Homology (mathematics)
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the term homology, originally introduced 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a
chain 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 contained in the kernel o ...
'', 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 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 ...
of a
cochain 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 contained in the kernel ...
, giving rise to various cohomology theories, in addition to the notion of the cohomology of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
.


Homology of chain complexes

To take the homology of a
chain 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 contained in the kernel o ...
, one starts with a chain complex, which is a sequence (C_\bullet, d_\bullet) of abelian groups C_ (whose elements are called chains) and group homomorphisms d_n (called boundary maps) such that the composition of any two consecutive maps is zero: : C_\bullet: \cdots \longrightarrow C_ \stackrel C_n \stackrel C_ \stackrel \cdots, \quad d_n \circ d_=0. The nth homology group H_ of this chain complex is then 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 ex ...
H_n = Z_n/B_n of cycles
modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
boundaries, where the n th group of cycles Z_n is given by the kernel subgroup Z_n := \ker d_n :=\, and the nth group of boundaries B_n is given by the
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subgroup B_n := \mathrm\, d_ :=\. One can optionally endow chain complexes with additional structure, for example by additionally taking the groups C_n to be modules over a coefficient ring R, and taking the boundary maps d_n to be R- module homomorphisms, resulting in homology groups H_ that are also quotient modules. Tools from
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
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 to 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 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-dimensional ...
, Morse homology, Khovanov homology, and
Hochschild homology In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a fiel ...
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 Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a homology theory is a type of
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from the
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * 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 In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of a ...
, 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 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-dimensional ...
(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 In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to a one-dimensional simplex, and a triangle-based pyramid is a 3-simplex.) Simplicial homology can in turn be generalized to
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-dimensional ...
, 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
ÄŒech cohomology In mathematics, specifically algebraic topology, ÄŒech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard ÄŒech. Moti ...
or (in the case of real coefficients)
De Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
.


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." For instance, a figure-eight shape has more holes than a circle S^1, and a 2-torus T^2 (a 2-dimensional surface shaped like an inner tube) has different holes from a 2-sphere 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 two embedded circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus T^2 and 2-sphere 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. Any formal sum of cycles is again called a cycle.


Cycles and boundaries (informal discussion)

Explicit constructions of homology groups are somewhat technical. As mentioned above, an explicit realization of the homology groups H_n(X) of a topological space X is defined in terms of the ''cycles'' and ''boundaries'' of a ''
chain 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 contained in the kernel o ...
'' (C_\bullet, d_\bullet) associated to X, where the type of chain complex depends on the choice of homology theory in use. These cycles and boundaries are elements of abelian groups, and are defined in terms of the boundary homomorphisms d_n: C_n \to C_ of the chain complex, where each C_n is an abelian group, and the d_n are group homomorphisms that satisfy d_ \circ d_n=0 for all n. Since such constructions are somewhat technical, informal discussions of homology sometimes focus instead on topological notions that parallel some of the group-theoretic aspects of cycles and boundaries. For example, in the context of chain complexes, a boundary is any element of the
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B_n := \mathrm\, d_ :=\ of the boundary homomorphism d_n: C_n \to C_, for some n. In topology, the boundary of a space is technically obtained by taking the space's closure minus its interior, but it is also a notion familiar from examples, e.g., the boundary of the unit disk is the unit circle, or more topologically, the boundary of D^2 is S^1. Topologically, the boundary of the closed interval ,1/math> is given by the disjoint union \ \, \amalg \, \ , and with respect to suitable orientation conventions, the oriented boundary of ,1/math> is given by the union of a positively-oriented \ with a negatively oriented \. The simplicial chain complex analog of this statement is that d_1( ,1 = \ - \ . (Since d_1 is a homomorphism, this implies d_1(k\cdot ,1 = k\cdot\ - k\cdot\ for any integer k .) In the context of chain complexes, a cycle is any element of the kernelZ_n := \ker d_n :=\, for some n. In other words, c \in C_n is a cycle if and only if d_n(c) = 0. The closest topological analog of this idea would be a shape that has "no boundary," in the sense that its boundary is the empty set. For example, since S^1, S^2 , and T^2 have no boundary, one can associate cycles to each of these spaces. However, the chain complex notion of cycles (elements whose boundary is a "zero chain") is more general than the topological notion of a shape with no boundary. It is this topological notion of no boundary that people generally have in mind when they claim that cycles can intuitively be thought of as detecting holes. The idea is that for no-boundary shapes like S^1, S^2, and T^2, it is possible in each case to glue on a larger shape for which the original shape is the boundary. For instance, starting with a circle S^1, one could glue a 2-dimensional disk D^2 to that S^1 such that the S^1 is the boundary of that D^2. Similarly, given a two-sphere S^2, one can glue a ball B^3 to that S^2 such that the S^2 is the boundary of that B^3. This phenomenon is sometimes described as saying that S^2 has a B^3-shaped "hole" or that it could be "filled in" with a B^3. More generally, any shape with no boundary can be "filled in" with a
cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...
, since if a given space Y has no boundary, then the boundary of the cone on Y is given by Y, and so if one "filled in" Y by gluing the cone on Y onto Y, then Y would be the boundary of that cone. (For example, a cone on S^1 is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to a disk D^2 whose boundary is that 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 In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
\mathbb^2 and complex projective plane \mathbb^2 have nontrivial
cobordism In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cob ...
classes and therefore cannot be "filled in" with manifolds. On the other hand, the boundaries discussed in the homology of a topological space X are different from the boundaries of "filled in" holes, because the homology of a topological space X has to do with the original space X, and not with new shapes built from gluing extra pieces onto X. For example, any embedded circle C in S^2 already bounds some embedded disk D in S^2, so such C gives rise to a boundary class in the homology of S^2. By contrast, no
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...
of S^1 into one of the 2 lobes of the figure-eight shape gives a boundary, despite the fact that it is possible to glue a disk onto a figure-eight lobe.


