Let $f\colon\thinspace E\to B$ be a Serre fibration whose fibre $F$ is $k-1$-connected, $k\geq 1$. Assume $B$ is a connected CW complex. Then the primary obstruction to the existence of a cross section of $f$ is defined; it is a cohomology class $$\mathfrak{o}(f)\in H^{k+1}(B;\tilde{H}\_{k}(F)).$$
Here the coefficients may be twisted by $\pi\_1(B)$. The definition involves choosing a section on the $k$-skeleton which you then try to extend, but the class itself is canonical (depends only on the fibration).


Meanwhile, there is the cohomology Leray-Serre spectral sequence of the fibration, with $$E\_2^{p,q}=H^p(B;H^q(F))\implies H^\*(E),$$ where again the coefficients in the $E\_2$ term may be twisted by the action of $\pi\_1(B)$.


Here is my question, which I'm a little embarrassed to ask:



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> Is there a canonical class in the $E\_2$ term which relates somehow to $\mathfrak{o}(f)$?
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Sorry for being (intentionally) vague.


**Edit:** As Grigory M points out in his answer, if we work over a field and assume the local system on the base formed from the homology of the fibres is trivial, then the first non-trivial differential $$d\_{k+1}\in \mathrm{Hom}(H^k(F),H^{k+1}(B))$$
is the linear dual of an element $$d\_{k+1}^\ast\in\mathrm{Hom}(H\_{k+1}(B),H\_k(F))\cong H^{k+1}(B;H\_{k}(F))$$ which should equal the obstruction class.



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> Has anyone seen a reference for this?
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> 
> Can anyone give a more general statement when the local coefficient system is non-trivial?
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Thanks.



Where does the primary obstruction of a fibration show up in its spectral sequence?
At least in the case $\pi\_1(B)=0$, $\mathfrak{o}(f)$ is just the first non-trivial differential, $d\_k$ in disguise (let's work over some field, for simplicity; then $d\_k\in\operatorname{Hom}(H^k(F),H^{k+1}(B))\cong H^{k+1}(B)\otimes H\_k(F)\ni \mathfrak o(f)$).


Reference (well, kind of: it doesn't even give precise statement, let alone proof): Mosher, Tangora. Cohomology operations and applications in homotopy theory (pp. 103, 109).


I'm afraid I can't say anything about non-simple case, though (not even sure what is the correct statement in this case).


