# List of cohomology theories

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This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For other sorts of homology theories see the links at the end of this article.

## Notation

• S = π = S0 is the sphere spectrum.
• Sn is the spectrum of the n-dimensional sphere
• SnY = SnY is the nth suspension of a spectrum Y.
• [X,Y] is the abelian group of morphisms from the spectrum X to the spectrum Y, given (roughly) as homotopy classes of maps.
• [X,Y]n = [SnX,Y]
• [X,Y]* is the graded abelian group given as the sum of the groups [X,Y]n.
• πn(X) = [Sn, X] = [S, X]n is the nth stable homotopy group of X.
• π*(X) is the sum of the groups πn(X), and is called the coefficient ring of X when X is a ring spectrum.
• XY is the smash product of two spectra.

If X is a spectrum, then it defines generalized homology and cohomology theories on the category of spectra as follows.

• Xn(Y) = [S, XY]n = [Sn, XY] is the generalized homology of Y,
• Xn(Y) = [Y, X]n = [SnY, X] is the generalized cohomology of Y

## Ordinary homology theories

These are the theories satisfying the "dimension axiom" of the Eilenberg–Steenrod axioms that the homology of a point vanishes in dimension other than 0. They are determined by an abelian coefficient group G, and denoted by H(X, G) (where G is sometimes omitted, especially if it is Z). Usually G is the integers, the rationals, the reals, the complex numbers, or the integers mod a prime p.

The cohomology functors of ordinary cohomology theories are represented by Eilenberg–MacLane spaces.

On simplicial complexes, these theories coincide with singular homology and cohomology.

### Homology and cohomology with integer coefficients.

Spectrum: H (Eilenberg–MacLane spectrum of the integers.)

Coefficient ring: πn(H) = Z if n = 0, 0 otherwise.

The original homology theory.

### Homology and cohomology with rational (or real or complex) coefficients.

Spectrum: HQ (Eilenberg–Mac Lane spectrum of the rationals.)

Coefficient ring: πn(HQ) = Q if n = 0, 0 otherwise.

These are the easiest of all homology theories. The homology groups HQn(X) are often denoted by Hn(X, Q). The homology groups H(X, Q), H(X, R), H(X, C) with rational, real, and complex coefficients are all similar, and are used mainly when torsion is not of interest (or too complicated to work out). The Hodge decomposition writes the complex cohomology of a complex projective variety as a sum of sheaf cohomology groups.

### Homology and cohomology with mod p coefficients.

Spectrum: HZp (Eilenberg–Maclane spectrum of the integers mod p.)

Coefficient ring: πn(HZp) = Zp (Integers mod p) if n = 0, 0 otherwise.

## K-theories

The simpler K-theories of a space are often related to vector bundles over the space, and different sorts of K-theories correspond to different structures that can be put on a vector bundle.

### Real K-theory

Spectrum: KO

Coefficient ring: The coefficient groups πi(KO) have period 8 in i, given by the sequence Z, Z2, Z2,0, Z, 0, 0, 0, repeated. As a ring, it is generated by a class η in degree 1, a class x4 in degree 4, and an invertible class v14 in degree 8, subject to the relations that 2η = η3 = ηx4 = 0, and x42 = 4v14.

KO0(X) is the ring of stable equivalence classes of real vector bundles over X. Bott periodicity implies that the K-groups have period 8.

### Complex K-theory

Spectrum: KU (even terms BU or Z × BU, odd terms U).

Coefficient ring: The coefficient ring K*(point) is the ring of Laurent polynomials in a generator of degree 2.

K0(X) is the ring of stable equivalence classes of complex vector bundles over X. Bott periodicity implies that the K-groups have period 2.

### Quaternionic K-theory

Spectrum: KSp

Coefficient ring: The coefficient groups πi(KSp) have period 8 in i, given by the sequence Z, 0, 0, 0,Z, Z2, Z2,0, repeated.

