Vector Bundle - K-theory

K-theory

The K-theory group

K(X)

of a manifold is defined as the abelian group generated by isomorphism classes of (complex) vector bundles modulo the relation that whenever we have an exact sequence

0 → A → B → C → 0

then

=+

in topological K-theory. KO-theory is a version of this construction which considers real vector bundles. K-theory with compact supports can also be defined, as well as higher K-theory groups.

The famous periodicity theorem of Raoul Bott asserts that the K-theory of any space X is isomorphic to that of the Cartesian product

X × S2,

where S2 denotes the 2-sphere.

In algebraic geometry, one considers the K-theory groups consisting of coherent sheaves on a scheme X, as well as the K-theory groups of vector bundles on the scheme with the above equivalence relation. The two constructs are the same provided that the underlying scheme is smooth.