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What does the statement actually mean? Could anyone please share some ideas on it? Also how does it follow from Künneth formula? Any help in this regard would be warmly appreciated.

Thanks for your time.

  • $\begingroup$ Related: math.stackexchange.com/questions/401528/… $\endgroup$ May 10, 2022 at 19:25
  • $\begingroup$ Does this answer your question? Leray Hirsch from Bott & Tu $\endgroup$ May 10, 2022 at 19:26
  • $\begingroup$ @MatthewLeingang What is $e_i$ and $i^{\ast} e_i\ $? What is $i$ first of all? Is $e_i$ some form on $E\ $? If we take any fiber under the projection $\pi : E \longrightarrow M$ then it will be is diffeomorphic to $F.$ Hence if $e_i$ is a form on $E$ then $i^{\ast} e_i$ will be a form on $F.$ Is it what is meant? $\endgroup$ May 11, 2022 at 5:42
  • $\begingroup$ @MatthewLeingang$:$ I don't know how to finish the theorem using Künneth formula. Please provide some idea if you have unless marking my question as duplicate. $\endgroup$ May 11, 2022 at 6:12
  • $\begingroup$ This theorem follows the Künneth formula in the text. KF is proven, but the remark just before the screenshot you clipped says “a similar argument shows...” This is the authors' way of saying that it's an exercise for the reader to adapt the proof of KF to Leray-Hirsch—and not that it's a corollary to KF. $\endgroup$ May 12, 2022 at 14:34


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