In this chapter we define the various K-theory groups associated to a topological space and study how they are related.

Definition of K0(X)

DefinitionLet X be compact Hausdorff. The Grothendieck completion of Vect(X) is denoted K0(X).

Thanks to Theorem 1.7.14, we may alternately define K0(X) as the Grothendieck completion of Idem(C(X)).

ExampleLet X be a single point. Then a vector bundle over X is just a vector space, and these are classified by rank. Thus Vect(X) ≅ ℤ+and so K0(X) ≅ ℤ.

ExampleLet X be the disjoint union of compact Hausdorff spaces X1, X2, …, Xk. A vector bundle on X is a choice of a vector bundle on each X1, X2, …, Xk, and the same is true for isomorphism classes of vector bundles on X. Therefore

By taking the Grothendieck completion, we obtain an isomorphism

In particular, if X consists of k distinct points in the discrete topology, then K0(X) ≅ ℤk.

We would certainly like to be able to compute K0(X) for a topological space more complicated than a finite set of points! To do this, we have to develop some machinery; this will occupy us for the remainder of the chapter.