2 - K-theory

Summary

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, …, X_k. A vector bundle on X is a choice of a vector bundle on each X₁, X₂, …, X_k, 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) ≅ ℤ^k.

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.