Homology groups

Given a sufficiently-nice topological space X, a choice of appropriate homology theory, and a chain complex (C_\bullet, d_\bullet) associated to X that is compatible with that homology theory, the nth homology group H_n(X) is then given by 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 ex ...
H_n(X)=Z_n/B_n of n-cycles (n-dimensional cycles) modulo n-dimensional boundaries. In other words, the elements of H_n(X), called ''homology classes'', are equivalence classes whose representatives are n-cycles, and any two cycles are regarded as equal in H_n(X) if and only if they differ by the addition of a boundary. This also implies that the "zero" element of H_n(X) is given by the group of n-dimensional boundaries, which also includes formal sums of such boundaries.


Informal examples

The homology of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' is a set of topological invariants of ''X'' represented by its ''homology groups'' H_0(X), H_1(X), H_2(X), \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 Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...
. Consequently, H_0(X) describes the path-connected components of ''X''. A one-dimensional
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
S^1 is a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
. 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\left(S^1\right) = \begin \Z & k = 0, 1 \\ \ & \text \end where \Z is the group of integers and \ is the trivial group. The group H_1\left(S^1\right) = \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 (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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\left(S^2\right) = \begin \Z & k = 0, 2 \\ \ & \text \end In general for an ''n''-dimensional sphere S^n, the homology groups are H_k\left(S^n\right) = \begin \Z & k = 0, n \\ \ & \text \end A two-dimensional
ball A ball is a round object (usually spherical, but sometimes 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 for s ...
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\left(B^2\right) = \Z. In general, for an ''n''-dimensional ball B^n, H_k\left(B^n\right) = \begin \Z & k = 0 \\ \ & \text \end The
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
is defined as a product of two circles 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 H_k(T^2) = \begin \Z & k = 0, 2 \\ \Z \times \Z & k = 1 \\ \ & \text \end If ''n'' products of a topological space ''X'' is written as X^n, then in general, for an ''n''-dimensional torus T^n = (S^1)^n, H_k(T^n) = \begin \Z^\binom & 0 \le k \le n \\ \ & \text \end (see ' and ' for more details). The two independent 1-dimensional holes form independent generators in a finitely-generated abelian group, expressed as the product group \Z \times \Z. For the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
''P'', a simple computation shows (where \Z_2 is the
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
of order 2): H_k(P) = \begin \Z & k = 0 \\ \Z_2 & k = 1 \\ \ & \text \end H_0(P) = \Z corresponds, as in the previous examples, to the fact that there is a single connected component. H_1(P) = \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, \ldots. connected by homomorphisms \partial_n : C_n \to C_, which are called boundary operators. That is, : \dotsb \overset C_n \overset C_ \overset \dotsb \overset C_1 \overset C_0 \overset 0 where 0 denotes the trivial group and C_i\equiv0 for ''i'' < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all ''n'', : \partial_n \circ \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 \mathrm(\partial_)\subseteq\ker(\partial_n), where \mathrm(\partial_) denotes the
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of the boundary operator and \ker(\partial_n) its kernel. Elements of B_n(X) = \mathrm(\partial_) are called boundaries and elements of Z_n(X) = \ker(\partial_n) are called cycles. Since each chain group ''Cn'' is abelian all its subgroups are normal. Then because \ker(\partial_n) is a subgroup of ''Cn'', \ker(\partial_n) is abelian, and since \mathrm(\partial_) \subseteq\ker(\partial_n) therefore \mathrm(\partial_) is 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 ...
of \ker(\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 ex ...
: H_n(X) := \ker(\partial_n) / \mathrm(\partial_) = Z_n(X)/B_n(X), called the ''n''th 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. 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 : \dotsb \overset C_n \overset C_ \overset \dotsb \overset C_1 \overset C_0 \overset \Z 0 where the boundary operator \epsilon is : \epsilon \left(\sum_i n_i \sigma_i\right) = \sum_i n_i for a combination \sum n_i \sigma_i, of points \sigma_i, which are the fixed generators of ''C''0. The reduced homology groups \tilde_i(X) coincide with H_i(X) for i \neq 0. The extra \Z in the chain complex represents the unique map emptyset\longrightarrow X 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 ''Hn''(''X'') of a ''
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
'' ''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 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-dimensional ...
'' 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 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 contained in the kernel ...
, 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 \ker\left(d^n\right) = Z^n(X) of ''cocycles'' and \mathrm\left(d^\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

