K-theory

Noun

K-theory (uncountable)

1. The study of rings R generated by the set of vector bundles over some topological space or scheme;
   • K-theory as an independent discipline is a fairly new subject, only about 35 years old.1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1
   • 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...2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix
2. (dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.
3. (countable) The cohomology generated by the set of vector bundles over some topological space or scheme.
   • 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...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...2007, Mathematical Reviews, American Mathematical Society, page 4338