reverse left and right. However it is wound a second time, which swaps right and left back again; it can be shrunk to a point and is homologous to c.

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 2a = 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 = 2b, 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 (2n-gons) can be glued to make different manifolds. Conversely, a closed surface with n non-zero classes can be cut into a 2n-gon. Variations are also possible, for example a hexagon may also be glued to form a torus.[4]

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 numbers of the manifold (Betti numbers are a refinement of the Euler characteristic). Classifying the non-orientable cycles requires additional information about torsion coefficients.[3]

The complete classification of 1- and 2-manifolds is given in the table.

Topological characteristics of closed 1- and 2-manifolds[5]
Manifold Euler no.,
χ
Orientability Betti numbers Torsion coefficient
(1-dimensional)
Symbol[4] Name b0 b1 b2
${\displaystyle S^{1}}$ Circle (1-manifold) 0 Orientable 1 1
${\displaystyle S^{2}}$ Sphere 2 Orientable 1 0 1 None
${\displaystyle T^{2}}$ Torus 0 Orientable 1 2 1 None
${\displaystyle P^{2}}$ Projective plane 1 Non-orientable 1 0 0 2
${\displaystyle K^{2}}$ Klein bottle 0 Non-orientable 1 1 0 2
2-holed torus −2 Orientable 1 4 1 None
g-holed torus (g is the genus) 2 − 2g Orientable 1 2g 1 None
Sphere with c cross-caps 2 − c Non-orientable 1 c − 1 0 2
2-Manifold with g holes and c cross-caps (c > 0) 2  (2g + c) Non-orientable 1 (2g + c)  1 0 2
Notes
1. For a non-orientable surface, a hole is equivalent to two cross-caps.
2. Any closed 2-manifold can be realised as the connected sum of g tori and c projective planes, where the 2-sphere ${\displaystyle S^{2}}$ is regarded as the empty connected sum. Homology is preserved by the operation of connected sum.

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 simplicial chain complex.[6][7] Chain complexes (since greatly generalized) form the basis for most modern treatments of homology.

Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28.[8][9][10] The new combinatorial topology formally treated topological classes as abelian groups. 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".[11] Algebraic homology remains the primary method of classifying manifolds.[12]

## Informal examples

The homology of a topological space X is a set of topological invariants of X represented by its homology groups ${\displaystyle H_{0}(X),H_{1}(X),H_{2}(X),\ldots }$ where the ${\displaystyle k^{\rm {th}}}$ homology group ${\displaystyle H_{k}(X)}$ 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, ${\displaystyle H_{0}(X)}$ describes the path-connected components of X.[13]

The circle or 1-sphere ${\displaystyle S^{1}}$
The 2-sphere ${\displaystyle S^{2}}$ is the outer shell, not the interior, of a ball

A one-dimensional sphere ${\displaystyle S^{1}}$ is a circle. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as ${\displaystyle H_{k}\left(S^{1}\right)={\begin{cases}\mathbb {Z} &k=0,1\\\{0\}&{\text{otherwise}}\end{cases}}}$ where ${\displaystyle \mathbb {Z} }$ is the group of integers and ${\displaystyle \{0\}}$ is the trivial group. The group ${\displaystyle H_{1}\left(S^{1}\right)=\mathbb {Z} }$ represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.[14]

A two-dimensional sphere ${\displaystyle S^{2}}$ 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[14][15] ${\displaystyle H_{k}\left(S^{2}\right)={\begin{cases}\mathbb {Z} &k=0,2\\\{0\}&{\text{otherwise}}\end{cases}}}$

In general for an n-dimensional sphere ${\displaystyle S^{n},}$the homology groups are ${\displaystyle H_{k}\left(S^{n}\right)={\begin{cases}\mathbb {Z} &k=0,n\\\{0\}&{\text{otherwise}}\end{cases}}}$

The solid disc or 2-ball ${\displaystyle B^{2}}$
The torus ${\displaystyle T=S^{1}\times S^{1}}$

