In mathematics, the term homology^{[a]}, originally introduced in algebraic topology, has three primary, closelyrelated 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 2dimensional 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.
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Homology of Chain Complexes
To take the homology of a chain complex, one starts with a chain complex, which is a sequence of abelian groups (whose elements are called chains) and group homomorphisms (called boundary maps) such that the composition of any two consecutive maps is zero:
The th homology group of this chain complex is then the quotient group of cycles modulo boundaries, where the th group of cycles is given by the kernel subgroup, and the th group of boundaries is given by the image subgroup. One can optionally endow chain complexes with additional structure, for example by additionally taking the groups to be modules over a coefficient ring , and taking the boundary maps to be module homomorphisms, resulting in homology groups 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 EilenbergSteenrod 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 1dimensional topological spaces, probably the simplest homology theory to use is graph homology, which could be regarded as a 1dimensional 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 onedimensional simplex, and a trianglebased pyramid is a 3simplex.) 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 lowdimensional shapes can be topologically distinguished by examining their "holes." For instance, a figureeight shape has more holes than a circle , and a 2torus (a 2dimensional surface shaped like an inner tube) has different holes from a 2sphere (a 2dimensional 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 figureeight shape provide examples of onedimensional cycles, or 1cycles, and the 2torus and 2sphere represent 2cycles. 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 of a topological space is defined in terms of the cycles and boundaries of a chain complex associated to , 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 of the chain complex, where each is an abelian group, and the are group homomorphisms that satisfy for all .
Since such constructions are somewhat technical, informal discussions of homology sometimes focus instead on topological notions that parallel some of the grouptheoretic aspects of cycles and boundaries.
For example, in the context of chain complexes, a boundary is any element of the imageof the boundary homomorphism , for some . 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 is .
Topologically, the boundary of the closed interval is given by the disjoint union , and with respect to suitable orientation conventions, the oriented boundary of is given by the union of a positivelyoriented with a negatively oriented The simplicial chain complex analog of this statement is that. (Since is a homomorphism, this implies for any integer .)
In the context of chain complexes, a cycle is any element of the kernel, for some . In other words, is a cycle if and only if . 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 , and 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 noboundary shapes like , , and , 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 , one could glue a 2dimensional disk to that such that the is the boundary of that . Similarly, given a twosphere , one can glue a ball to that such that the is the boundary of that . This phenomenon is sometimes described as saying that has a shaped "hole" or that it could be "filled in" with a .
More generally, any shape with no boundary can be "filled in" with a cone, since if a given space has no boundary, one can glue to the cone on , and then will be the boundary of that cone. (For example, a cone on is homeomorphic to a disk whose boundary is that .) 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 1cycles, 3cycles, and oriented 2cycles all admit manifoldshaped holes, but for example the real projective plane and complex projective plane have nontrivial cobordism classes and therefore cannot be "filled in" with manifolds.
On the other hand, the boundaries discussed in the homology of a topological space are different from the boundaries of "filled in" holes, because the homology of a topological space has to do with the original space , and not with new shapes built from gluing extra pieces onto . For example, any embedded circle in already bounds some embedded disk in , so such gives rise to a boundary class in the homology of . By contrast, no embedding of into one of the 2 lobes of the figureeight shape gives a boundary, despite the fact that it is possible to glue a disk onto a figureeight lobe.
Homology groups
Given a sufficientlynice topological space , a choice of appropriate homology theory, and a chain complex associated to that is compatible with that homology theory, the th homology group is then given by the quotient group of cycles (dimensional cycles) modulo dimensional boundaries. In other words, the elements of , called homology classes, are equivalence classes whose representatives are cycles, and any two cycles are regarded as equal in if and only if they differ by the addition of a boundary. This also implies that the "zero" element of is given by the group of dimensional boundaries, which also includes formal sums of such boundaries.