KSp0(X) is the ring of stable equivalence classes of quaternionic vector bundles over X. Bott periodicity implies that the K-groups have period 8.

### K theory with coefficients

Spectrum: KG

G is some abelian group; for example the localization Z(p) at the prime p. Other K-theories can also be given coefficients.

### Self conjugate K-theory

Spectrum: KSC

Coefficient ring:to be written...

The coefficient groups $\pi _{i}$ (KSC) have period 4 in i, given by the sequence Z, Z2, 0, Z, repeated. Introduced by Donald W. Anderson in his unpublished 1964 University of California, Berkeley Ph.D. dissertation, "A new cohomology theory".

### Connective K-theories

Spectrum: ku for connective K-theory, ko for connective real K-theory.

Coefficient ring: For ku, the coefficient ring is the ring of polynomials over Z on a single class v1 in dimension 2. For ko, the coefficient ring is the quotient of a polynomial ring on three generators, η in dimension 1, x4 in dimension 4, and v14 in dimension 8, the periodicity generator, modulo the relations that 2η = 0, x42 = 4v14, η3 = 0, and ηx = 0.

Roughly speaking, this is K-theory with the negative dimensional parts killed off.

### KR-theory

This is a cohomology theory defined for spaces with involution, from which many of the other K-theories can be derived.

## Bordism and cobordism theories

Cobordism studies manifolds, where a manifold is regarded as "trivial" if it is the boundary of another compact manifold. The cobordism classes of manifolds form a ring that is usually the coefficient ring of some generalized cohomology theory. There are many such theories, corresponding roughly to the different structures that one can put on a manifold.

The functors of cobordism theories are often represented by Thom spaces of certain groups.

### Stable homotopy and cohomotopy

Spectrum: S (sphere spectrum).

Coefficient ring: The coefficient groups πn(S) are the stable homotopy groups of spheres, which are notoriously hard to compute or understand for n > 0. (For n < 0 they vanish, and for n = 0 the group is Z.)

Stable homotopy is closely related to cobordism of framed manifolds (manifolds with a trivialization of the normal bundle).

### Unoriented cobordism

Spectrum: MO (Thom spectrum of orthogonal group)

Coefficient ring: π*(MO) is the ring of cobordism classes of unoriented manifolds, and is a polynomial ring over the field with 2 elements on generators of degree i for every i not of the form 2n−1. That is: $\mathbb {Z} _{2}[x_{2},x_{4},x_{5},x_{6},x_{8}\cdots ]$ where $x_{2n}$ can be represented by the classes of $\mathbb {RP} ^{2n}$ while for odd indices one can use appropriate Dold manifolds.

Unoriented bordism is 2-torsion, since 2M is the boundary of $M\times I$ .

MO is a rather weak cobordism theory, as the spectrum MO is isomorphic to H(π*(MO)) ("homology with coefficients in π*(MO)") – MO is a product of Eilenberg–MacLane spectra. In other words, the corresponding homology and cohomology theories are no more powerful than homology and cohomology with coefficients in Z/2Z. This was the first cobordism theory to be described completely.

### Complex cobordism

Spectrum: MU (Thom spectrum of unitary group)

Coefficient ring: π*(MU) is the polynomial ring on generators of degree 2, 4, 6, 8, ... and is naturally isomorphic to Lazard's universal ring, and is the cobordism ring of stably almost complex manifolds.

### Oriented cobordism

Spectrum: MSO (Thom spectrum of special orthogonal group)

Coefficient ring: The oriented cobordism class of a manifold is completely determined by its characteristic numbers: its Stiefel–Whitney numbers and Pontryagin numbers, but the overall coefficient ring, denoted $\Omega _{*}=\Omega (*)=MSO(*)$ is quite complicated. Rationally, and at 2 (corresponding to Pontryagin and Stiefel–Whitney classes, respectively), MSO is a product of Eilenberg–MacLane spectra$MSO_{\mathbf {Q} }=H(\pi _{*}(MSO_{\mathbf {Q} }))$ and $MSO=H(\pi _{*}(MSO))$ – but at odd primes it is not, and the structure is complicated to describe. The ring has been completely described integrally, due to work of John Milnor, Boris Averbuch, Vladimir Rokhlin, and C. T. C. Wall.