A two-dimensional ball ${\displaystyle B^{2}}$ 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 ${\displaystyle H_{0}\left(B^{2}\right)=\mathbb {Z} }$. In general, for an n-dimensional ball ${\displaystyle B^{n},}$[14]

${\displaystyle H_{k}\left(B^{n}\right)={\begin{cases}\mathbb {Z} &k=0\\\{0\}&{\text{otherwise}}\end{cases}}}$

The torus is defined as a product of two circles ${\displaystyle T^{2}=S^{1}\times S^{1}}$. 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[16] ${\displaystyle H_{k}(T^{2})={\begin{cases}\mathbb {Z} &k=0,2\\\mathbb {Z} \times \mathbb {Z} &k=1\\\{0\}&{\text{otherwise}}\end{cases}}}$

If n products of a topological space X is written as ${\displaystyle X^{n}}$, then in general, for an n-dimensional torus ${\displaystyle T^{n}=(S^{1})^{n}}$,

${\displaystyle H_{k}(T^{n})={\begin{cases}\mathbb {Z} ^{\binom {n}{k}}&0\leq k\leq n\\\{0\}&{\text{otherwise}}\end{cases}}}$

(see Torus#n-dimensional torus and Betti number#More examples for more details).

The two independent 1-dimensional holes form independent generators in a finitely-generated abelian group, expressed as the product group ${\displaystyle \mathbb {Z} \times \mathbb {Z} .}$

For the projective plane P, a simple computation shows (where ${\displaystyle \mathbb {Z} _{2}}$ is the cyclic group of order 2):[17] ${\displaystyle H_{k}(P)={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}$

${\displaystyle H_{0}(P)=\mathbb {Z} }$ corresponds, as in the previous examples, to the fact that there is a single connected component. ${\displaystyle H_{1}(P)=\mathbb {Z} _{2}}$ 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 chain complex C(X) encoding information about X. A chain complex is a sequence of abelian groups or modules ${\displaystyle C_{0},C_{1},C_{2},\ldots }$. connected by homomorphisms ${\displaystyle \partial _{n}:C_{n}\to C_{n-1},}$ which are called boundary operators.[16] That is,

${\displaystyle \dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\partial _{0}}{\longrightarrow \,}}0}$

where 0 denotes the trivial group and ${\displaystyle C_{i}\equiv 0}$ for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,

${\displaystyle \partial _{n}\circ \partial _{n+1}=0_{n+1,n-1},}$

i.e., the constant map sending every element of ${\displaystyle C_{n+1}}$ to the group identity in ${\displaystyle C_{n-1}.}$

The statement that the boundary of a boundary is trivial is equivalent to the statement that ${\displaystyle \mathrm {im} (\partial _{n+1})\subseteq \ker(\partial _{n})}$, where ${\displaystyle \mathrm {im} (\partial _{n+1})}$ denotes the image of the boundary operator and ${\displaystyle \ker(\partial _{n})}$ its kernel. Elements of ${\displaystyle B_{n}(X)=\mathrm {im} (\partial _{n+1})}$ are called boundaries and elements of ${\displaystyle Z_{n}(X)=\ker(\partial _{n})}$ are called cycles.

Since each chain group Cn is abelian all its subgroups are normal. Then because ${\displaystyle \ker(\partial _{n})}$ is a subgroup of Cn, ${\displaystyle \ker(\partial _{n})}$ is abelian, and since ${\displaystyle \mathrm {im} (\partial _{n+1})\subseteq \ker(\partial _{n})}$ therefore ${\displaystyle \mathrm {im} (\partial _{n+1})}$ is a normal subgroup of ${\displaystyle \ker(\partial _{n})}$. Then one can create the quotient group

${\displaystyle H_{n}(X):=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})=Z_{n}(X)/B_{n}(X),}$

called the nth homology group of X. The elements of Hn(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.[18]

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 nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.[19]

The reduced homology groups of a chain complex C(X) are defined as homologies of the augmented chain complex[20]

