Where does the primary obstruction of a fibration show up in its spectral sequence? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:35:21Z http://mathoverflow.net/feeds/question/67455 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67455/where-does-the-primary-obstruction-of-a-fibration-show-up-in-its-spectral-sequenc Where does the primary obstruction of a fibration show up in its spectral sequence? Mark Grant 2011-06-10T16:36:56Z 2011-06-13T11:35:16Z <p>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).</p> <p>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 $\pi_1(B)$.</p> <p>Here is my question, which I'm a little embarrassed to ask:</p> <blockquote> <p>Is there a canonical class in the $E_2$ term which relates somehow to $\mathfrak{o}(f)$?</p> </blockquote> <p>Sorry for being (intentionally) vague.</p> <p><strong>Edit:</strong> 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.</p> <blockquote> <p>Has anyone seen a reference for this?</p> <p>Can anyone give a more general statement when the local coefficient system is non-trivial?</p> </blockquote> <p>Thanks.</p> http://mathoverflow.net/questions/67455/where-does-the-primary-obstruction-of-a-fibration-show-up-in-its-spectral-sequenc/67457#67457 Answer by Grigory M for Where does the primary obstruction of a fibration show up in its spectral sequence? Grigory M 2011-06-10T17:40:23Z 2011-06-13T11:35:16Z <p>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 $d_k\in\operatorname{Hom}(H^k(F),H^{k+1}(B))\cong H^{k+1}(B)\otimes H_k(F)\ni \mathfrak o(f)$).</p> <p>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).</p> <p>I'm afraid I can't say anything about non-simple case, though (not even sure what is the correct statement in this case).</p>