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In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It also has some applications in operator algebras. It leads to the construction of families of K-functors, which contain useful but often hard-to-compute information.

In physics, K-theory and in particular twisted K-theory have appeared in Type II string theory 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).

Early history

The subject can be said to begin with Alexander Grothendieck (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 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 K(X) when only locally free sheaves are used, or G(X) when all coherent sheaves. Either of these two constructions is referred to as the Grothendieck group; K(X) has cohomological behavior and G(X) has homological behavior.

If X 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 bundles, Michael Atiyah and Friedrich Hirzebruch defined K(X) for a topological space X in 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory. It played a major role in the second proof of the Index Theorem (circa 1962). Furthermore this approach led to a noncommutative K-theory for C*-algebras.

Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is free; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.) In 1959, Serre formed the Grothendieck group 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.
Developments

The other historical origin of algebraic K-theory was the work of Whitehead and others on what later became known as Whitehead torsion.

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 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.

The corresponding constructions involving an auxiliary quadratic form received the general name L-theory. It is a major tool of surgery theory.

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

Algebraic K-theory
Topological K-theory
List of cohomology theories
K-theory (physics)
Operator K-theory
KK-theory
L-theory
Bott periodicity

Notes

^ http://arxiv.org/abs/math/0602082
^ by Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and Gregory Moore (http://www.physics.rutgers.edu/~gmoore) in K-theory and Ramond–Ramond Charge.

References

Atiyah, Michael Francis (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0-201-09394-0, MR1043170
Friedlander, Eric; Grayson, Daniel, eds. (2005), Handbook of K-Theory, Berlin, New York: Springer-Verlag, ISBN 978-3-540-30436-4, MR2182598
Swan, R. G. (1968), Algebraic K-Theory, Lecture Notes in Mathematics No. 76, Springer
Max Karoubi, K-theory, an introduction (1978) Springer-Verlag
Allen Hatcher, Vector Bundles & K-Theory, (2003)