The Goodwillie tower for S^1 and Kuhn's theorem

TL;DR

This paper studies the homological properties of the Goodwillie tower for S^1 at prime 2, revealing its connection to Kuhn's work on the Whitehead conjecture and establishing a homotopy contraction.

Contribution

It demonstrates that the attaching maps in the Goodwillie tower mimic James-Hopf maps, leading to a calculus-based proof of the Whitehead conjecture at prime 2.

Findings

Attaching maps in the tower behave like James-Hopf maps.

The Whitehead sequence acts as a contracting homotopy.

Provides a calculus formulation of the Whitehead conjecture.

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.