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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?


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Mark, in the case where $f$ is a sphere bundle (possibly non-orientable), Bredon's Sheaf Theory relates the differential in $E_2$ with the Gysin homomorphism. That this Gysin homomorphism equals the pushforward (as defined in B-R-S, ^_^) follows at least from the 5-lemma. That this pushforward equals the multiplication by the (twisted) Euler class is written up least in Lemma 2.1 in my paper on the van Kampen obstruction (it is on the arxiv). That the Euler class is the primary obstruction to the existence of a cross-section is treated in many books, I think. I hope this helps... – Sergey Melikhov Jun 13 '11 at 12:30
up vote 5 down vote accepted

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).

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Thanks Grigory, this is nice. Presumably we need to decorate the Hom's and tensors somehow when the coefficient system isn't simple? – Mark Grant Jun 10 '11 at 18:38
Actually, having thought about it, I think you mean that $d_k$ is the linear dual of $\mathfrak{o}(f)\in \mathrm{Hom}(H_{k+1}(B),H_k(F))$. (Since we assume field coefficients cohomology is dual to homology). By the way, do you know a reference for this? – Mark Grant Jun 13 '11 at 9:45

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