K-theory

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.

=Early history=

The subject takes its name from a particular construction applied by Alexander Grothendieck in his proof of the Grothendieck-Riemann-Roch theorem. In it, a commutative monoid of Sheaf (mathematics) of abelian groups under direct sum was converted into a group, by the formal addition of inverses (an explicit way of explaining a adjoint functor). This construction was taken up by Michael Atiyah and Friedrich Hirzebruch to define

: K ( X )

for a topological space X , by means on 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 Atiyah-Singer index theorem. Furthermore this approach led to a noncommutative topology 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.

= K-theory and physics =

In string theory, K-theory has proved to be a good description of the allowed charges of D-branes. Originally the spectrum of D-brane charges was thought to be described by homology (mathematics). However, the analyses of tachyon condensation (with possible non-trivial gauge fields) by Ashoke Sen has led Edward Witten to conjecture that K-theory is a better mathematical framework, and their construction was confirmed by subsequent research of many other physicists.

=External links=

*Allen Hatcher s book [http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory] is available free in PDF and PostScript formats.

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