The nth homotopy group \pi_n(X) of a topological space X is the group of homotopy classes of basepoint-preserving maps from the n-sphere S^n to X, under the group operation of concatenation. The most fundamental homotopy group is the fundamental group \pi_1(X). For connected X, the Hurewicz theorem describes a homomorphism h_*: \pi_n(X) \to H_n(X) called the Hurewicz homomorphism. For n>1, this homomorphism can be complicated, but when n=1, the Hurewicz homomorphism coincides with abelianization. That is, h_*: \pi_1(X) \to H_1(X) is surjective and its kernel is the commutator subgroup of \pi_1(X), with the consequence that H_1(X) is isomorphic to the abelianization of \pi_1(X). Higher homotopy groups are sometimes difficult to compute. 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. For an n=1 example, suppose X is the figure eight. As usual, its first homotopy group, or fundamental group, \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 isomorphic 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, \pi_1(X) \cong \mathbb * \mathbb, 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 H_1(X)\cong \mathbb \times \mathbb is abelian. To express this explicitly in terms of homology classes of cycles, one could take the homology class l of the lefthand cycle and the homology class r of the righthand cycle as basis elements of H_1(X), allowing us to write H_1(X)=\ .


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 from
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
: the simplicial homology of a
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
''X''. Here the chain group ''Cn'' is the free abelian group or
free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
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 \sigma as an ''n''-tuple (\sigma \sigma \dots, \sigma of its vertices listed in increasing order (i.e. \sigma < \sigma < \cdots < \sigma /math> in the complex's vertex ordering, where \sigma /math> is the ith vertex appearing in the tuple). The mapping \partial_n from ''Cn'' to ''Cn−1'' is called the and sends the simplex : \sigma = (\sigma \sigma \dots, \sigma to the formal sum : \partial_n(\sigma) = \sum_^n (-1)^i \left (\sigma \dots, \sigma -1 \sigma +1 \dots, \sigma \right ),\ which is evaluated as 0 if n = 0. This behavior on the generators induces a homomorphism on all of ''Cn'' as follows. Given an element c \in C_n, write it as the sum of generators c = \sum_ m_i \sigma_i, where 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 : \partial_n(c) = \sum_ 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 Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
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 In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, one uses homology to define
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s, 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 such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
homomorphism p_1 : F_1 \to X. Then one finds a free module F_2 and a surjective homomorphism p_2 : F_2 \to \ker\left(p_1\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 ...
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 ex ...
''G'', so that G = E / M.


