K-theory

Summary

K-theory studies a space, a ring or an operator algebra through the abelian groups built from its vector bundles, projective modules or projections. The first group $K_0$ is the Grothendieck group of the objects under direct sum. There are three main branches: algebraic K-theory of rings, topological K-theory of spaces and operator K-theory of C*-algebras. The theory links algebra, topology, number theory and index theory.