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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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Dear Michel, What sort of fibre bundle are you considering? E.g. a principal bundle for some group $G$ with a section is necessarily trivial, i.e. just isomorphic to $B \times G$, and so the cohomology is given by Kunneth. On the other hand, every vector bundle admits a section, and is homotopy equivalent to its base, so its cohomology coincides with that of its base. So to say something useful, it might help to give more details about the particular kind of bundles you care about. Regards, –  Matt E Aug 14 '12 at 5:20
    
Thank you for your response. I added more information about examples in my mind. –  Michel Aug 14 '12 at 5:46

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