Leray–Hirsch module isomorphism

By [F1] and [F3], the $E_2$ page is $H^p(B;R){\otimes }_R{H}^q(F;R)$ in the named basis. Each $e_i$ comes from total-space cohomology, so its fiber restriction is represented by the fiber edge and is a permanent cycle. The product in [F3] therefore makes the map induced by $\mathrm{\Phi }$ on $E_2$ the basis map ${⨁}_i{H}^p(B;R)[-\mathrm{\mid }{e}_i\mathrm{\mid }]\to H^p(B;R){\otimes }_R{H}^q(F;R)$. It is an isomorphism in every bidegree.

A morphism of spectral sequences that is an isomorphism on one page is an isomorphism on all subsequent pages, so step 1.1 gives the associated-graded isomorphism at $E_{\mathrm{\infty }}$. Applying [F2] to the actual filtered cup-product map proves that $\mathrm{\Phi }$ is an isomorphism in every total degree.

The formula gives $\mathrm{\Phi }(c{a}_i)=p^{\ast }c⌣\mathrm{\Phi }(a_i)$, so it is an $H^{\ast }(B;R)$-module map. For a map of fibrations carrying every specified target $e_i$ to the corresponding source $e_i$, contravariance of pullback and cup naturality make the two displayed formulas commute; this is the asserted, choice-of-basis-relative naturality. If the finite list is empty, all fiber cohomology is zero and both sides are zero; one basis element, the zero ring, degree-zero classes, and filtration endpoints are included in steps 1.1–2.1. No splitting or basis is selected, and AC is used exactly through [A1].