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.

Definitions

Let X be a compact Hausdorff space and k= \R or \Complex. Then K_k(X) is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional k-vector bundles over X under Whitney sum. Tensor product of bundles gives K-theory a commutative ring structure. Without subscripts, K(X) usually denotes complex K-theory whereas real K-theory is sometimes written as KO(X). The remaining discussion is focused on complex K-theory. As a first example, note that the K-theory of a point is the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natural numbers is the integers. There is also a reduced version of K-theory,, defined for X a compact pointed space (cf. reduced homology). This reduced theory is intuitively K(X) modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles E and F are said to be stably isomorphic if there are trivial bundles and, so that. This equivalence relation results in a group since every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, can be defined as the kernel of the map induced by the inclusion of the base point x0 into X. K-theory forms a multiplicative (generalized) cohomology theory as follows. The short exact sequence of a pair of pointed spaces (X, A) extends to a long exact sequence Let Sn be the n-th reduced suspension of a space and then define Negative indices are chosen so that the coboundary maps increase dimension. It is often useful to have an unreduced version of these groups, simply by defining: Here X_+ is X with a disjoint basepoint labeled '+' adjoined. Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

Properties

[, ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: Similarly, For real K-theory use BO . T(E) is the Thom space of the vector bundle E over X. This holds whenever E is a spin-bundle.

Bott periodicity

The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way: In real K-theory there is a similar periodicity, but modulo 8.

Applications

Topological K-theory has been applied in John Frank Adams’ proof of the “Hopf invariant one” problem via Adams operations. Adams also proved an upper bound for the number of linearly-independent vector fields on spheres.

Chern character

Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex X with its rational cohomology. In particular, they showed that there exists a homomorphism such that There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety X.