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Let $\tau_0$ be the element of dual Steenrod algebra $A_p^{*}$ at a prime $p$ which is dual to Bockstein $\beta \in A_p$. It is well known $\tau_0^2 =0$. Is it true/known that the elemnet $\xi_1$ belong to $p$-fold Massey product of $ \tau_0$, i.e. $$ \langle \tau_0, \ldots, \tau_0 \rangle \ni \xi_1 ?$$

If so what is the proof or a reference?

If the result is false feel free to hammer down this question.

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  • $\begingroup$ How do you define Massey products in $\mathcal{A}_p^\ast$? The only definition I know is for elements in the cohomology of a dga. Do you mean the vector space duals of Massey products in $\mathcal{A}_p$? $\endgroup$ – Mark Grant Jan 6 '14 at 18:12
  • $\begingroup$ I am not quite sure how I define the Massey product, which is also a reason why I posted this question. I remember Tyler Lawson mentioning this in a comment to one of my previous questions. mathoverflow.net/questions/119012/mod-3-moore-spectrum $\endgroup$ – Prasit Jan 6 '14 at 18:19
  • $\begingroup$ I do not think its the vector space dual of Massey product in $\mathcal{A}_p$ as the dimensions won't match up with that of $\xi_1$. $\endgroup$ – Prasit Jan 6 '14 at 18:25
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So far as definitions of Massey products, the dual Steenrod algebra is the homology $H_*(H\Bbb Z/p)$ of an $E_\infty$ ring spectrum, and so it has both Massey products and power operations.

Kraines, in Theorem 14 of "Massey higher products" from 1966, shows that the restricted power $\langle u \rangle^p \subset \langle u, u, \ldots, u\rangle$ can be identified with the power operation $-\beta P^m u$ when $u$ is an element in $H^{2m+1}(X; \Bbb Z/p)$. Kochman generalized this in "Symmetric Massey products and a Hirsch formula in homology" to certain differential graded Hopf algebras (of which the chains on the dual Steenrod algebra can be modeled by one, I think) showing that for such an algebra we have $\langle u \rangle^p = -\beta Q(u)$ plus a term involving iterated Browder brackets of $u$ and $\beta u$.

(In an ideal world, this would be an identification that's valid in the homology of any $E_\infty$ algebra, and not just one of these special forms.)

Then Theorem III.2.3 of Bruner-May-McClure-Steinberger's $H_\infty$ book proves that in the dual Steenrod algebra we have $\beta Q(\bar \tau_0) = \bar \xi_1$, and the conjugation in these degrees is thankfully just negation.

Thus $\langle \bar \tau_0,\ldots,\bar \tau_0\rangle = -\bar \xi_1$. This generalizes to $\tau_i$.

I learned this some years ago from Vigleik Angeltvelt (see his "Topological Hochschild homology and cohomology of $A_\infty$ ring spectra", Example 3.3).

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