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. BenoĆ®t

notassuming 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