K-theory

# K-theory

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, it is an extraordinary cohomology theory known as topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...

. In algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, it is referred to as algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

. It also has some applications in operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...

s. It leads to the construction of families of K-functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

s, which contain useful but often hard-to-compute information.

In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, K-theory and in particular twisted K-theory
Twisted K-theory
In mathematics, twisted K-theory is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory....

have appeared in Type II string theory
Type II string theory
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. These account for two of the five consistent superstring theories in ten dimensions. Both theories have the maximal amount of supersymmetry — namely 32 supercharges...

where it has been conjectured that they classify D-branes, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. For details, see also K-theory (physics)
K-theory (physics)
In string theory, the K-theory classification refers to a conjectured application of K-theory to superstrings, to classify the allowed Ramond-Ramond field strengths as well as the charges of stable D-branes....

.

## Early history

The subject can be said to begin with Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

(1957), who used it to formulate his Grothendieck–Riemann–Roch theorem. It takes its name from the German "Klasse", meaning "class". Grothendieck needed to work with coherent sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

on an algebraic variety X. Rather than working directly with the sheaves, he defined a group using (isomorphism classes of) sheaves as generators, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called when only locally free sheaves are used, or when all coherent sheaves. Either of these two constructions is referred to as the Grothendieck group
Grothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way...

; has cohomological behavior and has homological behavior.

If is a smooth variety, the two groups are the same. If it is a smooth affine variety, then all extensions of locally free sheaves split, so group has an alternative definition.

In topology, by applying the same construction to vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

s, Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

and Friedrich Hirzebruch
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...

defined for a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

in 1959, and using the Bott periodicity theorem
Bott periodicity theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy...

they made it the basis of an extraordinary cohomology theory. It played a major role in the second proof of the Index Theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...

(circa 1962). Furthermore this approach led to a noncommutative
Noncommutative topology
Noncommutative topology in mathematics is a term applied to the strictly C*-algebraic part of the noncommutative geometry program. The program has its origins in the Gel'fand duality between the topology of locally compact spaces and the algebraic structure of commutative C*-algebras.Several...

-theory for C*-algebras.

Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

had used the analogy of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

s with projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...

s to formulate Serre's conjecture
Quillen–Suslin theorem
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings...

, which states that every finitely generated projective module over a polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

is free
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...

; this assertion is correct, but was not settled until 20 years later. (Swan's theorem
Swan's 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 and gives rise to a common intuition throughout mathematics: "projective modules over commutative...

is another aspect of this analogy.) In 1959, Serre formed the Grothendieck group
Grothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way...

construction for rings, and used it to prove a weak form of the conjecture. This application was one of the beginnings of algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

.

## Developments

The other historical origin of algebraic K-theory was the work of Whitehead and others on what later became known as Whitehead torsion
In geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...

.

There followed a period in which there were various partial definitions of higher K-theory functors. Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972. A variant was also given by
Friedhelm Waldhausen
Friedhelm Waldhausen
Friedhelm Waldhausen is a German mathematician known for his work in algebraic topology.-Academic life:...

in order to study the algebraic K-theory of spaces, which is related to the study of pseudo-isotopies. Much modern research on higher K-theory is related to algebraic geometry and the study of motivic cohomology
Motivic cohomology
Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...

.

The corresponding constructions involving an auxiliary quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

L-theory
Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...

. It is a major tool of surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...

.

In string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

the K-theory classification of Ramond–Ramond field strengths and the charges of stable D-branes was first proposed in 1997.

• Algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

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

• List of cohomology theories
• K-theory (physics)
K-theory (physics)
In string theory, the K-theory classification refers to a conjectured application of K-theory to superstrings, to classify the allowed Ramond-Ramond field strengths as well as the charges of stable D-branes....

• Operator K-theory
Operator K-theory
In mathematics, operator K-theory is a variant of K-theory on the category of Banach algebras ....

• KK-theory
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...

• L-theory
L-theory
Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...

• Bott periodicity