Computation of Some K-Groups

Abstract

The purpose of this section is to define an isomorphism K ^q_ℂ (X) ≈ K ^q_ℂ (V), for any complex vector bundle V over a locally compact space X (note that K ^q_ℂ (V) ≈ K ^q_ℂ (B(V),S(V)), with respect to any metric on V; cf. II.5.12). For V trivial, we again obtain Bott periodicity in complex K-theory (cf. III.1.3 and III.2.1); however, Bott periodicity is actually an essential part of our proof. If X is compact, the one point compactification $\dot V$ of V is called the Thom space of V. Hence, the isomorphism $K_\mathbb{C}^q\left( X \right) \approx K_\mathbb{C}^q\left( V \right) \approx \tilde K_\mathbb{C}^q\left( {\dot V} \right)$ will enable us to compute the K-theory of the Thom space of a complex vector bundle. Before defining this isomorphism, we will first establish a general theorem (1.3), which will also be useful in next sections (* it is the analogous of the Leray-Hirsch-Dold theorem in the framework of K-theory *).