Other homology theories

* Borel–Moore homology * Cellular homology * Cyclic homology *
Hochschild homology In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a fiel ...
* Floer homology * Intersection homology * K-homology * Khovanov homology * Morse homology * Persistent homology * Steenrod homology


Homology functors

Chain complexes form a
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
: A morphism from the chain complex (d_n : A_n \to A_) to the chain complex (e_n : B_n \to B_) is a sequence of homomorphisms f_n : A_n \to B_n such that f_ \circ d_n = e_n \circ f_n for all ''n''. The ''n''-th homology ''Hn'' can be viewed as a covariant
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 \to Y induces a morphism from the chain complex of ''X'' to the chain complex of ''Y''), then the ''Hn'' are covariant
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 ''Hn'') form ''contravariant'' functors from the category that ''X'' belongs to into the category of abelian groups or modules.


Properties

If (d_n : A_n \to A_) 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 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's ...
'' : \chi = \sum (-1)^n \, \mathrm(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: : \chi = \sum (-1)^n \, \mathrm(H_n) and, especially in algebraic topology, this provides two ways to compute the important invariant \chi for the object ''X'' which gave rise to the chain complex. Every
short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
: 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 of chain complexes gives rise to a long exact sequence of homology groups : \cdots \to H_n(A) \to H_n(B) \to H_n(C) \to H_(A) \to H_(B) \to H_(C) \to H_(A) \to \cdots All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps H_n(C) \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 abel ...
. This lemma can be applied to homology in numerous ways that aid in calculating homology groups, such as the theories of
relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intu ...
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 a \in B^n with f(a) = a. * Invariance of domain: If ''U'' is an
open subset In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...
of \R^n and f : U \to \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 of its codomain; that is, implies (equivalently by contraposition, impl ...
continuous map In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
, then V = f(U) is open and ''f'' is a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
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 function, continuous tangent vector field on even-dimensional n‑sphere, ''n''-spheres. For the ord ...
: any continuous vector field on the 2-sphere (or more generally, the 2''k''-sphere for any k \geq 1) vanishes at some point. * The Borsuk–Ulam theorem: any
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
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 U \subseteq \R^m and V \subseteq \R^n are homeomorphic, then m = n.


Application in science and engineering

In topological data analysis, data sets are regarded as a
point cloud A point cloud is a discrete set of data Point (geometry), points in space. The points may represent a 3D shape or object. Each point Position (geometry), position has its set of Cartesian coordinates (X, Y, Z). Points may contain data other than ...
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 solution set, set of solutions of a system of polynomial equations over the real number, ...
embedded in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. 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 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 Network topology is the arrangement of the elements (Data link, links, Node (networking), nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, ...
to evaluate, for instance, holes in coverage. In
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s theory in
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, 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. 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 complexes
Linbox
is a C++ library for performing fast matrix operations, including Smith normal form; it interfaces with bot
Gap
an
MapleChompCAPD::Redhom
an
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 In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
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.


Some non-homology-based discussions of surfaces


Origins

Homology theory can be said to start with the Euler polyhedron formula, or
Euler 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's ...
. This was followed by Riemann's definition of
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
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.


Surfaces

On the ordinary
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
S^2, the curve ''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. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
''a'' can be shrunk in the same way. The Jordan curve theorem shows that any closed curve such as ''c'' can be similarly shrunk to a point. This implies that S^2 has trivial fundamental group, so as a consequence, it also has trivial first homology group. The
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
T^2 has closed curves 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. 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: K^2 is the
Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
, 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 In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), 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 Bened ...
. 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 (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
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 In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
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 Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
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 closed 2-manifold can be realised as the connected sum of ''g'' tori and ''c'' projective planes, where the 2-sphere 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 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 contained in the kernel o ...
. Chain complexes (since greatly generalized) form the basis for most modern treatments of homology.
Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
and, independently, Leopold Vietoris and Walther Mayer 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 commu ...
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 in the special case of surfaces, the torsion part of the homology group only occurs for non-orientable cycles. 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
". Algebraic homology remains the primary method of classifying manifolds.


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 *
De Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
* Eilenberg–Steenrod axioms * Extraordinary homology theory *
Homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
* Homological conjectures in commutative algebra * Homological connectivity * 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 This is a list of some of the ordinary and generalized cohomology theory, generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectrum (homotopy theory), spectr ...
– also has a list of homology theories *
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ...


References


Further reading

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


External links


''Homology group'' at Encyclopaedia of Mathematics


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