Informal examples
The homology of a topological space X is a set of topological invariants of X represented by its homology groups where the homology group describes, informally, the number of holes in X with a kdimensional boundary. A 0dimensionalboundary hole is simply a gap between two components. Consequently, describes the pathconnected components of X.^{[1]}
A onedimensional sphere is a circle. It has a single connected component and a onedimensionalboundary hole, but no higherdimensional holes. The corresponding homology groups are given as where is the group of integers and is the trivial group. The group represents a finitelygenerated abelian group, with a single generator representing the onedimensional hole contained in a circle.^{[2]}
A twodimensional sphere has a single connected component, no onedimensionalboundary holes, a twodimensionalboundary hole, and no higherdimensional holes. The corresponding homology groups are^{[2]}^{[3]}
In general for an ndimensional sphere the homology groups are
A twodimensional ball is a solid disc. It has a single pathconnected component, but in contrast to the circle, has no higherdimensional holes. The corresponding homology groups are all trivial except for . In general, for an ndimensional ball ^{[2]}
The torus is defined as a product of two circles . The torus has a single pathconnected component, two independent onedimensional holes (indicated by circles in red and blue) and one twodimensional hole as the interior of the torus. The corresponding homology groups are^{[4]}
If n products of a topological space X is written as , then in general, for an ndimensional torus ,
(see Torus#ndimensional torus and Betti number#More examples for more details).
The two independent 1dimensional holes form independent generators in a finitelygenerated abelian group, expressed as the product group
For the projective plane P, a simple computation shows (where is the cyclic group of order 2):^{[5]}
corresponds, as in the previous examples, to the fact that there is a single connected component. is a new phenomenon: intuitively, it corresponds to the fact that there is a single noncontractible "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 . connected by homomorphisms which are called boundary operators.^{[4]} That is,
where 0 denotes the trivial group and for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,
i.e., the constant map sending every element of to the group identity in
The statement that the boundary of a boundary is trivial is equivalent to the statement that , where denotes the image of the boundary operator and its kernel. Elements of are called boundaries and elements of are called cycles.
Since each chain group C_{n} is abelian all its subgroups are normal. Then because is a subgroup of C_{n}, is abelian, and since therefore is a normal subgroup of . Then one can create the quotient group
called the nth homology group of X. The elements of H_{n}(X) are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous.^{[6]}
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.^{[7]}
The reduced homology groups of a chain complex C(X) are defined as homologies of the augmented chain complex^{[8]}
where the boundary operator is
for a combination of points which are the fixed generators of C_{0}. The reduced homology groups coincide with for The extra in the chain complex represents the unique map from the empty simplex to X.
Computing the cycle and boundary groups is usually rather difficult since they have a very large number of generators. On the other hand, there are tools which make the task easier.
The simplicial homology groups H_{n}(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with C_{n}(X) the free abelian group generated by the nsimplices of X. See simplicial homology for details.
The singular homology groups H_{n}(X) are defined for any topological space X, and agree with the simplicial homology groups for a simplicial complex.
Cohomology groups are formally similar to homology groups: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted point in the direction of increasing n rather than decreasing n; then the groups of cocycles and of coboundaries follow from the same description. The nth cohomology group of X is then the quotient group
in analogy with the nth homology group.
Homology vs. homotopy
The nth homotopy group of a topological space is the group of homotopy classes of basepointpreserving maps from the sphere to , under the group operation of concatenation. The most fundamental homotopy group is the fundamental group . For connected , the Hurewicz theorem describes a homomorphism called the Hurewicz homomorphism. For , this homomorphism can be complicated, but when , the Hurewicz homomorphism coincides with abelianization. That is, is surjective and its kernel is the commutator subgroup of , with the consequence that is isomorphic to the abelianization of . 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 example, suppose is the figure eight. As usual, its first homotopy group, or fundamental group, 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 of rank 2, , 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 is abelian. To express this explicitly in terms of homology classes of cycles, one could take the homology class of the lefthand cycle and the homology class of the righthand cycle as basis elements of , allowing us to write .
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.^{[9]}
Simplicial homology
The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X. Here the chain group C_{n} is the free abelian group or module whose generators are the ndimensional oriented simplexes of X. The orientation is captured by ordering the complex's vertices and expressing an oriented simplex as an ntuple of its vertices listed in increasing order (i.e. in the complex's vertex ordering, where is the th vertex appearing in the tuple). The mapping from C_{n} to C_{n−1} is called the boundary mapping and sends the simplex
to the formal sum
which is considered 0 if This behavior on the generators induces a homomorphism on all of C_{n} as follows. Given an element , write it as the sum of generators where is the set of nsimplexes in X and the m_{i} are coefficients from the ring C_{n} is defined over (usually integers, unless otherwise specified). Then define
The dimension of the nth 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 C_{n} to be the free abelian group (or free module) whose generators are all continuous maps from ndimensional 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 and a surjective homomorphism Then one finds a free module and a surjective homomorphism Continuing in this fashion, a sequence of free modules and homomorphisms can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology of this complex depends only on F and X and is, by definition, the nth derived functor of F, applied to X.
