K-group

Definition

Let $X$ be a compact Hausdorff space. The (unreduced) K-group of $X$, denoted $K(X)$ is defined as the Grothendieck group of the following Abelian monoid:

• As a set, the set of all complex vector bundles over $X$, upto isomorphism
• The additive operation is direct sum of complex vector bundles
• The zero element is the zero-dimensional trivial vector bundle

Equivalently $K(X)$ can be viewed as the set of formal differences of complex vector bundles, upto stable isomorphism (we need stable isomorphism because the monoid of complex vector bundles per se is not cancellative).

$K(X)$ is also a ring, with the multiplication operation that induced by the tensor product of complex vector bundles.

Related notions

• Reduced K-group
• KO-group
• KSp-group