# Relation between cohomology operations and the Adams spectral sequence

$$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Cone}{Cone}$$ I'm trying to understand how higher order cohomology operations are related to the Adams spectral sequence. Specifically, in McCleary's book Proposition 9.2 states that if a nontrivial map in $$\pi_k^s$$ can be detected by an $$n$$^th order stable cohomology operation then it has Adams filtration $$m$$ for some $$m \leq n$$. He says this will become clear from the construction but I am unable to see why this is true.

In a minimal resolution, the relation $$Sq^3 Sq^1 + Sq^2 Sq^2 = 0$$ gives a generator of $$\Ext^{2,4}_\mathcal{A}(\Z/2, \Z/2)$$ corresponding to $$h_1^2$$, and this detects the class $$\eta^2$$ in $$\pi_2^s$$. The relation also gives a secondary cohomology operation $$\Phi_{1,1}$$ which supposedly also detects the map $$\eta^2$$, meaning $$\Phi_{1,1}$$ is nontrivial in $$\Cone(\eta^2)$$. I don't know how to prove this last result. Is it somehow implied by $$h_1^2$$ detecting $$\eta^2$$?

I believe the differentials in the Adams spectral sequence are also related to higher order cohomology operations. For example, the differential $$d_2(h_k) = h_0 h_{k-1}^2$$ is supposedly implied by Adams proof of the Hopf invariant one problem which involved decomposing $$Sq^{2^k} = \Sigma a_{i,j,k} \Phi_{i,j}$$ where $$\Phi_{i,j}$$ is the secondary operation associated to the Adem relation for $$Sq^{2^i} Sq^{2^j}$$ which also corresponds to the generator $$h_i h_j \in \Ext^{2}_\mathcal{A}(\Z/2, \Z/2)$$. I have not fully understood Adams proof yet, but am finding it difficult to see the connection between the decomposition and the differential.

Any help or reference would be appreciated!

Thanks.

Mauder, C. R. F.

Corollary 1 in Chapter 17 of Mosher-Tangora is about $$\Phi_{1,1}$$ detecting $$\eta^2$$.