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 = 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.
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 (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".