Mathematics Algebra Cohomology theories view content

List of cohomology theories

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description: Notation S = π = S0 is the sphere spectrum.Sn is the spectrum of the n-dimensional sphereSnY = Sn∧Y is the nth suspension of a spectrum Y. is the abelian group of morphisms from the spectrum X to th ...

S = π = S0 is the sphere spectrum.
Sn is the spectrum of the n-dimensional sphere
SnY = Sn∧Y 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.
X∧Y 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, X∧Y]n = [Sn, X∧Y] is the generalized homology of Y,
Xn(Y) = [Y, X]−n = [S−nY, 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.


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 πi(KSC) have period 4 in i, given by the sequence Z, Z2, 0, Z, repeated. Introduced by D. Anderson in his unpublished 1964 Berkeley PhD 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.

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.

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
Main article: 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
[icon]    This section requires expansion. (December 2009)
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[2] = H(\pi_*(MSO[2])) – 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 Milnor, Averbuch, 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:

Clifford algebra cobordism
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
Spectrum E(n)

Coefficient ring Z(2)[v1, ..., vn, 1/vn] where vi has degree 2(2i−1)

String cobordism

Coefficient ring:

Theories related to elliptic curves

Elliptic cohomology
Spectrum: Ell

[icon]    This section requires expansion. (December 2009)
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.

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