Volume 11, issue 4 (2011)

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

Mark Behrens

Algebraic & Geometric Topology 11 (2011) 2453–2475
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 $2$.

Keywords
Whitehead Conjecture, Goodwillie calculus, Dyer–Lashof operation
Mathematical Subject Classification 2010
Primary: 55P65
Secondary: 55Q40, 55S12