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The mod $p$ homology of $QX=\Omega ^{\infty}\Sigma ^{\infty}X$ for connected $X$ was computed by Dyer-Lashof Homology of Iterated Loop Spaces, Amer. J. of Math., vol.84, No.1 pp 35-88. 1962 It follows that, if we denote by $\sigma :H_*(Q\Sigma Y;Z/p) \rightarrow H_{*+1}(Q\Sigma ^2Y;Z/p)$ the suspension map, and restrict it to the primitives $PH_*(Q\Sigma Y;Z/p)$,then all elements in the kernel has to be in the image of $Q_0$ or $\beta Q_1$. (One can generalize this to spaces that are not suspensions, or to spaces that aren't even connected by replacing primitives with appropriate subsets of $H_*(QX;Z/p)$.)

In the absence of Bockstein, this fact was used by Ravenel-Wilson D. C. Ravenel and W. S. Wilson. The Hopf ring for complex cobordism. Journal of Pure and Applied Algebra, 9:241–280, 1977 to determine the kernel of the map $H_*(QS^i;Z/p)\rightarrow H_*(\Omega ^{\infty}\Sigma ^iBP;Z/p)$ induced by the unit map of the Brown-Peterson spectrum.

Now, my question is: has this fact appeared explicitly somewhere in literature?

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  • $\begingroup$ I suspect this might be in Madsen-Milgram. But I this it is easy to see. For the case of $p=2$, I think you have only $Q_0$, the squaring operation, right. So, let fix $p=2$. Now, I think by EMSS argument, there is an isomorphism $\sigma_*:\mathrm{Ind}H_*(Q_0 Y)\to\mathrm{Prim}H_*(Q\Sigma Y)$. Hence, you can write down an explicit formula for the primitives of $H_*(Q\Sigma Y)$ allowing to can show that the induced homomorphism $H_*(Q\Sigma Y)/\mathrm{Im}(Q_0)\to \mathrm{Prim}H_*(Q\Sigma^2 Y)$ is a mono. Similar technique works for $p>2$. $\endgroup$
    – user51223
    Commented Mar 1, 2016 at 17:56
  • $\begingroup$ Thank you, it is also easy from the properties of Dyer-Lashof operations, (The collapse of EMSS also works at odd $p$, but we still need to identify the transpotence elements with the image of $\beta Q_1$ to get the results mentioned. It happens that for our specific purpose, we only need the degrees of the elements in the kernel, so EMSS is good enough, but it is another matter.) But the question I am asking is whether there is any published reference. And it is not in "Classifying Spaces for Surgery and Corbordism of Manifolds" if that is what you mean by Madsen-Milgram. $\endgroup$
    – user43326
    Commented Mar 2, 2016 at 17:29
  • $\begingroup$ Yes, the Annals Lecture notes is the one I meant. $\endgroup$
    – user51223
    Commented Mar 2, 2016 at 17:55

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