The topic of K-theory 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.
The subject takes its name from a particular construction applied by Alexander Grothendieck in his proof of the Riemann-Roch theorem. In it, a commutative monoid of sheaves of abelian groups under direct sum was converted into a group, by the formal addition of inverses (an explicit way of explaining a left adjoint). 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 proof around 1962 of the Index Theorem.
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.
See also Swan's theorem.
K-theory and physics
In string theory, K-theory was 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. 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.
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