The Leray-Hirsch theorem: let $k$ be a field. Given a fibration $F \to E \to B $ with $F, B$ path connected and suppose system of local coefficient is zero and the following condition satisfied (a) $H^n(B;k) $ is finitely dimensional for each $n$ and (b) $ i^*:H^*(E;k) \to H^*(F;k)$ is onto, here $i$ is the inclusion of fiber. Then $$H^*(E;k) \cong H^*(B;k)\otimes_k H^*(F;k) $$ as vector spaces.

My question is: What are some extra condition do we need so that above becomes ring isomorphism. In particular for the field $k=\Bbb Z/2 \Bbb Z$ what extra condition do we need.

Thank you very much for your kind help.

  • $\begingroup$ The conditions of Proposition 8 are that $H_i(F;k) \to H_i(E;k)$ is injective for all $i \geq 0$ (or equivalently that $H^i(E;k) \to H^i(F;k)$ is surjective) and that each $H^i(F;k)$ and $H^i(B;k)$ are finite-dimensional. The only thing you haven't already assumed is that $H^i(F;k)$ is finite-dimensional. Once you assume that, you get what you want. $\endgroup$ – user98602 Jan 20 at 0:56
  • $\begingroup$ @MikeMiller I got a translation of the above mentioned J P Serre paper. Thanks. $\endgroup$ – Shivani Sengupta Jan 20 at 15:53

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