K-theory

In mathematics, K-theory is a tool used in several disciplines. 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 was originally discovered by Alexander Grothendieck (1957) so that he could formulate his Grothendieck-Riemann-Roch theorem. It takes its name from the German "Klasse", meaning "class" [http://arxiv.org/abs/math/0602082] . Grothendieck needed to work with sheaves on an algebraic variety "X". Rather than working directly with thesheaves, he gave two constructions. In the first, he used the operation of direct sum to convert the commutative monoid of
sheaves into a group "K(X)" by taking formal sums of classes of sheaves and formally adding inverses. (This is an explicit way of obtaining a left adjoint to a certain functor.) In the second construction, he imposed additional relations corresponding to extensions of sheaves to obtain a group now written as "G(X)". 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.

In topology, one has an analogous sum construction for vector bundles. Michael Atiyah and Friedrich Hirzebruch used the Grothendieck group construction to define "K(X)" for a topological space "X" in 1959 (the two constructions agree). This was the basis of the first extraordinary cohomology theory discovered in algebraic topology. It played a big 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 projective modules over the ring of polynomials over a field are free modules; this assertion is correct, but 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 show that projective modules are stably free. This application was the beginning of algebraic K-theory.

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 byFriedhelm Waldhausen in order to study the "algebraic K-theory of spaces," which is related to the study of pseudo-isotopies. Most modern research on higher K-theory is related to algebraic geometry and the study of motivic cohomology.

L-theory. The corresponding constructions involving an auxiliary quadratic form receive 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 by [http://string.lpthe.jussieu.fr/members.pl?key=7 Ruben Minasian] and Gregory Moore [http://www.physics.rutgers.edu/~gmoore] in [http://xxx.lanl.gov/abs/hep-th/9710230 K-theory and Ramond-Ramond Charge] . More details can be found at K-theory (physics).

ee also

*List of cohomology theories
*K-theory (physics)
*L-theory
*Bott periodicity

References

* | year=1989("Introductory lectures given at Harvard by Atiyah, published from notes taken by D. W. Anderson. Starts by defining vector bundles, assumes little advanced math.").
* Max Karoubi, [http://www.institut.math.jussieu.fr/~karoubi/KBook.html K-theory, an introduction] (1978) Springer-Verlag
* Allen Hatcher, " [http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory] ", (2003)
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* [http://math.jussieu.fr/~karoubi/ Max Karoubi's Page]