Homology of iterated loop spaces on odd--dimensional spheres

Question

For prime $p$ let $E_p[\dots]$ and $P_p[\dots]$ be the external and polynomial $\mathbb{Z}_p$--algebras.

It is known that for $n\geqslant 1$ and odd $p$ where is an isomorphism of primitively generated Hopf algebras $H_*(\Omega^2S^{2n+1};\mathbb{Z}_p)=E_p[x_0,x_1,x_2,\dots]\otimes P_p[y_1,y_2,\dots]$ where $\deg(x_i)=2p^in-1$ and $\deg(y_i)=2p^in-2$.

For $p=2$ where is an isomorphism of primitively generated Hopf algebras $H_*(\Omega^2S^{2n+1};\mathbb{Z}_2)=P_2[z_0,z_1,z_2,\dots]$ where $\deg(z_i)=2^{i+1}n-1$.

$\mathbf{Question:}$ How does a similar description of $H_*(\Omega^3S^{2n+1};\mathbb{Z}_p)$ looks like for $n\geqslant 2$ ?

Answer 1

For the case where $n=dp$, the calculation of $H_*(\Omega^3S^{2n+1};\mathbb{Z}/p)$ is reviewed in Section 4.2 of the paper The triple loop space approach to the telescope conjecture by Mahowald, Ravenel and Shick. I don't think that the assumption that $n=dp$ makes any real difference, although it will take a little work to straighten out all the indices for the general case. All the real work is done in the book Homology of iterated loop spaces by Cohen, Lada and May; the paper of Mahowald, Ravenel and Shick just spells out the simplifications that occur in the case at hand.