Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum

You can give a proof of multiplicativity by using that the smash product preserves connectivity. Here is a sketch proof.

(EDIT: Denis Nardin pointed me towards this reference by Dugger. This reference shows that I was a little too cavalier about pairings in the homotopy category vs. lifting them to the stable category, and I've tried to make some adjustments accordingly.)

For any $n$, let $\tau_{\geq n} E$ be the $(n-1)$-connected cover of $E$, assembling into the Whitehead tower $$ \dots \to \tau_{\geq 2} E \to \tau_{\geq 1} E \to \tau_{\geq 0} E \to \tau_{\geq -1} E \to \dots \to E. $$ The associated graded consists of the spectra $\tau_{\geq n} E / \tau_{\geq n+1} E = K(\pi_n E, n)$.

The smash product of an $(n-1)$-connected object with an $(m-1)$-connected object is $(n+m-1)$-connected, and so the composite maps $$ \tau_{\geq n} E \wedge \tau_{\geq m} E \to E \wedge E \to E $$ lift, essentially uniquely up to homotopy, to $\tau_{\geq n+m} E$. This means that, in the homotopy category, the tower $\{\tau_{\geq n} E\}$ forms a filtered algebra. Because these lifts are essentially unique, this multiplication can also be lifted in a more structured one, lifting the Whitehead tower to a filtered algebra in spectra. (To be precise, a filtered algebra is a lax symmetric monoidal functor from the symmetric monoidal poset $(\Bbb Z, \geq, +)$ to spectra.)

The smash product is symmetric monoidal. Therefore, if $F$ is a ring spectrum the tower $\{(\tau_{\geq n} E) \wedge F\}$ is also a filtered algebra, and the associated graded consists of the spectra $K(\pi_n E, n) \wedge F$ because the smash product preserves cofibers. Therefore, the associated-graded spectral sequence starts with the homology of $F$ with coefficients in the homotopy groups of $E$: this constructs the Atiyah-Hirzebruch spectral sequence, except that the indexing is slightly different (the $E_1$-term is a reindexed $E_2$-term of the Atiyah-Hirzebruch SS).

Thus, this boils down to the following assertion:

If $\{\dots \to R^{-1} \to R^0 \to R^1 \to \dots\}$ is a (commutative / associative / unital) filtered algebra, then the associated-graded spectral sequence with $$ E^1_{p,q} = \pi_{p+q}(R^p / R^{p-1}) $$ is multiplicative (and commutative / associative / unital).

This is a little more standard. Dugger's reference that I linked to above gives a proof; he also says that he "found the existing literature extremely frustrating," which I don't think will surprise many people.

Just a brief response to Tyler and John's answers- but not brief enough to fit into a comment.

Here is one way to make the filtered object $\{\tau_{\ge n}E\}$ as structured as you'd like (just spelling out exactly what Tyler indicated).

Consider the $\infty$-category $\mathsf{Fun}(\mathbb{Z}, \mathsf{Sp})$ of filtered spectra, where $\mathbb{Z}$ is regarded as a poset. This has a symmetric monoidal structure coming from Day convolution. We have a colimit functor $\mathsf{Fun}(\mathbb{Z}, \mathsf{Sp}) \to \mathsf{Sp}$ and its right adjoint the 'constant tower' functor. The constant tower functor can be promoted to a symmetric monoidal functor $\mathsf{Sp} \to \mathsf{Fun}(\mathbb{Z}, \mathsf{Sp})$ (this is true in general for Day convolution stuff, but here it's extra true since there's only one colimit preserving symmetric monoidal functor from $\mathsf{Sp}$ to any other stable, presentably symmetric monoidal $\infty$-category anyway...).

Consider the full subcategory $\mathcal{C} \subseteq \mathsf{Fun}(\mathbb{Z}, \mathsf{Sp})$ spanned by the objects $\{E_n\}$ such that $E_n$ is $n$-connective, i.e. $E_n = \tau_{\ge n}E_n$. Staring at the formula for Day convolution, we learn that this subcategory is closed under the symmetric monoidal structure because smashing $n$-connective and $m$-connective thing gets you an $(n+m)$-connective thing (and $k$-connective things are closed under hocolims).

Now general nonsense says we get a colocalization $\mathsf{Fun}(\mathbb{Z}, \mathsf{Sp}) \to \mathcal{C}$ which is canonically lax symmetric monoidal. (HA.2.2.1.1).

