2 problems

Let $E_r^{k,t}(S^1)$ denote the pages of the Goodwillie spectral sequence (GSS) for $S^1$. Goodwillie–Whitehead conjecture, weak form. The GSS for $S^1$ collapses at the $E_2$-page…

For each $k$, let … be the map in Kuhn's Kahn–Priddy sequence, and let … be the $k$-invariant map of the Goodwillie tower for $S^1$. Goodwillie–Whitehead conjecture, strong form. ……