Let be a compactHausdorff 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 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,, 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 Bott periodicity theorem as formulated below extends the theories to positive integers.
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,
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.
and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a 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.
