An abelian model for the Goodwillie tower of the circle

We study the Goodwillie tower of the identity evaluated at the circle. Our main result is the p-local equivalence $ΩP_{p^k}I(S^1) \simeq \mathbb{Z} \times Ω^{\infty+2k}τ_{>0}\mathbb{Sp}^{p^k}$ obtained from the work of Behrens and Kuhn on the Whitehead conjecture. In particular, this identifies the