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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.
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 ...
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.Geometric, algebraic, and arithmetic objects are assigned objects called K-groups.
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.
Coefficient ring: The coefficient ring K * (point) is the ring of Laurent polynomials in a generator of degree 2. K 0 (X) is the ring of stable equivalence classes of complex vector bundles over X. Bott periodicity implies that the K-groups have period 2.
The theory K(0) agrees with singular homology with rational coefficients, whereas K(1) is a summand of mod-p complex K-theory. The theory K(n) has coefficient ring F p [v n,v n −1] where v n has degree 2(p n − 1). In particular, Morava K-theory is periodic with this period, in much the same way that complex K-theory has period 2.
Adams operations ψ k on K theory (algebraic or topological) are characterized by the following properties. 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 ...
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.