Meaning of K-theory | Babel Free

Definitions

1. The study of rings R generated by the set of vector bundles over some topological space or scheme;
   uncountable
2. The study of rings R generated by the set of vector bundles over some topological space or scheme; (dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.
   dated, obsolete, uncountable
3. that part of algebraic topology comprising what is now called topological K-theory.
   dated, obsolete, uncountable
4. The cohomology generated by the set of vector bundles over some topological space or scheme.
   countable

Examples

“K'''-theory as an independent discipline is a fairly new subject, only about 35 years old.”

“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).”

“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.”

“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[…].”

“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.”

CEFR level