K-theory

K-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix

The theory of formal groups has found a number of rather spectacular applications in recent years in number theory, arithmetical algebraic geometry, algebraic geometry, and algebraic topology, ranging from congruences for the coefficients of modular forms and local class field theory to extraordinary K-theories and (indirectly) results on the homotopy groups of spheres.

1978, Michiel Hazewinkel, Formal Groups and Applications, Harcourt Brace Jovanovich (Academic Press), page xi

This has changed in recent years: on the one hand, bivariant K-theories were defined by the author for other categories of algebras [Doc. Math. 2 (1997), 139–182 (electronic); MR1456322 (98h: 19006)]; on the other hand, the local cyclic homology theory by M. Puschnigg works for small algebras and C^*-algebras alike[…].

2007, Mathematical Reviews, American Mathematical Society, page 4338

In particular, the Morava K-theory K(1) appears as a direct summand of 'complex K-theory modulo p'. However, the higher Morava K-theories #92;left#92;#123;K(n)#92;right#92;#125;#95;#123;n#92;ge 2#125; are much more mysterious from a geometric point of view. REMARK 5.8. For a fixed prime number p, the Morava K-theories K(n) are defined for 0lt;nlt;#92;infty.

2014, Stephan Stolz, Topology and Field Theories, American Mathematical Society, page 102