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Let$p:E\rightarrow B$ be an n dimensional vector bundle, R be a commutative ring. Assume that B is simplyconnected or char R=2. Then there is an element $U\in H^n(M(p);R)$ such that we have dual isomorphisms $\Phi_*:\tilde{H}_{*+n}(M(p);R)\cong H_*(B;R)$ and $\Phi^*:H^*(B;R)\cong \tilde{H}^{*+n}(M(p);R)$ defined by $\Phi_*(m)=p_*(U\cap m)$ and $\Phi^*(b)=p^*(b)\cup U$ for $m\in\tilde{H}_*(M(p);R)$ and$b\in H^*(B;R)$.

How can I see that the isomorphisms are given as above from the Serre spectral sequence? The spectral sequences have only one nontrival row and so collapse at $E_2$ and $E_2$ terms. It is clear that we have above isomorphisms ,but that they take cap-product and cup-product forms of the isomorphisms is not so obvious to me.

I have another question: How to construct a map of spectral sequences? It seems that so many commutative diagrams need to check.

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    $\begingroup$ I fixed the problems with Latex and capitalization, but I think it would be helpful if you motivated the question a bit, e.g., with what you know and what specifically is confusing you. $\endgroup$
    – S. Carnahan ♦
    Commented May 16, 2010 at 16:44
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    $\begingroup$ "How to construct a map of spectral sequences?" The disc bundle $DE$ has a sub-bundle $SE$ of unit spheres; the inclusion $i$ induces a map $i^\ast$ of filtered (singular cochain) complexes since the filtration is defined via the skeleta of $B$. So just on general principles, there's a map of Serre spectral sequences. More significantly, the relative cochains $C^\ast(DE,SE)$ inherit a filtration, whose spectral sequence should be as in Oscar's answer. $\endgroup$
    – Tim Perutz
    Commented May 17, 2010 at 14:55

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This follows immediately from a relative Serre spectral sequence. However, it is not the one usually called the "relative Serre spectral sequence", which has to do with a Serre fibration and its restriction over a subspace of the base space. Instead, given Serre fibrations $E \to B$ and $E' \to B$ with fibres $F$ and $F'$ respectively, and a cofibration $E' \to E$ over $B$, it is a spectral sequence $$E^2_{p,q} = H^p(B;\mathcal{H}^q(F, F')) \Rightarrow H^{p+q}(E, E'),$$ where the script letter denotes the system of local coefficients. One can prove the existence and algebra properties of this quite easily starting from the usual Serre spectral sequence (the key point is that $E/E' \to B$ is a Serre fibration with section), but I have never found a reference for it. The algebra structure of the spectral sequence gives you exactly what you want.

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    $\begingroup$ this is a great way to do it. you see the nature of the isomorphism right away too, the fundamental class of the relative homology of the fiber is sitting right there. Also, if you work with a simply connected base space the $E\_2$ term is easier. (but not by much) $\endgroup$
    – Sean Tilson
    Commented Aug 17, 2010 at 4:37
  • $\begingroup$ I see the isomorphism come out really quick this way. But could you say again how it follows this way that the iso is given by cupping with a Thom class? $\endgroup$
    – Urs Schreiber
    Commented Jun 2, 2016 at 20:51
  • $\begingroup$ Maybe like so: ncatlab.org/nlab/show/… $\endgroup$
    – Urs Schreiber
    Commented Jun 3, 2016 at 9:36

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