# K-theory

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 Atiyah and 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 made **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.