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I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere. I could try to do it myself but I really lack expertise, hence am afraid to miss something or ...

Here is one of the motivations for my question, when $p=2$. The homology of the spectrum $H\mathbb F_2$ as an algebra is generated by the Dyer Lashof operations on the single generator $\xi_1$ (and it ...