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 Thom isomophism $H^*(B)\rightarrow H^{*+n}(E,E_0)$ isisomorphisms are given by $x\mapsto p^*(x)\cup U$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.
