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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. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator ...
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch.
Topological K-theory was one of the first examples of an extraordinary cohomology theory: It associates to each topological space X (satisfying some mild technical constraints) a sequence of groups K n (X) which satisfy all the Eilenberg–Steenrod axioms except the normalization axiom. The setting of algebraic varieties, however, is much more ...
This is precisely the usual construction of topological K-theory. Thus the gauge bundles on stacks of D9's and anti-D9's are classified by topological K-theory. If Sen's conjecture is right, all D-brane configurations in type IIB are then classified by K-theory. Petr Horava has extended this conjecture to type IIA using D8-branes.
There are many generalized (or "extraordinary") homology and cohomology theories for topological spaces. K-theory and cobordism are the best-known. Unlike ordinary homology and cohomology, they typically cannot be defined using chain complexes. Thus Künneth theorems can not be obtained by the above methods of homological algebra.
In topology, a topological space is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other.
ψ k are ring homomorphisms. ψ k (l)= l k if l is the class of a line bundle. ψ k are functorial. The fundamental idea is that for a vector bundle V on a topological space X, there is an analogy between Adams operators and exterior powers, in which ψ k (V) is to Λ k (V) as the power sum Σ α k is to the k-th elementary symmetric function σ k
In mathematics, the Grothendieck group, or group of differences, [1] of a commutative monoid M is a certain abelian group.This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M.