The Atiyah–Segal completion theorem is a

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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 group, ...

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 ...

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

. 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

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

CW-complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...

. 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 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 represent ...

of ''G''. In the special case of ''X'' being 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 ...

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

action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...

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 In abstract algebra, a finite group is a group (mathematics), group whose underlying set is finite set, finite. Finite groups often arise when considering symmetry of Symmetry in mathematics, mathematical or Symmetry (physics), physical objects, ...

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 the ...

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

References

K-theory Theorems in topology {{topology-stub

See also

References