TITLE: Computing squaring operations in the Adams spectral sequence
QUESTION [9 upvotes]: This question is about the classical Adams spectral sequence.  Squaring operations are defined on its $E_2$ term.  I'd like to know how to compute some of the non-trivial operations, such as $Sq^2 ( c_0 ) = h_0 e_0$.  I feel like this ought to be doable in the May spectral sequence, but I don't know the details.
I'm aware of some work of Milgram on the subject, but there are some problems with his approach because of indeterminacies of Massey products.
Thanks!

REPLY [2 votes]: This is an answer Bob's question after my previous answer. It doesn't fit in a comment.
As a spectrum, $\mathrm{End}(H\mathbb{Z}/2)$ is known to be a product of Eilenberg-MacLane spectra, so it is the classifying spectrum $\mathbb{H} C_{*}$ of a chain complex $C_{*}$, i.e. it is in the image of the Eilenbeg-MacLane functor $\mathbb{H}$ from chain complexes to spectra. The work of Shipley shows that this functor is well behaved with respect to multipliciative structures, since the symmetric monoidal model category of chain complexes is Quillen equivalent to that of $H\mathbb{Z}$-algebra spectra. In particular  $\mathbb{H}$ takes chain algebras to ring spectra. However, we know that the ring spectrum $\mathrm{End}(H\mathbb{Z}/2)$ is not an $H\mathbb{Z}$-algebra, so it is the classifying spectrum of a chain complex, but not the classifying ring spectrum of a chain algebra.
Your question would be answered in full generality if we understood what a ring spectrum structure over the classifying spectrum of a chain complex is, i.e. what structure on a chain complex $C_{*}$ is equivalent to an $S$-algebra structure on $\mathbb{H} C_{*}$.
This is an achievable task, since there are nice models for the functor $\mathbb{H}$ taking values on symmetric spectra, and if $X$ is a symmetric spectrum, it is easy to describe a ring spectrum structure in terms of maps of simplicial sets $X_p\wedge X_q\rightarrow X_{p+q}$. The outcome should be some sort of non-additive multiplication on the chain complex $C_*$.
Now, let me remind you that Leif Kristensen constructed a long time ago a chain complex $C_{*}$ whose cohomology is the Steenrod algebra (reference below). This chain complex is endowed with a multiplication, which induces the product on the Steenrod algebra in cohomology. However, Kristensen's multiplication is not a chain algebra structure on $C_{*}$ since it is left distributive but not right distributive. This could be an approximation to a solution to the previous problem.
So far, I've been very lazy to pursue this project. There'd be a lot of work to do, but the outcome can be very valuable and surprising, and it looks like a feasible project.
MR0159333 (28 #2550) Kristensen, Leif On secondary cohomology operations. Math. Scand. 12 1963 57–82. (Reviewer: J. F. Adams)