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If one uses the Adams-Spectral sequence based on cohmology theories other than $BP$ it is possible to say a little more. In "A vanishing line in the $BP \langle 1 \rangle$-Adams spectral sequence" Jes …

Let $L_E$ denote Bousfield localisation with repsect to the cohomology theory $E$. I am trying to follow through some calculations in Hovey-Strickland's paper Morava $K$-theories and localisation Cla …

In Ravenel's paper on the Arf invariant, he shows (p. 439) that there is a composite of maps $$ \mathrm{Ext}_{BP_*BP}(BP_*,BP_*) \to H^*_c(\mathbb{S}_n,E_*) \to H^*(C_p,E_*/\frak{m}), $$ under which t …

In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that $$ K_i(BG_+) \simeq H_J^i(R(G)) $$ for $i=0,1$, where $J$ is the augmentation ideal and $H_ …

I've found a paper of Spanier's (Higher Order Operations) where he uses the theory of "carriers" to study $n$-th order operations. The set-up is rather general; for example a particular case defines t …

Just to close this off - thanks to Mike-Doherty it appears that the answer is yes (and in fact for spaces this goes back to the paper "The decomposition of stable homotopy" by Joel Cohen.) Using the …

It basically says that a CW complex has the coherent topology from its closed cells. This should be wrapped up into any definition of a CW complex that you see. For example in Massey's book it is eq …

An answer is given in Theorem 3.5 'On the computation of stable stems' by Kochman and Mahowald. In light of the recent work of Isaksen, Xu, Wang and others, I'm not sure how reliable this result is.

This has been shown by Daniel Cohen: On the Adem relations. Proc. Cambridge Philos. Soc. 57 1961 265–267.

Let $E = E_n$ be the $n$-th Morava $E$-theory with coefficient ring $$ E_* = \mathbb{W}(\mathbb{F}_{p^n})[\![u_1,\ldots,u_{n-1}]\!][u^{\pm 1}]. $$ It is usual to consider the completed co-operations …

There are several constructions of the Prüfer group $\mathbb{Z}/p^\infty$; here are two that are relevant for this question. It can be constructed via the short exact sequence $$ 0 \to \mathbb{Z} \ …

Let $D_1KO$ be the $K(1)$-local Spanier-Whitehead dual of $KO$, i.e. the spectrum $$ D_1KO = F(KO,L_{K(1)}S^0). $$ I am interested in what this is. In fact I know that $D_1KO = \Sigma^{-1} KO$. One w …

Let $R$ be a commutative ring, and $D(R)$ its derived category of unbounded chain complexes. I'm interested in when the residue fields $k(\mathfrak{p}) = \mathrm{Frac}(R/\frak p)$ for $\mathfrak p \in …

Regarding $E(n)_*E(n)$, see "On the Structure of the Hopf Algebroid $E(n)_*E(n)$" by Keith Johnson. Johnson shows that $$E(n)_*E(n) \otimes \mathbb{Q} \simeq \mathbb{Q}[v_1,\cdots,v_{n-1},v_n^{\pm 1}, …

I think the answer to your question is essentially unknown. As far as I'm aware the best known results are: $M(p)$ admits an $A_{p-1}$ structure but never an $A_p$-structure. I learnt the following …