Massey products in the Steenrod algebra

Question

When building $kU/2$ via its Postnikov tower, there are some interesting Massey products that show up in the Steenrod algebra, and I'd like to understand them. I bet these appear somewhere in the litterature, but I have not been able to find a reference. I will call $X = kU/2$, $H = H\mathbb{F}_2$ and I'll use $X_{\leq n}$ for the truncated guy, with homotopy in $\pi_0, \ldots, \pi_n$.

Start with $X_{\leq 0} = H$. The $k_0$-invariant to attach $\pi_2$ is the map $X_{\leq 0} =H \stackrel{Q_1}{\to} \Sigma^3 H$. We take its homotopy fiber to get the stage $X_{\leq 2}$.

Now we can rotate and we have $\Sigma^{-1} X_{\leq 0} \stackrel{Q_1}{\to} \Sigma^2 H \to X_{\leq 2}$. The $k_1$-invariant should be given by the extension of $\Sigma^2 H \stackrel{Q_1}{\to} \Sigma^5 H$ to $X_{\leq 2}$. Since $Q_1 Q_1 \simeq 0$ this map does extend to the Postnikov truncation $X_{\leq 2} \stackrel{k_1}{\to} \Sigma^5 H$ giving us the $k$-invariant. (here I am not sure about uniqueness, maybe it is easier to put an $HZ$ at the bottom $\pi_0$, because then $k_1$ would be unique as $[HZ, H] = A/Q_0$).

Anyways, we take the fiber again and get $X_{\leq 4}$. The $k_2$-invariant here will exists if and only if $Q_1 k_1 \simeq 0$. By drawing the diagram defining the Massey product $\langle Q_1, Q_1, Q_1 \rangle$ we see that the product $Q_1 k_1 = 0$ if and only if the product $\langle Q_1, Q_1, Q_1 \rangle$ is $Q_1$-divisible.

How to show this ? I am of course not interested in arguing that $kU$ exists and so this has to happen. Thank you for any help, or any reference.

Answer 1

As near as I've been able the find, the primary reference for a proof is probably Kristensen and Madsen's "On the structure of the operation algebra for certain cohomology theories." This result (in fact, its generalization to all the Milnor primitives) occurs just after Proposition 3.1 in this document; it appears to be a consequence of a general result for things in the kernel of the cap-product with $\xi_1$.

An alternative reference is in Baues' book "The algebra of secondary cohomology operations": he works out an algebraic method for computing triple Massey products in the Steenrod algebra, and in Table 1 (starting on page 456) computes all the (non-matric) Massey products of three homogeneous elements up through degree 22. In particular, your bracket $\langle Q_1, Q_1, Q_1\rangle$ is listed as $\langle 3+2.1, 3+2.1, 3+2.1 \rangle$ in the degree-9 portion of their table, and this bracket contains zero. (In fact, they only found one triple product that doesn't contain zero: $\langle Sq(0,2),Sq(0,2),Sq(0,2) \rangle$.) Their method, so far as I understand it, consists of using their machinery of "track algebra" to take a presentation of the Steenrod algebra and lift it up to a more enriched algebraic structure that also remembers homotopy data that enforces the Adem relations. To do this he needs to do some real work with Eilenberg-Mac Lane spaces. I would be misrepresenting things if I claimed that I understood how this works.

(A quick comment: You're correct that there's not uniqueness of the next k-invariant or the triple product. The indeterminacy amounts to the indeterminacy in the Massey product $\langle Q_1, Q_1, Q_1\rangle$, which in this case consists of multiples of $Q_1$, so you could equally well say that the next stage in the tower exists if and only of the Massey product contains zero.)

Answer 2

Here is a reference from way back: Kristensen, Leif. Massey products in Steenrod's algebra. 1970 Proc. Advanced Study Inst. on Algebraic Topology (Aarhus, 1970), Vol. II pp. 240–255 Mat. Inst., Aarhus Univ., Aarhus. No idea whether or not your specific question is answered. Around the time that was written I had a program for going further using Kristensen's structured cochains. The only way I could stop thinking about it was to throw away my notes.