The composite $\mathsf{Sp} \to \mathsf{Fun}(\mathbb{Z}, \mathsf{Sp}) \to \mathcal{C} \to \mathsf{Fun}(\mathbb{Z}, \mathcal{C})$ is then canonically lax symmetric monoidal and is given on objects by $E \mapsto \{\tau_{\ge n}E\}$.

From here you can get whatever you want! For example, this induces a lax symmetric monoidal functor on homotopy categories, which gives you the pairings you needed in Tyler's answer. But it does much more: it also tells you that if you start with $E$, and algebra over any operad $\mathcal{O}$, then the Whitehead tower is canonically a filtered algebra over that operad.

(This is not the end of the story, I think. I always get confused about this but, if I remember correctly, people often like to ask for some filtered version of the operad to act on this filtered gadget, in order to get the story of power operations in the spectral sequence? I might be confusing this with something else though... again- I never learned that story properly).

(A comment to Tyler's answer.) Strictifying pairings from the stable homotopy category to spectra can be tricky. To even get started with an inductive approach let me assume $E$ is connective, so that $E = \tau_{\ge0}E$. Let $p_n : \tau_{\ge n} E \to \tau_{\ge n-1} E$ be the maps in the Whitehead tower, and let $\mu = \mu_{0,0} : E \wedge E \to E$ be the given pairing. The composite $\mu_{0,0} (p_1 \wedge 1) : \tau_{\ge1} E \wedge E \to E$ factors up to homotopy though $p_1$. We may assume that each $p_n$ is a fibration (and that everything in sight is cofibrant), so by the homotopy lifting property there is also a strict factorization as $p_1 \mu_{1,0}$. Hence we can choose lifts $\mu_{1,0} : \tau_{\ge1} E \wedge E \to \tau_{\ge1} E$ and $\mu_{0,1} : E \wedge \tau_{\ge1} E \to \tau_{\ge1} E$ such that $p_1 \mu_{1,0} = \mu_{0,0} (p_1 \wedge 1)$ and $p_1 \mu_{0,1} = \mu_{0,0} (1 \wedge p_1)$. The composites $\mu_{1,0} (1 \wedge p_1) : \tau_{\ge1} E \wedge \tau_{\ge1} E \to \tau_{\ge1} E$ and $\mu_{0,1} (p_1 \wedge 1) : \tau_{\ge1} E \wedge \tau_{\ge1} E \to \tau_{\ge1} E$ agree when projected to $E$, but without further work they may not agree as maps to $\tau_{\ge1} E$. In particular, they may not have a common factorization as $p_2 \mu_{1,1}$ for some $\mu_{1,1} : \tau_{\ge1} E \wedge \tau_{\ge1} E$. So strictifying a pairing is not just a matter of obstruction theory or essential uniqueness of lifts. One may also need to change the models for the spectra involved.

In the case of spectra formed from simplicial sets, there may be a sufficiently functorial (and monoidal) construction of Whitehead towers of simplicial sets, hence also of symmetric spectra in simplicial sets, to ensure that $\mu$ induces compatible $\mu_{m,n} : \tau_{\ge m}E \wedge \tau_{\ge n}E \to \tau_{\ge m+n}E$ for all integers $m$ and $n$, but I do not recall checking this carefully, and if correct, it would be badly model-dependent.

There is a $2$-categorical approach that works. We can choose maps $\mu_{m,n} : \tau_{\ge m} E \wedge \tau_{\ge n} E \to \tau_{\ge m+n} E$, "horizontal" homotopies $h_{m,n} : \mu_{m-1,n} (p_m \wedge 1) \simeq p_{m+n} \mu_{m,n}$ and "vertical" homotopies $v_{m,n} : \mu_{m,n-1} (1 \wedge p_n) \simeq p_{m+n} \mu_{m+n}$. In the case of a Whitehead tower, one can find a $2$-homotopy between the composite homotopies $v_{m-1,n} h_{m,n}$ and $h_{m,n-1} v_{m,n}$, and this suffices to get a pairing of Cartan-Eilenberg systems, hence also a pairing of spectral sequences. These $1$- and $2$-homotopies produce a strict pairing of filtered spectra $Tel(E) \wedge Tel(E) \to E$, where $Tel$ denotes the mapping telescope.

This can surely be promoted to an $\infty$-categorical statement, but the $2$-categorical one suffices for pairings of spectral sequences.