$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Cone}{Cone}$ 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!



1 Answer 1


Maybe these references can help:

Mauder, C. R. F.
On the differentials in the Adams spectral sequence.
Proc. Cambridge Philos. Soc. 60 (1964), 409–420. 

Mosher, Robert E.; Tangora, Martin C.
Cohomology operations and applications in homotopy theory.
Harper & Row, Publishers, New York-London 1968 x+214 pp.

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


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