A common use of group (co)homology is to classify the possible extension groups E which contain a given Gmodule M as a normal subgroup and have a given quotient group G, so that
Other homology theories
Homology functors
Chain complexes form a category: A morphism from the chain complex () to the chain complex () is a sequence of homomorphisms such that for all n. The nth homology H_{n} 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 induces a morphism from the chain complex of X to the chain complex of Y), then the H_{n} 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 H^{n}) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.
Properties
If () is a chain complex such that all but finitely many A_{n} are zero, and the others are finitely generated abelian groups (or finitedimensional vector spaces), then we can define the Euler characteristic
(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:
and, especially in algebraic topology, this provides two ways to compute the important invariant for the object X which gave rise to the chain complex.
Every short exact sequence
of chain complexes gives rise to a long exact sequence of homology groups
All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps The latter are called connecting homomorphisms and are provided by the zigzag 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 MayerVietoris 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 B^{n} to itself, then there is a fixed point with
 Invariance of domain: If U is an open subset of and is an injective continuous map, then is open and f is a homeomorphism between U and V.
 The Hairy ball theorem: any continuous vector field on the 2sphere (or more generally, the 2ksphere for any ) vanishes at some point.
 The Borsuk–Ulam theorem: any continuous function from an nsphere into Euclidean nspace 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 nonempty open subsets and are homeomorphic, then ^{[10]}
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.^{[11]}
In sensor networks, sensors may communicate information via an adhoc 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.^{[12]}
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 infinitedimensional 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.^{[13]}
In one class of finite element methods, boundaryvalue problems for differential equations involving the HodgeLaplace 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.^{[14]}^{[15]}
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 Archived 20130715 at the Wayback Machine and Perseus are also written in C++. All three implement preprocessing algorithms based on simplehomotopy equivalence and discrete Morse theory to perform homologypreserving 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.^{[14]}
Some nonhomologybased discussions of surfaces
Origins
Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic.^{[16]} This was followed by Riemann's definition of genus and nfold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.^{[17]}
Surfaces
On the ordinary sphere , the curve 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 closed curve such as c can be similarly shrunk to a point. This implies that has trivial fundamental group, so as a consequence, it also has trivial first homology group.
The torus 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:
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 reglued surface must selfintersect (when immersed in Euclidean 3space). 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 reoriented in this way, the surface as a whole is said to be nonorientable.
The projective plane 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 nonshrinkable while c is. If b were only wound once, it would also be nonshrinkable 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 regluing it does not change the surface, so a + (−a) = 0.
But now consider two acycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the bcycle), and it will come back as −a. This is because the Klein bottle is made from a cylinder, whose acycle 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 bcycle around twice in the Klein bottle gives simply b + b = 2b, since this cycle lives in a torsionfree 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 evensided polygons (2ngons) can be glued to make different manifolds. Conversely, a closed surface with n nonzero classes can be cut into a 2ngon. Variations are also possible, for example a hexagon may also be glued to form a torus.^{[18]}
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 nonorientable cycles requires additional information about torsion coefficients.^{[19]}
The complete classification of 1 and 2manifolds is given in the table.
Manifold  Euler no., χ 
Orientability  Betti numbers  Torsion coefficient (1dimensional)  

Symbol^{[18]}  Name  b_{0}  b_{1}  b_{2}  
Circle (1manifold)  0  Orientable  1  1  —  —  
Sphere  2  Orientable  1  0  1  None  
Torus  0  Orientable  1  2  1  None  
Projective plane  1  Nonorientable  1  0  0  2  
Klein bottle  0  Nonorientable  1  1  0  2  
2holed torus  −2  Orientable  1  4  1  None  
gholed torus (g is the genus)  2 − 2g  Orientable  1  2g  1  None  
Sphere with c crosscaps  2 − c  Nonorientable  1  c − 1  0  2  
2Manifold with g holes and c crosscaps (c > 0)  2 − (2g + c)  Nonorientable  1  (2g + c) − 1  0  2 
 Notes
 For a nonorientable surface, a hole is equivalent to two crosscaps.
