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 (general) 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 formula_1 or formula_2. Then formula_3 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, formula_4 usually denotes complex -theory whereas real -theory is sometimes written as formula_5. The remaining discussion is focused on complex -theory.
As a first example, note that the -theory of a point are 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, formula_6, defined for a compact pointed space (cf. reduced homology). 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 formula_7 and formula_8, so that formula_9. 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, formula_6 can be defined as the kernel of the map formula_11 induced by the inclusion of the base point into .
-theory forms a multiplicative (generalized) 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 formula_16 is formula_17 with a disjoint basepoint labeled '+' adjoined.
Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.
Bott periodicity.
The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way:
In real -theory there is a similar periodicity, but modulo 8.
Applications.
The two most famous applications of topological -theory are both due to Frank Adams. First he solved the Hopf invariant one problem by doing a computation with his Adams operations. Then he 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 CW complex formula_17 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 formula_17.