Editing K-theory (section)

{{short description|Branch of mathematics}}
{{for|the hip hop group|K Theory}}
{{DISPLAYTITLE:''K''-theory}}
In [[mathematics]], '''K-theory''' is, roughly speaking, the study of a [[Ring (mathematics)|ring]] generated by [[vector bundles]] over a [[topological space]] or [[scheme (mathematics)|scheme]]. In [[algebraic topology]], it is a [[cohomology theory]] known as [[topological K-theory]]. In [[algebra]] and [[algebraic geometry]], it is referred to as [[algebraic K-theory]]. It is also a fundamental tool in the field of [[operator algebra]]s. It can be seen as the study of certain kinds of [[Invariant (mathematics)|invariants]] of large [[Matrix (mathematics)|matrices]].<ref>{{cite arXiv |last1=Atiyah |first1=Michael |author1-link=Michael Atiyah |year=2000 |title=K-Theory Past and Present |eprint=math/0012213}}</ref>

K-theory involves the construction of families of ''K''-[[functor]]s that map from topological spaces or schemes, or to be even more general: any object of a homotopy category to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to [[group (mathematics)|group]]s in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the [[Grothendieck–Riemann–Roch theorem]], [[Bott periodicity]], the [[Atiyah–Singer index theorem]], and the [[Adams operation]]s.

In [[high energy 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 structure|generalized complex manifolds]]. In [[condensed matter physics]] K-theory has been used to classify [[topological insulator]]s, [[superconductor]]s and stable [[Fermi surface]]s. For more details, see [[K-theory (physics)]].