${\displaystyle \dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\epsilon }{\longrightarrow \,}}\mathbb {Z} {\longrightarrow \,}0}$

where the boundary operator ${\displaystyle \epsilon }$ is

${\displaystyle \epsilon \left(\sum _{i}n_{i}\sigma _{i}\right)=\sum _{i}n_{i}}$

for a combination ${\displaystyle \sum n_{i}\sigma _{i},}$ of points ${\displaystyle \sigma _{i},}$ which are the fixed generators of C0. The reduced homology groups ${\displaystyle {\tilde {H}}_{i}(X)}$ coincide with ${\displaystyle H_{i}(X)}$ for ${\displaystyle i\neq 0.}$ The extra ${\displaystyle \mathbb {Z} }$ in the chain complex represents the unique map ${\displaystyle [\emptyset ]\longrightarrow X}$ from the empty simplex to X.

Computing the cycle ${\displaystyle Z_{n}(X)}$ and boundary ${\displaystyle B_{n}(X)}$ 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 Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with Cn(X) the free abelian group generated by the n-simplices of X. See simplicial homology for details.

The singular homology groups Hn(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 ${\displaystyle d_{n},}$ point in the direction of increasing n rather than decreasing n; then the groups ${\displaystyle \ker \left(d^{n}\right)=Z^{n}(X)}$ of cocycles and ${\displaystyle \mathrm {im} \left(d^{n-1}\right)=B^{n}(X)}$ of coboundaries follow from the same description. The nth cohomology group of X is then the quotient group

${\displaystyle H^{n}(X)=Z^{n}(X)/B^{n}(X),}$

in analogy with the nth homology group.

## Homology vs. homotopy

Homotopy groups 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 ${\displaystyle \pi _{1}(X)}$ and the first homology group ${\displaystyle H_{1}(X)}$: the latter is the abelianization of the former. Hence, it is said that "homology is a commutative alternative to homotopy".[21]: 4:00  The higher homotopy groups are abelian and are related to homology groups by the Hurewicz theorem, but can be vastly more complicated. For instance, the homotopy groups of spheres 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. As usual, its first homotopy group, or fundamental group, ${\displaystyle \pi _{1}(X)}$ is the group of homotopy classes of directed loops starting and ending at a predetermined point (e.g. its center). It is equivalent to the free group of rank 2, ${\displaystyle \pi _{1}(X)\cong \mathbb {Z} *\mathbb {Z} }$, which is not commutative: looping around the lefthand cycle and then around the righthand cycle is different from looping around the righthand cycle and then looping around the lefthand cycle. By contrast, the figure eight's first homology group ${\displaystyle H_{1}(X)\cong \mathbb {Z} \times \mathbb {Z} }$ is abelian. To express this explicitly in terms of homology classes of cycles, one could take the homology class ${\displaystyle l}$ of the lefthand cycle and the homology class ${\displaystyle r}$ of the righthand cycle as basis elements of ${\displaystyle H_{1}(X)}$, allowing us to write ${\displaystyle H_{1}(X)=\{a_{l}l+a_{r}r\,|\;a_{l},a_{r}\in \mathbb {Z} \}}$.

## 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.[22]

### Simplicial homology

The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X. Here the chain group Cn is the free abelian group 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 ${\displaystyle \sigma }$ as an n-tuple ${\displaystyle (\sigma [0],\sigma [1],\dots ,\sigma [n])}$ of its vertices listed in increasing order (i.e. ${\displaystyle \sigma [0]<\sigma [1]<\cdots <\sigma [n]}$ in the complex's vertex ordering, where ${\displaystyle \sigma [i]}$ is the ${\displaystyle i}$th vertex appearing in the tuple). The mapping ${\displaystyle \partial _{n}}$ from Cn to Cn−1 is called the boundary mapping and sends the simplex

${\displaystyle \sigma =(\sigma [0],\sigma [1],\dots ,\sigma [n])}$

to the formal sum

${\displaystyle \partial _{n}(\sigma )=\sum _{i=0}^{n}(-1)^{i}\left(\sigma [0],\dots ,\sigma [i-1],\sigma [i+1],\dots ,\sigma [n]\right),}$

which is considered 0 if ${\displaystyle n=0.}$ This behavior on the generators induces a homomorphism on all of Cn as follows. Given an element ${\displaystyle c\in C_{n}}$, write it as the sum of generators ${\textstyle c=\sum _{\sigma _{i}\in X_{n}}m_{i}\sigma _{i},}$ where ${\displaystyle X_{n}}$ is the set of n-simplexes in X and the mi are coefficients from the ring Cn is defined over (usually integers, unless otherwise specified). Then define

${\displaystyle \partial _{n}(c)=\sum _{\sigma _{i}\in X_{n}}m_{i}\partial _{n}(\sigma _{i}).}$

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 Smith normal form.

