15
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ 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... $\endgroup$ – Sergey Melikhov Jun 13 '11 at 12:30
7
$\begingroup$

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).

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks Grigory, this is nice. Presumably we need to decorate the Hom's and tensors somehow when the coefficient system isn't simple? $\endgroup$ – Mark Grant Jun 10 '11 at 18:38
  • $\begingroup$ 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? $\endgroup$ – Mark Grant Jun 13 '11 at 9:45
9
$\begingroup$

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 $H^{k+1}(B;H^0(F;\pi_k(F))$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ This is great, thank you! Do you know of anywhere in the literature where this is proved and/or used? $\endgroup$ – Mark Grant Aug 2 '16 at 18:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.