This is related to this question (edit: now answered, see below). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ is some multiplicative extraordinary cohomology theory (satisfting the wedge axiom), and I assume
that $X\to B$ is a Serre fibration over a CW complex with typical fibre $F$. Then consider the Leray-Serre / Atiyah-Hirzebruch / Whitehead spectral sequence
$$ E^{p,q}_2=H^p(B;h^q(F))\Rightarrow h^{p+q}(X)\;.$$
Several books state that there is a cup product on each page $E_k$ such that $d_k$ satisfies a Leipniz rule, and the cup product on $E_{k+1}$ is the induced one. However, I only found a proof in G. W. Whitehead's "elements of homotopy theory", which looks rather scary. Is there a more accessible account?
Edit (again) The answers to the question mentioned above name two papers: one by Massey and one by Douady. None of these contains an actual proof, so I am still looking for a nice reference.
Best Answer
We follow Douady's approach using Cartan-Eilenberg systems, see here.
Let $B$ be a CW complex and $\pi\colon X\to B$ a Serre fibration. Put $X^k=\pi^{-1}(B^k)$. A cellular approximation~$\Delta_B\colon B\to B\times B$ of the diagonal can be lifted to an approximation $\Delta\colon X\to X\times X$ of the diagonal such that $$X^k\stackrel\Delta\longrightarrow\bigcup_{m+n=k}X^m\wedge X^n\;.$$
Let $(\tilde h^\bullet,\delta,\wedge)$ be a reduced multiplicative generalised cohomology theory. We define a Cartan-Eilenberg system $(H,\eta,\partial)$ by $$H(p,q)=\tilde h^\bullet(X^{q-1}/X^{p-1})$$ for~$p\le q$ with the obvious maps $\eta\colon H(p',q')\to H(p,q)$ for $p\le p'$, $q\le q'$. The corresponding exact sequences take the form $$\cdots\to\tilde h^\bullet(X^{r-1},X^{q-1})\to\tilde h^\bullet(X^{r-1},X^{p-1}) \to\tilde h^\bullet(X^{q-1},X^{p-1})\stackrel\delta\to \tilde h^\bullet(X^{r-1},X^{q-1})\to\cdots$$ We ignore the grading; it is easy to fill in.
To define a spectral product $\mu\colon(H,\eta,\partial)\times(H,\eta,\partial)\to(H,\eta,\partial)$ we consider the map \begin{multline*} F_{m,n,r}\colon(X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1} \cong\bigcup_{a+b=m+n+r-1}(X^a\wedge X^b)\Bigm/ \bigcup_{c+d=m+n-1}(X^c\wedge X^d)\\ \begin{aligned} \twoheadrightarrow\mathord{}&\bigcup_{a+b=m+n+r-1}(X^a\wedge X^b)\Bigm/ \Bigl(\bigcup_{a=0}^m(X^{a-1}\wedge X^{m+n+r-a}) \cup\bigcup_{b=0}^n(X^{m+n+r-b}\wedge X^{b-1})\\ \cong\mathord{}&\bigcup_{a=m+1}^{m+r}(X^{a-1}\wedge X^{m+n+r-a})\Bigm/ \bigl(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1}\bigr)\\ \hookrightarrow\mathord{}& X^{m+r-1}\wedge X^{n+r-1}\bigm/ (X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1})\\ \cong\mathord{}&(X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})\;. \end{aligned} \end{multline*}Together with the diagonal map $\Delta$, for $r\ge 1$, we define \begin{multline*} \mu_r\colon H(m,m+r)\otimes H(n,n+r) \cong\tilde h(X^{m+r-1}/X^{m-1})\otimes\tilde h(X^{n+r-1}/X^{n-1})\\ \begin{aligned} &\stackrel\wedge\longrightarrow\tilde h\bigl((X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})\bigr)\\ &\stackrel{F_{m,n,r}^*}\longrightarrow\tilde h\bigl((X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1}\bigr)\\ &\stackrel{\Delta_X^*}\longrightarrow\tilde h(X^{m+n+r-1}/X^{m+n-1})=H(m+n,m+n+r)\;. \end{aligned} \end{multline*}
Proposition For all $m$, $n$, $r\ge 1$, the following diagram commutes $\require{AMScd}$ \begin{CD} H(m,m+1)\otimes H(n,n+1)@>\mu_1>>H(m+n,m+n+1)\\ @A\eta\oplus A\eta A@AA\eta A\\ H(m,m+r)\otimes H(n,n+r)@>\mu_r>>H(m+n,m+n+r)\\ @V\partial\otimes\eta\oplus V\eta\otimes\partial V@VV\partial V\\ {\begin{matrix}H(m+r,m+r+1)\otimes H(n,n+1)\\\oplus\\H(m,m+1)\otimes H(n+r,n+r+1)\end{matrix}}@>\mu_1\pm\mu_1>>H_{p+q-1}(m+n+r,m+n+r+1)\rlap{;,} \end{CD}
As explained here, this Proposition allows us to define a multiplicative structure on the associated spectral sequence.
Proof. The upper square commutes because the maps~$F_{m,n,r}$ are defined sufficiently naturally. For the lower square, we consider the boundary morphism $\delta$ of the triple $$(X^{m+r}\wedge X^{n+r-1}\cup X^{m+r-1}\wedge X^{n+r}, X^{m+r}\wedge X^{n-1}\cup X^{m+r-1}\wedge X^{n+r-1}\cup X^{m-1}\wedge X^{n+r},\\ X^{m+r}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r})\;.$$ The following diagram commutes: \begin{CD} \tilde h^{-p}(X^{m+r-1}/X^{m-1})\otimes\tilde h^{-q}(X^{n+r-1}/X^{n-1}) @>\wedge>> \tilde h^{-p-q}\bigl((X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})\bigr)\\ @V\delta\wedge\mathrm{id}\oplus V\mathrm{id}\wedge\delta V @VV\delta V\\ {\begin{matrix} \tilde h^{1-p}(X^{m+r}/X^{m+r-1})\otimes\tilde h^{-q}(X^{n+r-1}/X^{n-1})\\ \oplus\\ \tilde h^{-p}(X^{m+r-1}/X^{m-1})\otimes\tilde h^{1-q}(X^{n+r}/X^{n+r-1}) \end{matrix}} @>\wedge\oplus\wedge>> {\begin{matrix} \tilde h^{1-p-q}\bigl((X^{m+r}/X^{m+r-1})\wedge(X^{n+r-1}/X^{n-1})\bigr)\\ \oplus\\ \tilde h^{1-p-q}\bigl((X^{m+r-1}/X^{m-1})\wedge(X^{n+r}/X^{n+r-1})\bigr) \end{matrix}} \end{CD} We extend this diagram to the right using the maps $F_{m,n,r}$ and conclude that the lower square also commutes.