I haven't references about a proof of this theorem: Let $p: Y \rightarrow X$ be a fiber bundle. If for all $x \in X$ $p^{-1}(x)$ satisfies $H^{\ast}(p^{-1}(x)) \simeq \mathbb{R} $, then $p$ induces an isomorphism $ H^{\ast}(X) \simeq H^{*}(Y)$. How could I prove this theorem? A spectral sequences argument for the fiber bundle?
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2$\begingroup$ Sure just write down the Serre spectral sequence and it's clear. There is the more classical Leray-Hirsch theorem as well that can work for this situation. $\endgroup$– Daniel PomerleanoCommented Jan 16, 2013 at 19:33
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$\begingroup$ How could I write Serre spectral sequence here? $\endgroup$– MiliskWallCommented Jan 17, 2013 at 0:31
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2$\begingroup$ The $E_2$-page just consists of a single horizontal line -- the homology of $Y$; i.e. there are no further differentials or filtrations, and you can read the homology of $X$ off directly from this line. $\endgroup$– Tarje BargheerCommented Jan 17, 2013 at 3:01
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$\begingroup$ I haven't know how can I build the page $E_{2}$ and why there is only a horizontal line. What are $E_{1}$ and $E_{0}$ pages? $\endgroup$– MiliskWallCommented Jan 17, 2013 at 21:08
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