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Stable range condition

In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring is the smallest integer such that whenever in generate the unit ideal (they form a unimodular row), th

Additive K-theory

In mathematics, additive K-theory means some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl. It is n

K-theory of a category

In algebraic K-theory, the K-theory of a category C (usually equipped with some kind of additional data) is a sequence of abelian groups Ki(C) associated to it. If C is an abelian category, there is n

Atiyah–Segal completion theorem

The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem then states that th

KR-theory

In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by , motivated by applications to the Atiyah–Singer index theorem for real ellip

Twisted K-theory

In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and ope

Λ-ring

In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natura

Kuiper's theorem

In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible boun

Goncharov conjecture

In mathematics, the Goncharov conjecture is a conjecture introduced by Goncharov suggesting that the cohomology of certain motivic complexes coincides with pieces of K-groups. It extends a conjecture

Weibel's conjecture

In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by Charles Weibel and proven in full generality by using methods

Bott cannibalistic class

In mathematics, the Bott cannibalistic class, introduced by Raoul Bott, is an element of the representation ring of a compact Lie group that describes the action of the Adams operation on the Thom cla

Birch–Tate conjecture

The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate.

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

Snaith's theorem

In algebraic topology, a branch of mathematics, Snaith's theorem, introduced by , identifies the complex K-theory spectrum with the of the suspension spectrum of away from the Bott element.

Atiyah–Hirzebruch spectral sequence

In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by Michael Atiyah and Friedrich Hirzebruch in the special case of topo

Milnor K-theory

In mathematics, Milnor K-theory is an algebraic invariant (denoted for a field ) defined by John Milnor as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped thi

Circle bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle . Oriented circle bundles are also known as principal U(1)-bundles. In physics, circle bundles are the natural geometric

Assembly map

In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a

Serre–Swan theorem

In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules a

Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group of a ring is the universal central extension of the commutator subgroup of the stable general linear group of . It is named after Rob

Milnor conjecture

In mathematics, the Milnor conjecture was a proposal by John Milnor of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois (o

Spin structure

In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. S

Steinberg symbol

In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg. For

Calkin algebra

In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the

K-theory (physics)

In string theory, K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond–Ramond field stre

Norm variety

In mathematics, a norm variety is a particular type of algebraic variety V over a field F, introduced for the purposes of algebraic K-theory by Voevodsky. The idea is to relate Milnor K-theory of F to

Weyl–von Neumann theorem

In mathematics, the Weyl–von Neumann theorem is a result in operator theory due to Hermann Weyl and John von Neumann. It states that, after the addition of a compact operator or Hilbert–Schmidt operat

Operator K-theory

In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras.

K-homology

In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space.

Karoubi conjecture

In mathematics, the Karoubi conjecture is a conjecture by Max Karoubi that the algebraic and topological K-theories coincide on C* algebras spatially tensored with the algebra of compact operators. It

KK-theory

In mathematics, KK-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician

Farrell–Jones conjecture

In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms. The

Baum–Connes conjecture

In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of p

Grothendieck group

In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense

Topological K-theory

In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were i

Whitehead's lemma

Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form is equivalent to the identity matrix by elementary transformations (that is,

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