# Moore spectra are not E-infinity (oldest known proof)

Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map $$S^0 \overset{p^i}\longrightarrow S^0$$ where $S^0$ be the sphere spectrum. In the Mathoverflow discussion, it was pointed out that $M_p(i)$ is not $E_{\infty}$. Reference to a proof was given.

Is this the first/only known proof of Moore spectrum $M_p(i)$ (where $i> 1$ ) not being $E_{\infty}$? If not, when was it first proved and by whom (Please include references if possible)? Feel free to write /refer to a proof which is different from the one that is given here.

• I suggest the paper of Kraines and Lada on the transfer conjecture, titled `A Counterexample to the Transfer Conjecture'. I don't think it is stated as a statement you quote above, but their result on a 3-stage Postnikov tower of an Eilenberg MacLane space, I guess, implies the statement you look for. – user51223 Jul 1 '15 at 9:34
• If you don't mind, can you point me out to some of the details of this paper that are relevant to the question above? May be a sketch! – Prasit Jul 2 '15 at 0:25
• I think I have been a little vague here, so what follows might be vague too, and maybe wonrg!! Often, you start with a Moore space and then attach cells to get an EM space. So, somehow reverse, if we can think off a Moore space as the $\infty$-stage of Postnikov system for an EM space, then you see that in their Kraines-Lada paper, what they do is to consider spaces of different stages in a Postnikov system built over an EM space and show that these spaces cannot have $A_k$ structure, so on. So, if you feel happy reading their paper in this way, I thought it would imply what you are after. – user51223 Jul 2 '15 at 18:40