An abelian model for the Goodwillie tower of the circle

Abstract: We study the Goodwillie tower of the identity evaluated at the circle. Our main result is the p-local equivalence $ΩP_{p^k}I(S¹⁾ \simeq \mathbb{Z} \times Ω<sup>{\infty+2k}τ_{&gt;0}\mathbb{Sp}<sup>{p<sup>k}$ obtained from the work of Behrens and Kuhn on the Whitehead conjecture. In particular, this identifies the Goodwillie approximations of the circle as infinite loop spaces, a phenomenon which does not hold in general. The equivalence also reduces the computation of their homotopy groups to a stable problem, making them accessible to tools such as the Adams spectral sequence and to computer calculations. We place these computations in a more general framework by developing a Goodwillie-calculus analogue of the Gray sequence, which relates the homotopy groups of successive stages of the tower.