Search results
Results from the WOW.Com Content Network
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.In algebraic topology, it is a cohomology theory known as topological K-theory.
The Grothendieck–Riemann–Roch theorem says that these are equal. When Y is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the Grothendieck–Riemann–Roch theorem specializes to Hirzebruch's theorem. The group K(X) is now known as K 0 (X).
The Thom isomorphism theorem in topological K-theory is () ~ (()), where T(E) is the Thom space of the vector bundle E over X. This holds whenever E is a spin-bundle. The Atiyah-Hirzebruch spectral sequence allows computation of K -groups from ordinary cohomology groups.
In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to [] or [,]. The theorem was first proved by Hyman Bass for K 0 , K 1 {\displaystyle K_{0},K_{1}} and was later extended to higher K -groups by Daniel Quillen .
Waldhausen Localization Theorem [2] — Let be the category with cofibrations, equipped with two categories of weak equivalences, () (), such that (,) and (,) are both Waldhausen categories. Assume ( A , w ) {\displaystyle (A,w)} has a cylinder functor satisfying the Cylinder Axiom, and that w ( A ) {\displaystyle w(A)} satisfies the Saturation ...
A theorem of Ravindra Bapat, generalizing Sperner's lemma, [5]: chapter 16, pp. 257–261 implies the KKM lemma extends to connector-free coverings (he proved his theorem for =). The connector-free variant also has a permutation variant, so that both these generalizations can be used simultaneously.
Thus Künneth theorems can not be obtained by the above methods of homological algebra. Nevertheless, Künneth theorems in just the same form have been proved in very many cases by various other methods. The first were Michael Atiyah's Künneth theorem for complex K-theory and Pierre Conner and Edwin E. Floyd's result in cobordism.
Operator K-theory resembles topological K-theory more than algebraic K-theory. In particular, a Bott periodicity theorem holds. So there are only two K-groups, namely K 0, which is equal to algebraic K 0, and K 1. As a consequence of the periodicity theorem, it satisfies excision.