Where does the primary obstruction of a fibration show up in its spectral sequence?

Question

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:

Is there a canonical class in the $E_2$ term which relates somehow to $\mathfrak{o}(f)$?

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.

Has anyone seen a reference for this?

Can anyone give a more general statement when the local coefficient system is non-trivial?

Thanks.

Answer 1

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).

Answer 2

In the general case of integral coefficients and possibly non-trivial local coefficient system, let $\pi=\pi_1(B)$. A cocycle for the obstruction class is an element $o\in Hom_{\mathbb Z\pi}(C_{k+1}\tilde{B},\pi_kF)$. Now $o$ induces a group homomorphism $$ H^0\left(B;{H^k(F)}\right)\cong Hom_{\mathbb Z}(\pi_kF,\mathbb Z)^\pi \cong Hom_{\mathbb Z\pi}(\pi_kF,\mathbb Z)\to Hom_{\mathbb Z\pi}(C_{k+1}\tilde{B},\mathbb Z).$$ Here we use the universal coefficient theorem and the fact that $H^0(B,M)=M^\pi$ for connected $B$. Since $o$ is a cocycle, the group homomorphism factors through cocycles $C^{k+1}(B;\mathbb Z)$, so, since $H^0(F)=\mathbb Z$ there is an induced map $$ H^0\left(B;{H^k(F)}\right) \to H^{k+1}(B;H^0(F))$$ which is the $d_{k+1}$-differential $$E_{2}^{0,k}= E_{k+1}^{0,k}\to E_{k+1}^{k+1,0}=E_2^{k+1,0}$$ in the spectral sequence.

Or to put it differently, there is a map $$H^{k+1}(B,\pi_k F) \to Hom(H^0(B;H^k(F)),H^{k+1}(B;H^0(F)))$$ adjoint to the composition of cup product and Hurewicz and Kronecker maps $$H^{k+1}(B,\pi_k F) \otimes H^0(B;H^k(F)) \to H^{k+1}(B;\pi_k F \otimes H^k(F))\to H^{k+1}(B;\mathbb Z)$$ which sends the obstruction class to the $d_{k+1}$-differential.

Moreover if we consider the spectral sequence $H^*(B;H^*(F;\pi_k(F)))\to H^*(E;\pi_k(F))$ the connection is even more direct: then $H^0(B;H^k(F;\pi_k(F))$ contains the distinguished element corresponding to the identity of $\pi_k F$ under the universal coefficient theorem, whose image under $d_{k+1}$ is the obstruction class in $H^{k+1}(B;H^0(F;\pi_k(F))$.