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