K-theory
From Wikipedia, the free encyclopedia
In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. It also includes algebraic K-theory. It spans the subjects of algebraic topology, abstract algebra and some areas of application like operator algebras and algebraic geometry. 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).
[edit] Early history
The subject was originally discovered by Alexander Grothendieck so that he could formulate his Grothendieck-Riemann-Roch theorem. It takes its name from the German "Klassen", meaning "class" [1]. Grothendieck needed to convert the commutative monoid of sheaves with the operation of direct sum into a group. Instead of attempting to work with the sheaves directly, he took formal sums of certain classes of sheaves and formally added inverses. (This is an explicit way of obtaining a left adjoint to a certain functor.) This construction, now called the Grothendieck group, was taken up by Michael Atiyah and Friedrich Hirzebruch to define
- K(X)
for a topological space X, by means of the analogous sum construction for vector bundles. This was the basis of the first of the extraordinary cohomology theories of algebraic topology. It played a big role in the second proof around 1962 of the Index Theorem. Furthermore this approach led to a noncommutative K-theory for C*-algebras.
In turn, Jean-Pierre Serre used the analogy of vector bundles with projective modules to found in 1959 what became algebraic K-theory. He formulated Serre's conjecture, that projective modules over the ring of polynomials over a field are free modules; this resisted proof for 20 years.
There followed a period in which there were various partial definitions of higher K-functors; until a comprehensive definition was given by Daniel Quillen using homotopy theory.
The corresponding constructions involving an auxiliary quadratic form receive the general name L-theory. It is a major tool of surgery theory.
See also Swan's theorem.
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 Ruben Minasian and Gregory Moore in K-theory and Ramond-Ramond Charge. More details can be found at K-theory (physics).
[edit] See also
[edit] References
- M. F. Atiyah, K-Theory, (1967) W.A. Benjamin, Inc. New York. (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, "K-theory", an introduction (1978) Springer-Verlag
- Allen Hatcher, Vector Bundles & K-Theory, (2003)
- K-theory on PlanetMath
- Examples of K-theory groups on PlanetMath
- Algebraic K-theory on PlanetMath
- Examples of algebraic K-theory groups on PlanetMath
- Fredholm module on PlanetMath
- K-homology on PlanetMath
- Max Karoubi's Page
