The Atiyah–Segal completion theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

in mathematics about

equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...

in homotopy theory. Let ''G'' be a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Lie group and let ''X'' be a ''G''-

CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...

. The theorem then states that the projection map :

\pi\colon X\times EG\to X

induces an isomorphism of prorings :

\pi^*\colon K_G^*(X)_ \to K^*((X\times EG)/G).

Here, the induced map has as domain the completion of the ''G''-equivariant K-theory of ''X'' with respect to ''I'', where ''I'' denotes the

augmentation ideal In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If ''G'' is a group and ''R'' a commutative ring, there is a ring homomorphism \varepsilon, called the augmentation map, from the group ring R /math> to R, define ...

of the

representation ring In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear represen ...

of ''G''. In the special case of ''X'' a point, the theorem specializes to give an isomorphism

K^*(BG)\cong R(G)_

between the K-theory of the

classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...

of ''G'' and the completion of the representation ring. The theorem can be interpreted as giving a comparison between the geometrical process of taking the homotopy quotient of a ''G''-space, by making the

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free before passing to the quotient, and the algebraic process of completing with respect to an ideal. The theorem was first proved for

finite groups Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...

in 1961, and a proof of the general case was published by Atiyah together with Graeme Segal in 1969. Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups. The corresponding statement for algebraic K-theory was proven by

Alexander Merkurjev Aleksandr Sergeyevich Merkurjev (russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev ...

, holding in the case that the group is algebraic over the complex numbers.

References

K-theory Theorems in topology {{topology-stub

See also

References