Topological K-theory

In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck. The early work on topological -theory is due to Michael Atiyah and Friedrich Hirzebruch.

Definitions

Let be a compact Hausdorff space and or. Then is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, usually denotes complex -theory whereas real -theory is sometimes written as. The remaining discussion is focused on complex -theory.
As a first example, note that the -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 -theory,, defined for a compact pointed space. This reduced theory is intuitively modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles and 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 into.
-theory forms a multiplicative cohomology theory as follows. The short exact sequence of a pair of pointed spaces
extends to a long exact sequence
Let be the -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 is with a disjoint basepoint labeled '+' adjoined.
Finally, the periodicity theorem">Frequency">periodicity theorem as formulated below extends the theories to positive integers.

Properties

is a contravariant functor from the homotopy category of spaces to the category of commutative rings. Thus, for instance, the -theory over contractible spaces is always
The spectrum of -theory is , i.e. where denotes pointed homotopy classes and is the colimit of the classifying spaces of the unitary groups: Similarly, For real -theory use.
There is a natural ring homomorphism the Chern character, such that is an isomorphism.
The equivalent of the Steenrod operations in -theory are the Adams operations. They can be used to define characteristic classes in topological -theory.
The Splitting principle of topological -theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
The Thom isomorphism theorem in topological -theory is where is the Thom space of the vector bundle over. This holds whenever is a spin-bundle.
The Atiyah-Hirzebruch spectral sequence allows computation of -groups from ordinary cohomology groups.
Topological -theory can be generalized vastly to a functor on C*-algebras, see operator K-theory and KK-theory.

Bott periodicity

The phenomenon of periodicity named after Raoul Bott can be formulated this way:

and where H is the class of the tautological bundle on i.e. the Riemann sphere.

In real -theory there is a similar periodicity, but modulo 8.

Applications

Topological -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

and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex 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.