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.
is a spectrum, then it defines generalized homology and cohomology theories on the category of spectra as follows: 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.
On simplicial complexes, these theories coincide with singular homology and cohomology.
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.
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.
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.
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.
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.
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.
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.
Spectrum: KSC Coefficient ring: to be written...
Introduced by Donald W. Anderson in his unpublished 1964 University of California, Berkeley Ph.D. dissertation, "A new cohomology theory".
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.
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.
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.
Stable homotopy is closely related to cobordism of framed manifolds (manifolds with a trivialization of the normal bundle).
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.
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.
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
Rationally, and at 2 (corresponding to Pontryagin and Stiefel–Whitney classes, respectively), MSO is a product of Eilenberg–MacLane spectra –
– 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.
Spectrum: MSp (Thom spectrum of symplectic group) Coefficient ring: 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.
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.