### Singular homology

Using simplicial homology example as a model, one can define a singular homology for any topological space X. A chain complex for X is defined by taking Cn to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into X. The homomorphisms ∂n arise from the boundary maps of simplices.

### Group homology

In abstract algebra, 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 ${\displaystyle F_{1}}$ and a surjective homomorphism ${\displaystyle p_{1}:F_{1}\to X.}$ Then one finds a free module ${\displaystyle F_{2}}$ and a surjective homomorphism ${\displaystyle p_{2}:F_{2}\to \ker \left(p_{1}\right).}$ Continuing in this fashion, a sequence of free modules ${\displaystyle F_{n}}$ and homomorphisms ${\displaystyle p_{n}}$ can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology ${\displaystyle H_{n}}$ 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 ${\displaystyle H^{2}(G,M)}$is to classify the possible extension groups E which contain a given G-module M as a normal subgroup and have a given quotient group G, so that ${\displaystyle G=E/M.}$

## Homology functors

Chain complexes form a category: A morphism from the chain complex (${\displaystyle d_{n}:A_{n}\to A_{n-1}}$) to the chain complex (${\displaystyle e_{n}:B_{n}\to B_{n-1}}$) is a sequence of homomorphisms ${\displaystyle f_{n}:A_{n}\to B_{n}}$ such that ${\displaystyle f_{n-1}\circ d_{n}=e_{n}\circ f_{n}}$ for all n. The n-th homology Hn 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 ${\displaystyle X\to Y}$ induces a morphism from the chain complex of X to the chain complex of Y), then the Hn are covariant functors 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 Hn) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.

## Properties

If (${\displaystyle d_{n}:A_{n}\to A_{n-1}}$) is a chain complex such that all but finitely many An are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the Euler characteristic

${\displaystyle \chi =\sum (-1)^{n}\,\mathrm {rank} (A_{n})}$

(using the 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:

${\displaystyle \chi =\sum (-1)^{n}\,\mathrm {rank} (H_{n})}$

and, especially in algebraic topology, this provides two ways to compute the important invariant ${\displaystyle \chi }$ for the object X which gave rise to the chain complex.

Every short exact sequence

${\displaystyle 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0}$

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

${\displaystyle \cdots \to H_{n}(A)\to H_{n}(B)\to H_{n}(C)\to H_{n-1}(A)\to H_{n-1}(B)\to H_{n-1}(C)\to H_{n-2}(A)\to \cdots }$

All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps ${\displaystyle H_{n}(C)\to H_{n-1}(A)}$ The latter are called connecting homomorphisms and are provided by the zig-zag lemma. This 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.

## Applications

### Application in pure mathematics

Notable theorems proved using homology include the following:

• The Brouwer fixed point theorem: If f is any continuous map from the ball Bn to itself, then there is a fixed point ${\displaystyle a\in B^{n}}$ with ${\displaystyle f(a)=a.}$
• Invariance of domain: If U is an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle f:U\to \mathbb {R} ^{n}}$ is an injective continuous map, then ${\displaystyle V=f(U)}$ is open and f is a homeomorphism between U and V.
• The Hairy ball theorem: any continuous vector field on the 2-sphere (or more generally, the 2k-sphere for any ${\displaystyle k\geq 1}$) vanishes at some point.
• The Borsuk–Ulam theorem: any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points 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 ${\displaystyle U\subseteq \mathbb {R} ^{m}}$ and ${\displaystyle V\subseteq \mathbb {R} ^{n}}$ are homeomorphic, then ${\displaystyle m=n.}$[23]

