K-theory
246748
221443052
2008-06-24T14:50:34Z
Vyznev Xnebara
1119743
Fixing temporary "arxiv.org/PS_cache" and obsolete "arxiv.org/ftp" URLs to link to abstract page with download links instead (with [[User:Ilmari Karonen/fixarxivlinks.js|script assistance]])
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 algebra]]s. It leads to the construction of families of ''K''-[[functor]]s, 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|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 [[sheaf (mathematics)|sheaves]] on an algebraic variety ''X''. Rather than working directly with the
sheaves, he gave two constructions. In the first, he used the operation of direct sum to convert the commutative [[monoid]] of
[[sheaf (mathematics)|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 [[adjoint functor|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 bundle]]s. [[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 [[Atiyah-Singer index theorem|Index Theorem]] (circa 1962). Furthermore this approach led to a [[noncommutative topology|noncommutative]] ''K''-theory for [[C*-algebra]]s.
Already in 1955, [[Jean-Pierre Serre]] had used the analogy of [[vector bundle]]s with [[projective module]]s to formulate [[Quillen-Suslin theorem|Serre's conjecture]], which states that projective modules over the ring of [[polynomial]]s over a field are [[free module]]s; 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 by
Friedhelm 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)]].
==See also==
*[[List of cohomology theories]]
*[[K-theory (physics)]]
*[[L-theory]]
*[[Bott periodicity]]
==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, [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)
* {{planetmath reference|id=3338|title=K-theory}}
* {{planetmath reference|id=4049|title=Examples of K-theory groups}}
* {{planetmath reference|id=4117|title=Algebraic K-theory}}
* {{planetmath reference|id=4118|title=Examples of algebraic K-theory groups}}
* {{planetmath reference|id=3329|title=Fredholm module}}
* {{planetmath reference|id=3330|title=K-homology}}
* [http://math.jussieu.fr/~karoubi/ Max Karoubi's Page]
[[Category:Algebra]]
[[Category:Algebraic topology]]
[[Category:K-theory|*]]
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