The Goodwillie tower for $S^1$ and Kuhn's Theorem

Abstract

We analyze the homological behavior of the attaching maps in the $2$–local Goodwillie tower of the identity evaluated at $S^1$. We show that they exhibit the same homological behavior as the James–Hopf maps used by N Kuhn to prove the $2$–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 $S^1$ at the prime $2$.