We analyze the homological behavior of the attaching maps in the –local Goodwillie tower of the identity evaluated at . We show that they exhibit the same homological behavior as the James–Hopf maps used by N Kuhn to prove the –primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie tower of at the prime .
"The Goodwillie tower for $S^1$ and Kuhn's Theorem." Algebr. Geom. Topol. 11 (4) 2453 - 2475, 2011. https://doi.org/10.2140/agt.2011.11.2453