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Let us consider the following situation: $\pi:X\rightarrow B$ is a locally trivial fibration between smooth manifolds, its fiber being a torus $T$. My question is two-fold:

1) what is the obstruction for $\pi$ to have a global section ? It seems to me that it lives in $H^2(B,R^1\pi_*\mathbb{Z})$ but I'm not sure. If it is the case, can this obstruction be described in term of the Leray spectral sequence of $\mathbb{Z}$ with respect to $\pi$ ?

2) Assume furthermore that the fundamental groups (of $X,B$ and $T$) fit into an exact sequence: $1\rightarrow\pi_1(T)\rightarrow\pi_1(X)\rightarrow\pi_1(B)\rightarrow 1$ (it is of course not always the case). Is the latter obstruction related to the extension class $e\in H^2(\pi_1(B),\pi_1(T))$ associated with this short exact sequence? If $e$ is a torsion element, is it true that $\pi$ admits a section after a suitable finite cover of the base?

Thanks in advance. BenoƮt

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In particular, you're not assuming that the bundle is principal? Or else the obstruction is in $H^2(B;$ the weight lattice$)$, a way of collecting together the first Chern classes of all the associated line bundles. – Allen Knutson Dec 3 '13 at 15:39
that's it: this is merely a fibration. The cohomology group I wrote is in some sense a "locally constant" version of the one you mentionned. – Benoit Dec 3 '13 at 16:00
What is $R^1\pi_*\mathbb Z$? I gather it is some kind of local system associated with the fibration; which local system is it? Also by a fibration do you mean a Serre fibration? What does locally trivial mean in this context? Or are you really talking about a locally trivial smooth fiber bundle with the structure group $Diff(T^n)$? As written the question makes no sense, I think. – Igor Belegradek Dec 4 '13 at 0:47
This local system is the one given by $(H^1(X_b,\mathbb{Z}))_{b\in B}$ ($X_b$ being the fiber over $b$): the fibration being locally topologically trivial (locally a product over $B$), it is without any doubt a local system. Since the situation I'm interested in comes from complex geometry, it is likely that I can add hypothesis on the fibration but even in this setting I think that it makes sense (or there is something I missed...). – Benoit Dec 4 '13 at 10:43
How is your local system related to obstructions? Normally, the primary obstruction lives in $H^{n+1}(B,\pi_n(F))$ where $F$ is the fiber, $n$ is the least integer such that $\pi_n(F)\neq 0$, and $\pi_n(F)$ is the local system of $n$-th homotopy groups. There are also higher obstructions to the existence of a section, see e.g. "Higher obstructions to sectioning a special type of fibre bundles" [Transactions AMS, 1964, by W. Hsiang]. – Igor Belegradek Dec 4 '13 at 13:18

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