### Application in science and engineering

In topological data analysis, data sets are regarded as a point cloud sampling of a manifold or algebraic variety embedded in Euclidean space. 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.[24]

In sensor networks, 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 to evaluate, for instance, holes in coverage.[25]

In dynamical systems theory in physics, Poincaré was one of the first to consider the interplay between the invariant manifold of a dynamical system and its topological invariants. Morse theory 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 that can be investigated using Floer homology.[26]

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.[27][28]

## Software

Various software packages have been developed for the purposes of computing homology groups of finite cell complexes. Linbox is a C++ library for performing fast matrix operations, including Smith normal form; it interfaces with both Gap and Maple. Chomp, CAPD::Redhom and Perseus are also written in C++. All three implement pre-processing algorithms based on simple-homotopy equivalence and discrete Morse theory to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebra. Kenzo is written in Lisp, and in addition to homology it may also be used to generate presentations of homotopy groups of finite simplicial complexes. Gmsh includes a homology solver for finite element meshes, which can generate Cohomology bases directly usable by finite element software.[27]

## Notes

1. ^ in part from Greek ὁμός homos "identical"

## References

1. ^ Stillwell 1993, p. 170
2. ^ Weibel 1999, pp. 2–3 (in PDF)
3. ^ a b Richeson 2008, p. 254
4. ^ a b Weeks, Jeffrey R. (2001). The Shape of Space. CRC Press. ISBN 978-0-203-91266-9.
5. ^ Richeson 2008
6. ^ Richeson 2008, p. 258
7. ^ Weibel 1999, p. 4
8. ^ Hilton 1988, p. 284
9. ^ For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), in French, note 41, explicitly names Noether as inventing the homology group.
10. ^ Hirzebruch, Friedrich, Emmy Noether and Topology in Teicher 1999, pp. 61–63.
11. ^ Bourbaki and Algebraic Topology by John McCleary (PDF) Archived 2008-07-23 at the Wayback Machine gives documentation (translated into English from French originals).
12. ^ Richeson 2008, p. 264
13. ^ Spanier 1966, p. 155
14. ^ a b c Gowers, Barrow-Green & Leader 2010, pp. 390–391
15. ^ Wildberger, Norman J. (2012). "More homology computations". YouTube. Archived from the original on 2021-12-11.
16. ^ a b Hatcher 2002, p. 106
17. ^ Wildberger, Norman J. (2012). "Delta complexes, Betti numbers and torsion". YouTube. Archived from the original on 2021-12-11.
18. ^ Hatcher 2002, pp. 105–106
19. ^ Hatcher 2002, p. 113
20. ^ Hatcher 2002, p. 110
21. ^ Wildberger, N. J. (2012). "An introduction to homology". YouTube. Archived from the original on 2021-12-11.
22. ^ Spanier 1966, p. 156
23. ^ Hatcher 2002, p. 126.
24. ^ "CompTop overview". Retrieved 16 March 2014.
25. ^ "Robert Ghrist: applied topology". Retrieved 16 March 2014.
26. ^ van den Berg, J.B.; Ghrist, R.; Vandervorst, R.C.; Wójcik, W. (2015). "Braid Floer homology" (PDF). Journal of Differential Equations. 259 (5): 1663–1721. Bibcode:2015JDE...259.1663V. doi:10.1016/j.jde.2015.03.022. S2CID 16865053.
27. ^ a b Pellikka, M; S. Suuriniemi; L. Kettunen; C. Geuzaine (2013). "Homology and Cohomology Computation in Finite Element Modeling" (PDF). SIAM J. Sci. Comput. 35 (5): B1195–B1214. Bibcode:2013SJSC...35B1195P. CiteSeerX 10.1.1.716.3210. doi:10.1137/130906556.
28. ^ Arnold, Douglas N.; Richard S. Falk; Ragnar Winther (16 May 2006). "Finite element exterior calculus, homological techniques, and applications". Acta Numerica. 15: 1–155. Bibcode:2006AcNum..15....1A. doi:10.1017/S0962492906210018. S2CID 122763537.