History
Alexander Grothendieck discovered K-theory in the mid-1950s as a framework to state his far-reaching generalization of the Riemann-Roch theorem. Within a few years, its topological counterpart was considered by Michael Atiyah and Hirzebruch and is now known as topological K-theory.
Applications of K-groups were found from 1960 onwards in surgery theory for manifolds, in particular; and numerous other connections with classical algebraic problems were brought out.
A little later a branch of the theory for operator algebras was fruitfully developed, resulting in operator K-theory and KK-theory. It also became clear that K-theory could play a role in algebraic cycle theory in algebraic geometry (Gersten's conjecture): here the higher K-groups become connected with the higher codimension phenomena, which are exactly those that are harder to access. The problem was that the definitions were lacking (or, too many and not obviously consistent). Using Robert Steinberg's work on universal central extensions of classical algebraic groups, John Milnor defined the group K2(A) of a ring A as the center, isomorphic to H2(E(A),Z), of the universal central extension of the group E(A) of infinite elementary matrices over A. (Definitions below.) There is a natural bilinear pairing from K1(A) × K1(A) to K2(A). In the special case of a field k, with K1(k) isomorphic to the multiplicative group GL(1,k), computations of Hideya Matsumoto showed that K2(k) is isomorphic to the group generated by K1(A) × K1(A) modulo an easily described set of relations.
Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by Quillen (1973, 1974), who gave several definitions of Kn(A) for arbitrary non-negative n, via the +-construction and the Q-construction.
Read more about this topic: Algebraic K-theory
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