Leray-Hirsch theorem for K-theory

Statement

Let $p:E\to B$ be a fiber bundle with both $E$ and $B$ compact Hausdorff spaces and with fiber space $F$, such that :

• $K^{*}(F)$ is free and finitely generated
• There exist classes $c_{1},c_{2},\ldots ,c_{k}\in K^{*}(E)$ that restrict to a freely generating set for $K^{*}(F)$ for each fiber $F$

And suppose one of these conditions holds:

1. $B$ is a finite cell complex
2. $F$ is a finite cell complex having only cells of even dimension

Then $K^{*}(E)$ is free as a module over $K^{*}(B)$, with basis $\{c_{1},c_{2},\ldots ,c_{k}\}$.

References

• Vector bundles and K-theory by Allen Hatcher