### Special unitary cobordism

Spectrum: MSU (Thom spectrum of special unitary group)

Coefficient ring:

### Spin cobordism (and variants)

Spectrum: MSpin (Thom spectrum of spin group)

Coefficient ring: See (D. W.Anderson,E. H. Brown&F. P. Peterson  1967 ).

### Symplectic cobordism

Spectrum: MSp (Thom spectrum of symplectic group)

Coefficient ring:

### PL cobordism and topological cobordism

Spectrum: MPL, MSPL, MTop, MSTop

Coefficient ring:

The definition is similar to cobordism, except that one uses piecewise linear or topological instead of smooth manifolds, either oriented or unoriented. The coefficient rings are complicated.

### Brown–Peterson cohomology

Spectrum: BP

Coefficient ring: π*(BP) is a polynomial algebra over Z(p) on generators vn of dimension 2(pn  1) for n  1.

Brown–Peterson cohomology BP is a summand of MUp, which is complex cobordism MU localized at a prime p. In fact MU(p) is a sum of suspensions of BP.

### Morava K-theory

Spectrum: K(n) (They also depend on a prime p.)

Coefficient ring:Fp[vn, vn−1], where vn has degree 2(pn -1).

These theories have period 2(pn  1). They are named after Jack Morava.

### Johnson–Wilson theory

SpectrumE(n)

Coefficient ringZ(2)[v1, ..., vn, 1/vn] where vi has degree 2(2i1)

### String cobordism

Spectrum:

Coefficient ring:

Spectrum: Ell

### Topological modular forms

Spectra: tmf, TMF (previously called eo2.)

The coefficient ring π*(tmf) is called the ring of topological modular forms. TMF is tmf with the 24th power of the modular form Δ inverted, and has period 242=576. At the prime p = 2, the completion of tmf is the spectrum eo2, and the K(2)-localization of tmf is the Hopkins-Miller Higher Real K-theory spectrum EO2.

## Related Research Articles

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces. In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold, then the kth cohomology group of M is isomorphic to the th homology group of M, for all integers k

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different categories of spectra, but they all determine the same homotopy category, known as the stable homotopy category.

In mathematics, and algebraic topology in particular, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

In mathematics, the Thom space,Thom complex, or Pontryagin–Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.

In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.

In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.

In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958). Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Originally developed for differentiable manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson (1966), depending on a choice of prime p. It is described in detail by Douglas Ravenel . Its representing spectrum is denoted by BP.

In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by Dennis Sullivan (1977) and Daniel Quillen (1969). This simplification of homotopy theory makes calculations much easier.

In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X, a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information. If the dimension of X is 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov.

In mathematics, and particularly homology theory, Steenrod's Problem is a problem concerning the realisation of homology classes by singular manifolds.

In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.

This is a glossary of properties and concepts in algebraic topology in mathematics.

• Stable Homotopy and Generalised Homology (Chicago Lectures in Mathematics) by J. Frank Adams, University Of Chicago Press; Reissue edition (February 27, 1995) ISBN   0-226-00524-0
• Anderson, Donald W.; Brown, Edgar H. Jr.; Peterson, Franklin P. (1967), "The Structure of the Spin Cobordism Ring", Annals of Mathematics , Second Series, 86 (2): 271–298, doi:10.2307/1970690, JSTOR   1970690
• Notes on cobordism theory, by Robert E. Stong, Princeton University Press (1968) ASIN B0006C2BN6
• Elliptic Cohomology (University Series in Mathematics) by Charles B. Thomas, Springer; 1 edition (October, 1999) ISBN   0-306-46097-1