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Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map $ f^{*}:H^{*}(B,\mathbb{Z})\longrightarrow H^{*}(E,\mathbb{Z}) $ is injective. I would like to understand the cohomology groups of $E$. What can we say about the cohomology groups (mod torsions is enough) of this fiber bundle with sections?

Edit Examples in my mind are 2-torus (i.e.$(S^1)^2$) fibration over fourfold $B$. We may also assume that we know both the cohomology groups $H^{*}(B,\mathbb{Z})$ and the monodromy representation $\pi(B)\rightarrow GL(H^{*}((S^1)^2),\mathbb{Z}))$.

Of course, we may use spectral sequence with these information, but local coefficient computation is quite hard. I wonder if the existence of section may make things simpler.

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    Thank you for your response. I added more information about examples in my mind.2012-08-14

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