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The $D$ with $D(xy) = xD(y) + D(x)y$ are the primitives in the Steenrod algebra $A$, which are dual to the indecomposables $\xi_i$ in $A_* = F_2[\xi_i \mid i\ge1]$, so there is one such $D$ in each de …

The initial example of such an $E$ is the Thom spectrum $M\xi$ associated to the $E_1$-map $\Omega \Sigma BU(1) \to BU$, studied by Baker and Richter in "Quasisymmetric functions from a topological po …

You can learn about this, and more, in Part II of Adams' "Stable Homotopy and Generalised Homology" (1974), especially section 4, in the special case $E = KU$.

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. Cohomolo …

In the context of functors with smash product (FSP), or symmetric spectra, or orthogonal spectra, the spectrum with $\Sigma_2$-action $X \wedge X$ prolongs essentially uniquely to a $\Sigma_2$-spectru …

For $3$-primary homotopy of $S$ there is early work by Nakamura, Osamu Some differentials in the mod 3 Adams spectral sequence. Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. No. 19 (1975), 1– …

Not an answer, but a possible approach: Using Dundas' cartesian square with corners $A(*)$, $K(Z)$, $TC(*)$ and $TC(Z)$ you can see that $Wh^{Diff}_3(*) = Z/2$ comes from $TC_4(Z) = Z/8$ (plus odd tor …

The $E$-based Adams spectral sequence is the homotopy spectral sequence associated to the tower of spectra $\dots \to Y_2 \to Y_1 \to Y_0 = S$ with $Y_{s+1} \to Y_s \to E \wedge Y_s$ a homotopy fiber …

Let me write $HH^S(B) = THH(B) = B \wedge_{B^e} B$ for topological Hochschild homology, and $HH_S(B) = F_{B^e}(B, B)$ for topological Hochschild cohomology, where $B^e = B \wedge_S B^{op}$. For $B$ c …

A concise formula $$ \mu(\beta) = \beta(c(t^F)) $$ for the $BP_* BP$-coaction $\mu$ on $BP_* CP^\infty$ is given in the "Note added in proof" on page 279 of Ravenel, Douglas C.; Wilson, W. Stephen The …

Perhaps it helps to first think about how you can construct a map $\alpha_k : \Sigma^k S(\pi_k M) \to M$ inducing an isomorphism on $\pi_k$. Choose a free resolution $$ 0 \to \bigoplus_{j \in J} \mat …

The $2$-completed stable $20$-stem $\pi_{20}(S)_2$ is cyclic of order $8$. Mimura and Toda (1963, Lemma 15.4) mr=157384 show the existence of a class $\bar\kappa_7 \in \pi_{27}(S^7)$ whose stable cla …

Just as $ku$ is quite close to $H\mathbb Z$ in the sequence $S \to ku \to H\mathbb Z$ of commutative ring spectra, and $KU$ is quite close to $H\mathbb Q$, the classifying space $BGL(ku)$ for $2$-vect …

The $A(2)$-module structure on $A(2)//A(1)$ does not extend to an $A$-module structure. In particular, there is no spectrum $X$ with $H^*(X; F_2) = A(2)//A(1)$ as an $A(2)$-module. Additively, $A(2) …

Let $R$ be bounded below, bounded above and nontrivial, and work in $R$-modules. Let $X = \bigvee_{n\in\mathbb{Z}} \Sigma^n R$. Then $X \overset{\simeq}\to D(DX)$ is an equivalence, but $X$ is not d …