# Leray-Hirsch theorem for K-theory

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## 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