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.Comment: v2: 23 pages, clarified exposition in many parts, to appear in AG