 Any closed 2manifold can be realised as the connected sum of g tori and c projective planes, where the 2sphere 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.^{[21]}^{[22]} 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.^{[23]}^{[24]}^{[25]} 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 in the special case of surfaces, the torsion part of the homology group only occurs for nonorientable cycles.
The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".^{[26]} Algebraic homology remains the primary method of classifying manifolds.^{[27]}
Notes
See also
 Betti number
 Cycle space
 De Rham cohomology
 Eilenberg–Steenrod axioms
 Extraordinary homology theory
 Homological algebra
 Homological conjectures in commutative algebra
 Homological connectivity
 Homological dimension
 Homotopy group
 Künneth theorem
 List of cohomology theories  also has a list of homology theories
 Poincaré duality
References
 ^ Spanier 1966, p. 155
 ^ ^{a} ^{b} ^{c} Gowers, BarrowGreen & Leader 2010, pp. 390–391
 ^ Wildberger, Norman J. (2012). "More homology computations". YouTube. Archived from the original on 20211211.
 ^ ^{a} ^{b} Hatcher 2002, p. 106
 ^ Wildberger, Norman J. (2012). "Delta complexes, Betti numbers and torsion". YouTube. Archived from the original on 20211211.
 ^ Hatcher 2002, pp. 105–106
 ^ Hatcher 2002, p. 113
 ^ Hatcher 2002, p. 110
 ^ Spanier 1966, p. 156
 ^ Hatcher 2002, p. 126.
 ^ "CompTop overview". Archived from the original on 22 June 2007. Retrieved 16 March 2014.
 ^ "Robert Ghrist: applied topology". Retrieved 16 March 2014.
 ^ 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.
 ^ ^{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.
 ^ 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.
 ^ Stillwell 1993, p. 170
 ^ Weibel 1999, pp. 2–3 (in PDF)
 ^ ^{a} ^{b} Weeks, Jeffrey R. (2001). The Shape of Space. CRC Press. ISBN 9780203912669.
 ^ Richeson 2008, p. 254
 ^ Richeson 2008
 ^ Richeson 2008, p. 258
 ^ Weibel 1999, p. 4
 ^ Hilton 1988, p. 284
 ^ 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.
 ^ Hirzebruch, Friedrich, Emmy Noether and Topology in Teicher 1999, pp. 61–63.
 ^ Bourbaki and Algebraic Topology by John McCleary (PDF) Archived 20080723 at the Wayback Machine gives documentation (translated into English from French originals).
 ^ Richeson 2008, p. 264
Further reading
 Cartan, Henri Paul; Eilenberg, Samuel (1956). Homological Algebra. Princeton mathematical series. Vol. 19. Princeton University Press. ISBN 9780674079779. OCLC 529171.
 Eilenberg, Samuel; Moore, J.C. (1965). Foundations of relative homological algebra. Memoirs of the American Mathematical Society number. Vol. 55. American Mathematical Society. ISBN 9780821812556. OCLC 1361982.
 Gowers, Timothy; BarrowGreen, June; Leader, Imre, eds. (2010), The Princeton Companion to Mathematics, Princeton University Press, ISBN 9781400830398.
 Hatcher, A. (2002), Algebraic Topology, Cambridge University Press, ISBN 0521795400. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
 Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine, 60 (5), Mathematical Association of America: 282–291, doi:10.1080/0025570X.1988.11977391, JSTOR 2689545
 Richeson, D. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University.
 Spanier, Edwin H. (1966), Algebraic Topology, Springer, p. 155, ISBN 0387906460.
 Stillwell, John (1993), "Homology Theory and Abelianization", Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72, Springer, pp. 169–184, doi:10.1007/9781461243724_6, ISBN 9780387979700.
 Teicher, M., ed. (1999), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, BarIlan University/American Mathematical Society/Oxford University Press, ISBN 9780198510451, OCLC 223099225
 Weibel, Charles A. (1999), "28. History of Homological Algebra" (PDF), in James, I. M. (ed.), History of Topology, Elsevier, ISBN 9780080534077.
External links
 Homology group at Encyclopaedia of Mathematics
 [1] N.J. Windberger intro to algebraic topology, last six lectures with an easy intro to homology
 [2] Algebraic topology Allen Hatcher  Chapter 2 on homology