List of cohomology theories

This is a list of important ideas called homology and cohomology theories used in a part of math called algebraic topology. These theories help us understand shapes and spaces by turning them into numbers and groups. They work with special kinds of shapes called CW complexes or spectra. Some of these theories are called ordinary, while others are called generalized (or extraordinary). At the end, you can find more links to learn about other types of homology theories.

Notation

This section explains some special symbols used in the study of shapes and spaces. These symbols help mathematicians describe and understand complex structures.

• S represents the sphere spectrum, a basic building block in this area of math.
• Sⁿ refers to the spectrum of an n-dimensional sphere, which is like a circle stretched into higher dimensions.
• [X, Y] shows the group of connections between two spectra, X and Y, which helps describe how they relate to each other.
• πₙ(X) is the n-th stable homotopy group of X, capturing important information about the structure of X.
• X ∧ Y is called the smash product of two spectra, a way to combine them while keeping track of their relationships.

When X is a special kind of structure called a spectrum, it can be used to define two important ideas: generalized homology and generalized cohomology. These help mathematicians study properties of spaces in new and powerful ways.

Ordinary homology theories

Ordinary homology theories follow special rules called the Eilenberg–Steenrod axioms. These theories help us understand shapes by looking at points and how they connect. They use different groups of numbers to measure these connections.

These theories are linked to spaces called Eilenberg–MacLane spaces and work well with simplicial complexes, which are made of triangles and their higher-dimensional versions. There are different types based on the numbers used, such as whole numbers, fractions, real numbers, complex numbers, or numbers mod a prime.

K-theories

The simpler K-theories of a space are linked to vector bundles over that space. Different types of K-theories match different structures added to these vector bundles.

Real K-theory

Spectrum: KO

Coefficient ring: The coefficient groups π_i(KO) repeat every 8 steps, following the pattern Z, Z₂, Z₂, 0, Z, 0, 0, 0.

KO⁰(X) represents the stable equivalence classes of real vector bundles over X. Bott periodicity shows that these groups repeat every 8 steps.

Complex K-theory

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

Coefficient ring: The coefficient ring K^*(point) is made 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 means these groups repeat every 2 steps.

Quaternionic K-theory

Spectrum: KSp

Coefficient ring: The coefficient groups π_i(KSp) repeat every 8 steps, following the pattern Z, 0, 0, 0, Z, Z₂, Z₂, 0.

KSp⁰(X) represents the stable equivalence classes of quaternionic vector bundles over X. Bott periodicity shows these groups repeat every 8 steps.

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 have coefficients added.

Self conjugate K-theory

Spectrum: KSC

Coefficient ring: to be written...

The coefficient groups π_i (KSC) repeat every 4 steps, following the pattern Z, Z₂, 0, Z. This was introduced by Donald W. Anderson in his 1964 University of California, Berkeley Ph.D. dissertation.

Connective K-theories

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

Coefficient ring: For ku, the coefficient ring is made of polynomials over Z using a single class v₁ in dimension 2. For ko, the coefficient ring is a special type of polynomial ring with three generators that follow specific rules.

This version of K-theory removes the parts with negative dimensions.

KR-theory

KR-theory is a cohomology theory for spaces with a special kind of symmetry. Many other K-theories come from it.

Bordism and cobordism theories

Cobordism looks at special shapes called manifolds. A manifold is simple if it can be the edge of another shape. The ways these shapes can be simple form a ring, used in a type of math called generalized cohomology theory. There are many of these theories, each matching different structures on manifolds.

These theories are often shown using things called Thom spaces linked to certain groups.

Stable homotopy and cohomotopy

Spectrum: S (sphere spectrum).

Coefficient ring: The groups π_n(S) are the stable homotopy groups of spheres, which are very hard to figure out for n > 0.

MO is one of the simplest cobordism theories ever found.

Complex cobordism

Main article: Complex cobordism

Spectrum: MU (Thom spectrum of unitary group)

Coefficient ring: π_*(MU) matches what is called Lazard's universal ring, and it relates to shapes called almost complex manifolds.

Oriented cobordism

Spectrum: MSO (Thom spectrum of special orthogonal group)

Coefficient ring: The oriented cobordism of a shape is known from certain numbers linked to it, but the full ring of these numbers is very tricky. Smart people like John Milnor, Boris Averbuch, Vladimir Rokhlin, and C. T. C. Wall figured it out completely.

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 (/wiki/List_of_cohomology_theories#CITEREFAndersonBrownPeterson1967)).

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 idea is like cobordism but uses different kinds of shapes instead of smooth ones. The rings of numbers for these are hard to understand.

Brown–Peterson cohomology

Spectrum: BP

Coefficient ring: π_*(BP) is built from certain numbers linked to a prime p.

Brown–Peterson cohomology BP is part of something called MU_p, which comes from complex cobordism.

Morava K-theory

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 repeat every 2(pⁿ − 1) steps. They are named after Jack Morava.

Johnson–Wilson theory

Spectrum E(n)

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

String cobordism

Spectrum:

Coefficient ring:

Theories related to elliptic curves

Main article: Elliptic cohomology

Spectrum: Ell

Spectra: tmf, TMF (previously called eo₂.)

The coefficient ring π_*(tmf) is called the ring of topological modular forms.