List of cohomology theories

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

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

The original homology theory.

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

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

These are the easiest of all homology theories. The homology groups HQ_n(X) are often denoted by H_n(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.

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

Coefficient ring: π_n(HZ_p) = Z_p (Integers mod p) if n = 0, 0 otherwise.

Spectrum: KO

Coefficient ring: The coefficient groups π_i(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.

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.

Spectrum: KSp

Coefficient ring: The coefficient groups π_i(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...

The coefficient groups ${\displaystyle \pi _{i}}$(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 Introduced by Donald W. Anderson in his unpublished 1964 University of California, Berkeley Ph.D. dissertation, "A new cohomology theory".

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.

Spectrum: S (sphere spectrum).

Coefficient ring: The coefficient groups π_n(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).

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: ${\displaystyle \mathbb {Z} _{2}[x_{2},x_{4},x_{5},x_{6},x_{8}\cdots ]}$ where ${\displaystyle x_{2n}}$ can be represented by the classes of ${\displaystyle \mathbb {RP} ^{2n}}$ while for odd indices one can use appropriate Dold manifolds.

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

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.

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 ${\displaystyle \Omega _{*}=\Omega (*)=MSO(*)}$ is quite complicated. Rationally, and at 2 (corresponding to Pontryagin and Stiefel–Whitney classes, respectively), MSO is a product of Eilenberg–MacLane spectra – ${\displaystyle MSO_{\mathbf {Q} }=H(\pi _{*}(MSO_{\mathbf {Q} }))}$ and ${\displaystyle MSO[2]=H(\pi _{*}(MSO[2]))}$ – 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 John Milnor, Boris Averbuch, Vladimir Rokhlin, and C. T. C. Wall.

Spectrum: MSU (Thom spectrum of special unitary group)

Coefficient ring:

Spectrum: MSpin (Thom spectrum of spin group)

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

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. The coefficient rings are complicated.

Spectrum: BP

Coefficient ring: π_*(BP) is a polynomial algebra over Z_(p) on generators v_n of dimension 2(pⁿ − 1) for n ≥ 1.

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

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

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

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

Spectrum E(n)

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

Spectrum:

Coefficient ring:

Spectrum: Ell

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.